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family="Arial Cyr" charset="204" size="12">ешить уравнение вида: </f> <region region-id="30" left="252" top="21.75" width="79.5" height="36.75" align-x="293.25" align-y="42.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="2"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> для заданных начальных условий<sp count="2"/> </inlineAttr> </f> <region region-id="31" left="72.75" top="59.25" width="11.25" height="17.25" align-x="77.25" align-y="70.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">o</ml:id> </ml:apply> </math> <rendering item-idref="3"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <region region-id="32" left="90.75" top="59.25" width="54" height="17.25" align-x="125.25" align-y="70.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">o</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">o</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="4"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, численно методом конечных разностей.</inlineAttr> </f> </text> </region> <region region-id="27" left="24" top="96.75" width="150.75" height="13.5" align-x="24" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Число точек дискретизации:</f> </text> </region> <region region-id="28" left="180" top="96.75" width="89.25" height="15.75" align-x="193.5" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:plus/> <ml:real>10</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="48" left="318" top="96.75" width="42" height="15.75" align-x="328.5" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="34" left="24" top="120.75" width="97.5" height="13.5" align-x="24" align-y="132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Задание функции:</f> </text> </region> <region region-id="35" left="132" top="120.75" width="72" height="15.75" align-x="169.5" align-y="132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="22" left="24" top="144.75" width="216" height="16.5" align-x="24" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Область определения функции:<sp count="2"/>x=[x</f> <f family="Arial Cyr" charset="204" size="12"> o </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12">b]</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="25" left="258" top="144.75" width="30.75" height="15.75" align-x="272.25" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">b</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="14" left="24" top="168.75" width="111.75" height="13.5" align-x="24" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Начальные условия:</f> </text> </region> <region region-id="15" left="150" top="168.75" width="35.25" height="17.25" align-x="168.75" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="16" left="198" top="168.75" width="35.25" height="17.25" align-x="216.75" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="693" left="24" top="192.75" width="395.25" height="13.5" align-x="24" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Просмотр на экране дисплея численного значения функции в этой точке:</f> </text> </region> <region region-id="694" left="438" top="192.75" width="70.5" height="17.25" align-x="486.75" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="11"/> </region> <region region-id="37" left="24" top="228.75" width="80.25" height="13.5" align-x="24" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Дискрет по "х":</f> </text> </region> <region region-id="21" left="120" top="219" width="56.25" height="35.25" align-x="133.5" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="695" left="24" top="258.75" width="244.5" height="13.5" align-x="24" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Для выше выбранных параметров он равен:</f> </text> </region> <region region-id="696" left="288" top="258.75" width="43.5" height="15.75" align-x="300.75" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.1</ml:real> </result> </ml:eval> </math> <rendering item-idref="13"/> </region> <region region-id="697" left="24" top="282.75" width="258" height="13.5" align-x="24" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Дискретное представление самой переменной:</f> </text> </region> <region region-id="698" left="300" top="282.75" width="37.5" height="17.25" align-x="315.75" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="41" left="24" top="324.75" width="420.75" height="13.5" align-x="24" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Таким образом, для вычисления У(х) по методу Эйлера<sp count="2"/>получим выражение:</f> </text> </region> <region region-id="699" left="78" top="348.75" width="102.75" height="17.25" align-x="105" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="700" left="450" top="354.75" width="15.75" height="15.75" align-x="456.75" align-y="366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="16"/> </region> <region region-id="704" left="24" top="372.75" width="259.5" height="13.5" align-x="24" align-y="384" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Справочно приведем некоторые его значения: </f> </text> </region> <region region-id="709" left="66" top="396.75" width="40.5" height="15.75" align-x="78.75" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>10</ml:real> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="708" left="144" top="396.75" width="66.75" height="17.25" align-x="162" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>3.1874849202</ml:real> </result> </ml:eval> </math> <rendering item-idref="18"/> </region> <region region-id="710" left="234" top="396.75" width="60.75" height="17.25" align-x="252" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>5</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.72102</ml:real> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <region region-id="45" left="24" top="420.75" width="302.25" height="13.5" align-x="24" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Точное решение этого дифференциального уравнения:</f> </text> </region> <region region-id="47" left="342" top="415.5" width="93" height="21" align-x="372" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Q</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="789" left="24" top="450.75" width="207" height="13.5" align-x="24" align-y="462" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Значение функции на правой границе:</f> </text> </region> <region region-id="790" left="252" top="450.75" width="81" height="15.75" align-x="284.25" align-y="462" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>3.43656365691809</ml:real> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <region region-id="52" left="36" top="498" width="242.25" height="135.75" align-x="36" align-y="498" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="22"/> <rendering item-idref="23"/> </region> <region region-id="864" left="36" top="660.75" width="400.5" height="13.5" align-x="36" align-y="672" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Рис. 1.<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Возможна анимация графика </f> <f family="Arial Cyr" charset="204" size="12">по параметру "n".</f> </text> </region> <region region-id="56" left="30" top="720.75" width="469.5" height="135" align-x="30" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Разностная схема (метод) не всегда работает.<sp count="2"/>Если f(x,y)=f(y) и на каком-то шаге f(y</f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">)=0, то далее </f> <f family="Arial Cyr" charset="204" size="12">в силу (1) последующие значения y</f> <f family="Arial Cyr" charset="204" size="12"> i+1 </f> <f family="Arial Cyr" charset="204" size="12"> будут равны y</f> <f family="Arial Cyr" charset="204" size="12">i </f> <f family="Arial Cyr" charset="204" size="12">, что не всегда так</f> <f family="Arial Cyr" charset="204" size="12"> . Примером тому<sp count="2"/>может служить процесс, описываемый уравнением<sp count="3"/> </f> <region region-id="67" left="30" top="768" width="111.75" height="44.25" align-x="126.75" align-y="794.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="24"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">или </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">уравнением (2)</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> с заданными начальными условиями: z(0)=zo, z'(0)=q. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Независимо от значения "q" , z(t) это изменяемая во времени функция.<sp count="2"/>Однако, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">при q=0 разностная схема<sp count="2"/>дает другой результат - z(t)=const. При </inlineAttr> </f> <region region-id="74" left="30" top="840" width="28.5" height="15.75" align-x="43.5" align-y="851.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:notEqual/> <ml:id xml:space="preserve">q</ml:id> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="25"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> картина получается достоверной.</inlineAttr> </f> </text> </region> <region region-id="795" left="36" top="873" width="213" height="42.75" align-x="72" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">q</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </math> <rendering item-idref="26"/> </region> <region region-id="794" left="282" top="888.75" width="30.75" height="15.75" align-x="296.25" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">q</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="793" left="336" top="879.75" width="39.75" height="34.5" align-x="354" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="792" left="444" top="888.75" width="15.75" height="15.75" align-x="450.75" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="29"/> </region> <region region-id="130" left="36" top="930.75" width="423.75" height="27" align-x="36" align-y="942" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Если не ввести отдельно обозначение константы, то возникнут трудности при </f> <f family="Arial Cyr" charset="204" size="12">записи<sp count="2"/>f1(z).<sp count="2"/>Поэтому </f> </text> </region> <region region-id="423" left="168" top="972.75" width="81.75" height="17.25" align-x="202.5" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">CON</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="424" left="456" top="972.75" width="15.75" height="15.75" align-x="462.75" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="31"/> </region> <region region-id="87" left="30" top="1008.75" width="485.25" height="27" align-x="30" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Пусть "z" и "t" принадлежат соответственно интервалам [-zo,zo] и [0,T]. Запишем переменную t в дискретном виде (значение Т подобрано как полупериод) </f> </text> </region> <region region-id="801" left="36" top="1056.75" width="91.5" height="15.75" align-x="52.5" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <ml:apply> <ml:minus/> <ml:real>50</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="800" left="144" top="1056.75" width="65.25" height="15.75" align-x="156" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="799" left="228" top="1056.75" width="40.5" height="15.75" align-x="243" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">T</ml:id> <ml:real>3.3</ml:real> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="798" left="276" top="1048.5" width="37.5" height="33.75" align-x="291.75" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">δ</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="797" left="330" top="1056" width="39.75" height="18" align-x="344.25" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">δ</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="796" left="396" top="1056.75" width="52.5" height="15.75" align-x="416.25" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">q1</ml:id> <ml:real>-0.2</ml:real> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="809" left="30" top="1110" width="204.75" height="16.5" align-x="30" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Введем обозначение (</f> <region region-id="185" left="153" top="1110.75" width="70.5" height="15.75" align-x="208.5" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">CON</ml:id> <ml:id xml:space="preserve">q</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="38"/> </region> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="808" left="264" top="1095" width="173.25" height="41.25" align-x="295.5" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:id xml:space="preserve">CON</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">q</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="807" left="480" top="1110.75" width="15.75" height="15.75" align-x="486.75" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="40"/> </region> <region region-id="825" left="24" top="1164.75" width="220.5" height="27" align-x="24" align-y="1176" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Здесь (1.1) может быть записано в виде</f> <f family="Arial Cyr" charset="204" size="12">первых 2-х слагаемых ряда Тейлора:</f> </text> </region> <region region-id="822" left="276" top="1176" width="97.5" height="18" align-x="303" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">δ</ml:id> <ml:apply> <ml:id xml:space="preserve">f1</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="823" left="396" top="1176.75" width="66" height="17.25" align-x="413.25" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>4</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.5707963267948966</ml:real> </result> </ml:eval> </math> <rendering item-idref="42"/> </region> <region region-id="824" left="480" top="1176.75" width="15.75" height="15.75" align-x="486.75" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </ml:parens> </math> <rendering item-idref="43"/> </region> <region region-id="867" left="30" top="1200.75" width="239.25" height="16.5" align-x="30" align-y="1212" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Где f1(z</f> <f family="Arial Cyr" charset="204" size="12"> j </f> <f family="Arial Cyr" charset="204" size="12">) - первая производная на<sp count="2"/>j-м шаге.</f> </text> </region> <region region-id="717" left="36" top="1230.75" width="438.75" height="30" align-x="36" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">В результате получаем ошибочное решение Z</f> <f family="Arial Cyr" charset="204" size="12"> j </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> представляемое прямой линией </f> <f family="Arial Cyr" charset="204" size="12">в виде графика рис. 2</f> </text> </region> <region region-id="156" left="36" top="1290" width="258" height="16.5" align-x="36" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">С другой стороны при </f> <region region-id="152" left="158.25" top="1290.75" width="56.25" height="15.75" align-x="199.5" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:notEqual/> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">q</ml:id> <ml:id xml:space="preserve">q1</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="44"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>будем иметь:</inlineAttr> </f> </text> </region> <region region-id="175" left="318" top="1275" width="179.25" height="41.25" align-x="349.5" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:id xml:space="preserve">CON</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">q1</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="162" left="42" top="1326.75" width="44.25" height="17.25" align-x="66" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z2</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="163" left="114" top="1326" width="115.5" height="18" align-x="147" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z2</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z2</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">δ</ml:id> <ml:apply> <ml:id xml:space="preserve">f2</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z2</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="718" left="42" top="1362.75" width="444.75" height="13.5" align-x="42" align-y="1374" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Это решение также представлено на рис. 2 и является единственно правильным.</f> </text> </region> <region region-id="865" left="48" top="1446" width="165" height="135.75" align-x="48" align-y="1446" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="48"/> <rendering item-idref="49"/> </region> <region region-id="826" left="42" top="1608.75" width="330" height="13.5" align-x="42" align-y="1620" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Рис. 2.<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Возможна анимация графика.</f> </text> </region> <region region-id="719" left="48" top="1632" width="335.25" height="16.5" align-x="48" align-y="1644" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> Упражнение: </f> <f family="Arial Cyr" charset="204" size="12"> 1) снять клип по параметру "q1" :<sp count="2"/> </f> <region region-id="722" left="309" top="1632.75" width="73.5" height="15.75" align-x="366.75" align-y="1644" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>-0.2</ml:real> <ml:id xml:space="preserve">q1</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="50"/> </region> </text> </region> <region region-id="724" left="48" top="1656.75" width="465" height="40.5" align-x="48" align-y="1668" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">2) привести другие примеры неустойчивой работы этого метода.</f> <f family="Arial Cyr" charset="204" size="12">3) решить дифференциаальное уравнение y'(x)=cos(y(x)) при заданных начальных</f> <f family="Arial Cyr" charset="204" size="12">условиях х=0, у(0)=0.</f> </text> </region> <region region-id="188" left="48" top="1715.25" width="432.75" height="197.25" align-x="48" align-y="1728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="14">1.2 Метод Адамса.</f> <f family="Arial Cyr" charset="204" size="12">Решить уравнение вида: </f> <region region-id="237" left="184.5" top="1731.75" width="79.5" height="36.75" align-x="221.25" align-y="1752.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="51"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> для заданных начальных</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> условий </inlineAttr> </f> <region region-id="238" left="96.75" top="1769.25" width="9" height="17.25" align-x="100.5" align-y="1780.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="52"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <region region-id="239" left="112.5" top="1769.25" width="51.75" height="17.25" align-x="144.75" align-y="1780.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="53"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">,<sp count="2"/>численно методом конечных разностей. </f> </inlineAttr> <region region-id="194" left="393.75" top="1769.25" width="51" height="17.25" align-x="429" align-y="1780.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </math> <rendering item-idref="54"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">Запишем формулу Адамса с четырьмя членами [2]: </f> </inlineAttr> <region region-id="189" left="48" top="1800.75" width="231" height="33.75" align-x="80.25" align-y="1820.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:real>12</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">Δ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="55"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">,<sp count="44"/>(6)</f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">где </f> </inlineAttr> <region region-id="190" left="69" top="1841.25" width="72.75" height="17.25" align-x="81.75" align-y="1852.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="56"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">, </f> </inlineAttr> <region region-id="191" left="148.5" top="1840.5" width="106.5" height="18" align-x="190.5" align-y="1852.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="57"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <region region-id="192" left="261.75" top="1835.25" width="142.5" height="23.25" align-x="313.5" align-y="1852.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">Δ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="58"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <region region-id="193" left="48" top="1859.25" width="125.25" height="17.25" align-x="107.25" align-y="1870.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X'</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:sequence> </math> <rendering item-idref="59"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">Формула Адамса дает возможность, зная </f> </inlineAttr> <region region-id="195" left="279" top="1877.25" width="75.75" height="17.25" align-x="325.5" align-y="1888.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:sequence> </math> <rendering item-idref="60"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">,<sp count="2"/>определить </f> </inlineAttr> <region region-id="196" left="431.25" top="1877.25" width="25.5" height="17.25" align-x="443.25" align-y="1888.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="61"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">.</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Таким образом, зная </inlineAttr> </f> <region region-id="197" left="165" top="1895.25" width="53.25" height="17.25" align-x="200.25" align-y="1906.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real font="0">1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real font="0">2</ml:real> </ml:apply> </ml:sequence> </math> <rendering item-idref="62"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,<sp count="2"/>мы можем определить </inlineAttr> </f> <region region-id="198" left="355.5" top="1895.25" width="14.25" height="17.25" align-x="361.5" align-y="1906.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>3</ml:real> </ml:apply> </math> <rendering item-idref="63"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> и далее </f> </inlineAttr> <region region-id="199" left="418.5" top="1895.25" width="46.5" height="17.25" align-x="434.25" align-y="1906.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:sequence> <ml:real>5</ml:real> <ml:id xml:space="preserve" font="1">...</ml:id> </ml:sequence> </ml:apply> </ml:sequence> </math> <rendering item-idref="64"/> </region> </text> </region> <region region-id="869" left="48" top="1920" width="430.5" height="27.75" align-x="48" align-y="1932" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Моделирование алгоритма Адамса в системе MathCAD. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">На примере задачи, решенной выше.)</inlineAttr> </f> </text> </region> <region region-id="226" left="42" top="1968.75" width="30.75" height="13.5" align-x="42" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Дано:</f> </text> </region> <region region-id="225" left="96" top="1968" width="80.25" height="16.5" align-x="141" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">ψ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="65"/> </region> <region region-id="332" left="210" top="1968.75" width="33" height="17.25" align-x="226.5" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="66"/> </region> <region region-id="331" left="258" top="1968.75" width="38.25" height="17.25" align-x="279.75" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="67"/> </region> <region region-id="903" left="312" top="1960.5" width="110.25" height="33.75" align-x="325.5" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:minus/> <ml:real>0.5</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>50</ml:real> </ml:apply> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="68"/> </region> <region region-id="904" left="444" top="1968.75" width="39" height="15.75" align-x="457.5" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="69"/> </region> <region region-id="905" left="42" top="1998.75" width="413.25" height="13.5" align-x="42" align-y="2010" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Для проведения анимации необходимо убрать h:=0.1, заменив его на<sp count="2"/>h=0.1</f> </text> </region> <region region-id="898" left="54" top="2037" width="93" height="35.25" align-x="71.25" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">K</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="70"/> </region> <region region-id="877" left="156" top="2046.75" width="49.5" height="15.75" align-x="170.25" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">K</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="71"/> </region> <region region-id="876" left="234" top="2046.75" width="63.75" height="17.25" align-x="250.5" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="72"/> </region> <region region-id="733" left="48" top="2076.75" width="174" height="13.5" align-x="48" align-y="2088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Просмотр численного значения:</f> </text> </region> <region region-id="879" left="234" top="2076.75" width="44.25" height="15.75" align-x="250.5" align-y="2088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">K</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>10</ml:real> </result> </ml:eval> </math> <rendering item-idref="73"/> </region> <region region-id="731" left="294" top="2076.75" width="35.25" height="15.75" align-x="307.5" align-y="2088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">b</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="74"/> </region> <region region-id="737" left="48" top="2109" width="450" height="72" align-x="48" align-y="2136" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Вычисляем значение n-й производной: </f> <region region-id="251" left="261" top="2109.75" width="113.25" height="44.25" align-x="309" align-y="2136" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">Xn</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:id xml:space="preserve">n</ml:id> </ml:degree> </ml:apply> </ml:apply> </math> <rendering item-idref="75"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> . В системе Maple достаточно предыдущей символьной записи, а в MathCAD - каждую производную надо записывать отдельно:</inlineAttr> </f> </text> </region> <region region-id="880" left="42" top="2202.75" width="189.75" height="13.5" align-x="42" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Первая производная уже имеется:</f> </text> </region> <region region-id="886" left="258" top="2202" width="96.75" height="16.5" align-x="307.5" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">X1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="76"/> </region> <region region-id="739" left="42" top="2256.75" width="123" height="13.5" align-x="42" align-y="2268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Вторая производная:<sp count="2"/> </f> </text> </region> <region region-id="741" left="180" top="2247" width="228.75" height="36.75" align-x="229.5" align-y="2268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">X2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="77"/> </region> <region region-id="742" left="42" top="2292.75" width="121.5" height="13.5" align-x="42" align-y="2304" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Третья производная:<sp count="2"/> </f> </text> </region> <region region-id="242" left="48" top="2337.75" width="378" height="87" align-x="97.5" align-y="2364" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">X3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> </ml:function> <ml:parens> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:define> </math> <rendering item-idref="78"/> </region> <region region-id="743" left="36" top="2454.75" width="452.25" height="54" align-x="36" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Запись последующих производных в MathCAD становится обременительной. В </f> <f family="Arial Cyr" charset="204" size="12">Maple эту процедуру символьного дифференцирования выполняет сама система, </f> <f family="Arial Cyr" charset="204" size="12">что позволяет записывать производные практически любого порядка и вести </f> <f family="Arial Cyr" charset="204" size="12">далее с ними работу.</f> </text> </region> <region region-id="346" left="36" top="2520.75" width="454.5" height="27" align-x="36" align-y="2532" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Можно осуществить проверку<sp count="2"/>правильности </f> <f family="Arial Cyr" charset="204" size="12">вычислен</f> <f family="Arial Cyr" charset="204" size="12">ия производных для выбранных значений функции в отдельных точках:</f> </text> </region> <region region-id="744" left="36" top="2556.75" width="82.5" height="17.25" align-x="96.75" align-y="2568" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">X3</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.9999999999997282</ml:real> </result> </ml:eval> </math> <rendering item-idref="79"/> </region> <region region-id="745" left="120" top="2556.75" width="82.5" height="17.25" align-x="180.75" align-y="2568" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">X2</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0000000000000089</ml:real> </result> </ml:eval> </math> <rendering item-idref="80"/> </region> <region region-id="748" left="36" top="2580.75" width="352.5" height="13.5" align-x="36" align-y="2592" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Вывод: вычисляет правильно, в том числе и для других функций.</f> </text> </region> <region region-id="262" left="36" top="2610.75" width="122.25" height="13.5" align-x="36" align-y="2622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Введем обозначения </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="274" left="168" top="2610" width="87" height="18" align-x="195.75" align-y="2622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X1</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="81"/> </region> <region region-id="275" left="264" top="2610.75" width="91.5" height="17.25" align-x="291.75" align-y="2622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X2</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X2</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="82"/> </region> <region region-id="276" left="360" top="2610.75" width="91.5" height="17.25" align-x="387.75" align-y="2622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X3</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X3</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="83"/> </region> <region region-id="755" left="36" top="2646" width="307.5" height="14.25" align-x="36" align-y="2658" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Просмотр численных значений и проверка: </f> </text> </region> <region region-id="756" left="36" top="2670.75" width="48.75" height="17.25" align-x="63" align-y="2682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X1</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="84"/> </region> <region region-id="757" left="102" top="2670.75" width="48.75" height="17.25" align-x="129" align-y="2682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X2</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0000000000000089</ml:real> </result> </ml:eval> </math> <rendering item-idref="85"/> </region> <region region-id="758" left="174" top="2670.75" width="48.75" height="17.25" align-x="201" align-y="2682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X3</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.9999999999997282</ml:real> </result> </ml:eval> </math> <rendering item-idref="86"/> </region> <region region-id="370" left="36" top="2694" width="174" height="14.25" align-x="36" align-y="2706" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> Проверка завершена успешно. </f> </text> </region> <region region-id="280" left="36" top="2718" width="462" height="36" align-x="36" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Используя разложение </f> <region region-id="281" left="164.25" top="2718.75" width="21.75" height="15.75" align-x="172.5" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </math> <rendering item-idref="87"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> в ряд Тейлора в окрестности точки </inlineAttr> </f> <region region-id="282" left="383.25" top="2718.75" width="9" height="17.25" align-x="387" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="88"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> и беря его первые</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">четыре члена, определим таким образом </inlineAttr> </f> <region region-id="283" left="264.75" top="2736.75" width="57.75" height="17.25" align-x="285.75" align-y="2748" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="89"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="3"/>и<sp count="2"/> </inlineAttr> </f> <region region-id="284" left="344.25" top="2736.75" width="57.75" height="17.25" align-x="365.25" align-y="2748" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="90"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> как </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">:</inlineAttr> </f> </text> </region> <region region-id="293" left="42" top="2759.25" width="188.25" height="39" align-x="63.75" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X1</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X2</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:real>6</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X3</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="91"/> </region> <region region-id="292" left="252" top="2759.25" width="235.5" height="39" align-x="273.75" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X1</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X2</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> <ml:real>6</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X3</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="92"/> </region> <region region-id="889" left="36" top="2808.75" width="454.5" height="50.25" align-x="36" align-y="2820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Значения X</f> <f family="Arial Cyr" charset="204" size="12"> o </f> <f family="Arial Cyr" charset="204" size="12">, X</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12">, X</f> <f family="Arial Cyr" charset="204" size="12"> 2 </f> <f family="Arial Cyr" charset="204" size="12"> необходимы только вначале для запуска алгоритма Адамса.</f> <f family="Arial Cyr" charset="204" size="12">Вычисления последующих X</f> <f family="Arial Cyr" charset="204" size="12">k</f> <f family="Arial Cyr" charset="204" size="12">,<sp count="2"/> </f> <region region-id="888" left="202.5" top="2826" width="29.25" height="15.75" align-x="216" align-y="2837.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:greaterOrEqual/> <ml:id xml:space="preserve">k</ml:id> <ml:real>3</ml:real> </ml:apply> </math> <rendering item-idref="93"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> идет автоматически по рекуррентной формуле</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">через разности.</inlineAttr> </f> </text> </region> <region region-id="760" left="42" top="2874" width="176.25" height="14.25" align-x="42" align-y="2886" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> Проверка численных значений: </f> </text> </region> <region region-id="761" left="42" top="2904.75" width="69.75" height="17.25" align-x="63" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.1103333333333334</ml:real> </result> </ml:eval> </math> <rendering item-idref="94"/> </region> <region region-id="762" left="114" top="2904.75" width="69.75" height="17.25" align-x="135" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>2</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.2426666666666666</ml:real> </result> </ml:eval> </math> <rendering item-idref="95"/> </region> <region region-id="763" left="186" top="2904" width="105" height="18" align-x="242.25" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.2103333333333335</ml:real> </result> </ml:eval> </math> <rendering item-idref="96"/> </region> <region region-id="764" left="306" top="2904" width="105" height="18" align-x="362.25" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.4426666666666665</ml:real> </result> </ml:eval> </math> <rendering item-idref="97"/> </region> <region region-id="766" left="42" top="2928.75" width="99" height="13.5" align-x="42" align-y="2940" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Первые разности:</f> </text> </region> <region region-id="364" left="42" top="2958.75" width="46.5" height="15.75" align-x="56.25" align-y="2970" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="98"/> </region> <region region-id="357" left="108" top="2958" width="189.75" height="18" align-x="154.5" align-y="2970" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX1</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="99"/> </region> <region region-id="361" left="312" top="2958" width="52.5" height="56.25" align-x="330.75" align-y="2970" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX1</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="2" cols="1"> <ml:real>0.21033333333333348</ml:real> <ml:real>0.23233333333333306</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="100"/> </resultFormat> </math> <rendering item-idref="101"/> </region> <region region-id="767" left="48" top="3024.75" width="97.5" height="13.5" align-x="48" align-y="3036" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Вторые разности:</f> </text> </region> <region region-id="385" left="48" top="3054.75" width="30.75" height="15.75" align-x="62.25" align-y="3066" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="102"/> </region> <region region-id="374" left="114" top="3054" width="146.25" height="18" align-x="160.5" align-y="3066" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Δ2X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX1</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX1</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="103"/> </region> <region region-id="375" left="282" top="3054" width="88.5" height="18" align-x="327.75" align-y="3066" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Δ2X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.021999999999999575</ml:real> </result> </ml:eval> </math> <rendering item-idref="104"/> </region> <region region-id="371" left="42" top="3084" width="174" height="14.25" align-x="42" align-y="3096" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Проверка завершена успешно. </f> </text> </region> <region region-id="294" left="42" top="3114.75" width="418.5" height="27" align-x="42" align-y="3126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Далее все последующие значения "х" будем вычислять по формуле Адамса,</f> <f family="Arial Cyr" charset="204" size="12">имеющей вид:</f> </text> </region> <region region-id="426" left="48" top="3162.75" width="49.5" height="15.75" align-x="62.25" align-y="3174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>2</ml:real> <ml:id xml:space="preserve">K</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="105"/> </region> <region region-id="425" left="126" top="3154.5" width="359.25" height="33.75" align-x="159" align-y="3174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>23</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:real>12</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:real>12</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="106"/> </region> <region region-id="892" left="48" top="3198.75" width="423.75" height="13.5" align-x="48" align-y="3210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Результаты вычислений представим в виде вектора X и графически на рис. 3</f> </text> </region> <region region-id="908" left="48" top="3228.75" width="408.75" height="31.5" align-x="70.5" align-y="3246" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:transpose/> <ml:id xml:space="preserve">X</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="1" cols="12"> <ml:real>1</ml:real> <ml:real>1.1103333333333334</ml:real> <ml:real>1.2426666666666666</ml:real> <ml:real>1.3994666666666666</ml:real> <ml:real>1.5832727777777778</ml:real> <ml:real>1.7969156157407404</ml:real> <ml:real>2.043532516165123</ml:real> <ml:real>2.32659053207208</ml:real> <ml:real>2.64992086588641</ml:real> <ml:real>3.0177574824119078</ml:real> <ml:real>3.4347794899256723</ml:real> <ml:real>3.9061579305851053</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="107"/> </resultFormat> </math> <rendering item-idref="108"/> </region> <region region-id="893" left="120" top="3294.75" width="47.25" height="15.75" align-x="132" align-y="3306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">K</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="109"/> </region> <region region-id="894" left="210" top="3294.75" width="49.5" height="17.25" align-x="237.75" align-y="3306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">XX</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="110"/> </region> <region region-id="910" left="330" top="3295.5" width="111" height="21" align-x="372" align-y="3312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">QQ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">tt</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:id xml:space="preserve">tt</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">tt</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="111"/> </region> <region region-id="829" left="66" top="3354" width="201.75" height="153.75" align-x="66" align-y="3354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="112"/> <rendering item-idref="113"/> </region> <region region-id="830" left="318" top="3414.75" width="42.75" height="13.5" align-x="318" align-y="3426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Рис. 3</f> </text> </region> <region region-id="524" left="48" top="3522.75" width="435" height="31.5" align-x="48" align-y="3534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Сравнительный анализ, полученных численных результатов разными методами</f> <f family="Arial Cyr" charset="204" size="12">(метод Эйлера - </f> <region region-id="435" left="139.5" top="3537" width="11.25" height="17.25" align-x="144" align-y="3548.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </math> <rendering item-idref="114"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> , </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">метод</f> <sp/> <f family="Arial Cyr" charset="204" size="12">Адамса -</f> <f family="Arial Cyr" charset="204"> <sp/> </f> <region region-id="436" left="249.75" top="3537" width="14.25" height="17.25" align-x="255.75" align-y="3548.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </math> <rendering item-idref="115"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,точное решение - </inlineAttr> </f> <region region-id="437" left="368.25" top="3537" width="24.75" height="15.75" align-x="377.25" align-y="3548.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Q</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </math> <rendering item-idref="116"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">).</inlineAttr> </f> </text> </region> <region region-id="834" left="36" top="3564.75" width="117" height="13.5" align-x="36" align-y="3576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Значение в точке 0.4:</f> </text> </region> <region region-id="839" left="168" top="3564.75" width="66.75" height="17.25" align-x="186" align-y="3576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>4</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.5282</ml:real> </result> </ml:eval> </math> <rendering item-idref="117"/> </region> <region region-id="842" left="312" top="3564.75" width="69.75" height="17.25" align-x="333" align-y="3576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>4</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.5832727777777778</ml:real> </result> </ml:eval> </math> <rendering item-idref="118"/> </region> <region region-id="831" left="420" top="3564.75" width="90" height="15.75" align-x="461.25" align-y="3576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.4</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.5836493952825408</ml:real> </result> </ml:eval> </math> <rendering item-idref="119"/> </region> <region region-id="837" left="36" top="3600.75" width="83.25" height="27" align-x="36" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Относительная</f> <f family="Arial Cyr" charset="204" size="12">ошибка в %:</f> </text> </region> <region region-id="838" left="168" top="3591" width="138" height="35.25" align-x="257.25" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.4</ml:real> 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<ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.4</ml:real> </ml:apply> </ml:apply> <ml:real>100</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.023781621480418645</ml:real> </result> </ml:eval> </math> <rendering item-idref="121"/> </region> <region region-id="539" left="36" top="3666.75" width="117" height="13.5" align-x="36" align-y="3678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Значение в точке 0.6:</f> </text> </region> <region region-id="531" left="168" top="3666.75" width="66.75" height="17.25" align-x="186" align-y="3678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>6</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.943122</ml:real> </result> </ml:eval> </math> <rendering item-idref="122"/> </region> <region region-id="534" left="318" top="3666.75" width="69.75" height="17.25" align-x="339" align-y="3678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>6</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.043532516165123</ml:real> </result> </ml:eval> </math> <rendering item-idref="123"/> </region> <region region-id="507" left="426" top="3666.75" width="90" height="15.75" align-x="467.25" align-y="3678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.6</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0442376007810177</ml:real> </result> </ml:eval> </math> <rendering item-idref="124"/> </region> <region region-id="540" left="36" top="3720.75" width="83.25" height="27" align-x="36" align-y="3732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Относительная</f> <f family="Arial Cyr" charset="204" size="12">ошибка в %:</f> </text> </region> <region region-id="532" left="168" top="3711" width="138" height="35.25" align-x="257.25" align-y="3732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.6</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.6</ml:real> </ml:apply> </ml:apply> <ml:real>100</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>4.9463722192755686</ml:real> </result> </ml:eval> </math> <rendering item-idref="125"/> </region> <region region-id="533" left="318" top="3711" width="141" height="35.25" align-x="410.25" align-y="3732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.6</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.6</ml:real> </ml:apply> </ml:apply> <ml:real>100</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.034491324082156972</ml:real> </result> </ml:eval> </math> <rendering item-idref="126"/> </region> <region region-id="896" left="30" top="3759.75" width="472.5" height="88.5" align-x="30" align-y="3774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Задание </f> <f family="Arial Cyr" charset="204" size="12">. 1) Решить уравнение x''(t)=-t</f> <f family="Arial Cyr" charset="204" size="12"> 2 </f> <f family="Arial Cyr" charset="204" size="12">*x(t), преобразовав его к виду </f> <region region-id="897" left="30" top="3777" width="130.5" height="44.25" align-x="98.25" align-y="3803.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">t</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="127"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>методом припасовывания при условии х(0)=1, x'(0)=0.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">2) Решить уравнение x'(t)=t*x(t) при условии х(0)=1 методом Адамса.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">3) </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Решить уравнение x'=cos(x) при условии х(0)=0 методом Адамса.</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> </text> </region> <region region-id="543" left="36" top="3875.25" width="459" height="191.25" align-x="36" align-y="3888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="14">1.3 Метод разложения в ряд Тейлора<sp count="2"/>(метод припасовывания).</f> <f family="Arial Cyr" charset="204" size="12">В основе метода лежит представление решения рядом Тейлора. Решение д.б. гладким и иметь непрерывные производные до </f> <f family="Arial Cyr" charset="204" size="12">"n+1"-го порядка включительно. Ряд Тейлора заменяется его первыми </f> <f family="Arial Cyr" charset="204" size="12">"n+1" членами разложения в окрестности точки </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <region region-id="542" left="71.25" top="3944.25" width="9" height="17.25" align-x="75" align-y="3955.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </math> <rendering item-idref="128"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">на каждом участке "k". Конечные значения аргументов на этом участке становятся начальными на следующем участке "k+1" (подробнее см.[3]).</f> <f family="Arial Cyr" charset="204" size="12">На примере ранее рассмотренного уравнения </f> <region region-id="544" left="292.5" top="3975.75" width="86.25" height="36.75" align-x="332.25" align-y="3996.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="129"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>с начальными условиями </inlineAttr> </f> <region region-id="546" left="96.75" top="4013.25" width="42.75" height="17.25" align-x="107.25" align-y="4024.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:sequence> </math> <rendering item-idref="130"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> покажем как работает этот метод.</inlineAttr> </f> <inlineAttr font-weight="normal" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> Дано (см. выше)<sp count="2"/> </f> </inlineAttr> <region region-id="563" left="132.75" top="4031.25" width="207.75" height="35.25" align-x="325.5" align-y="4052.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:sequence> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">K</ml:id> <ml:real>6</ml:real> </ml:apply> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">k</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> <ml:id xml:space="preserve">K</ml:id> </ml:range> </math> <rendering item-idref="131"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10"> <sp/> </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">:</inlineAttr> </f> </text> </region> <region region-id="567" left="36" top="4086" width="78" height="16.5" align-x="78.75" align-y="4098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">φ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">V</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="132"/> </region> <region region-id="568" left="150" top="4086.75" width="33" height="17.25" align-x="166.5" align-y="4098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="133"/> </region> <region region-id="569" left="198" top="4086.75" width="38.25" height="17.25" align-x="219.75" align-y="4098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="134"/> </region> <region region-id="580" left="42" top="4122.75" width="78" height="13.5" align-x="42" align-y="4134" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Вычисляем </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="578" left="150" top="4113" width="222" height="36.75" align-x="199.5" align-y="4134" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">V2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id 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<ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:define> </math> <rendering item-idref="136"/> </region> <region region-id="845" left="48" top="4260" width="57.75" height="14.25" align-x="48" align-y="4272" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Проверка: </f> </text> </region> <region region-id="614" left="54" top="4290.75" width="30.75" height="15.75" align-x="68.25" align-y="4302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="137"/> </region> <region region-id="602" left="102" top="4290.75" width="91.5" height="17.25" align-x="129.75" align-y="4302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V2</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">V2</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="138"/> </region> <region region-id="603" left="210" top="4290.75" width="91.5" height="17.25" align-x="237.75" align-y="4302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V3</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">V3</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="139"/> </region> <region region-id="604" left="330" top="4290.75" width="48.75" height="17.25" align-x="357" align-y="4302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V2</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0000000000000089</ml:real> </result> </ml:eval> </math> <rendering item-idref="140"/> </region> <region region-id="605" left="390" top="4290.75" width="48.75" height="17.25" align-x="417" align-y="4302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V3</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.9999999999997282</ml:real> </result> </ml:eval> </math> <rendering item-idref="141"/> </region> <region region-id="774" left="42" top="4326" width="427.5" height="28.5" align-x="42" align-y="4338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Численные значения производных в заданных точках подтверждены. Проверка завершена успешно. </f> </text> </region> <region region-id="775" left="48" top="4374" width="356.25" height="28.5" align-x="48" align-y="4386" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Алгоритм вычислений. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">(некоторые параметры следует переопределить еще раз) :</inlineAttr> </f> </text> </region> <region region-id="617" left="48" top="4428.75" width="33" height="17.25" align-x="64.5" align-y="4440" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="dd62d7d7-9936-4965-89ae-cf836ae1798f" 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<inlineAttr font-weight="normal" underline="false">недопустимо определение </inlineAttr> </f> <region region-id="674" left="213" top="5136.75" width="61.5" height="17.25" align-x="235.5" align-y="5148" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Q</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="162"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false" line-through="false">!<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Формула в тексте идет как </inlineAttr> </f> <inlineAttr font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">исполняемая формула далее!</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Перемести выше формулу и убедись в этом!</inlineAttr> </f> </text> </region> <region region-id="780" left="36" top="5190.75" width="419.25" height="27" align-x="36" align-y="5202" show-border="false" show-highlight="false" is-protected="true" 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optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">o</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">o</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="4"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, численно методом конечных разностей.</inlineAttr> </f> </text> </region> <region region-id="27" left="24" top="96.75" width="150.75" height="13.5" align-x="24" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Число 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xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">b</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="14" left="24" top="168.75" width="111.75" height="13.5" align-x="24" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Начальные условия:</f> </text> </region> <region region-id="15" left="150" top="168.75" width="35.25" height="17.25" align-x="168.75" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="16" left="198" top="168.75" width="35.25" height="17.25" align-x="216.75" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="693" left="24" top="192.75" width="395.25" height="13.5" align-x="24" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Просмотр на экране дисплея численного значения функции в этой точке:</f> </text> </region> <region region-id="694" left="438" top="192.75" width="70.5" height="17.25" align-x="486.75" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="11"/> </region> <region region-id="37" left="24" top="228.75" width="80.25" height="13.5" align-x="24" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Дискрет по "х":</f> </text> </region> <region region-id="21" left="120" top="219" width="56.25" height="35.25" align-x="133.5" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="695" left="24" top="258.75" width="244.5" height="13.5" align-x="24" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Для выше выбранных параметров он равен:</f> </text> </region> <region region-id="696" left="288" top="258.75" width="43.5" height="15.75" align-x="300.75" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.1</ml:real> </result> </ml:eval> </math> <rendering item-idref="13"/> </region> <region region-id="697" left="24" top="282.75" width="258" height="13.5" align-x="24" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Дискретное представление самой переменной:</f> </text> </region> <region region-id="698" left="300" top="282.75" width="37.5" height="17.25" align-x="315.75" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="41" left="24" top="324.75" width="420.75" height="13.5" align-x="24" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Таким образом, для вычисления У(х) по методу Эйлера<sp count="2"/>получим выражение:</f> </text> </region> <region region-id="699" left="78" top="348.75" width="102.75" height="17.25" align-x="105" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="700" left="450" top="354.75" width="15.75" height="15.75" align-x="456.75" align-y="366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="16"/> </region> <region region-id="704" left="24" top="372.75" width="259.5" height="13.5" align-x="24" align-y="384" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Справочно приведем некоторые его значения: </f> </text> </region> <region region-id="709" left="66" top="396.75" width="40.5" height="15.75" align-x="78.75" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>10</ml:real> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="708" left="144" top="396.75" width="66.75" height="17.25" align-x="162" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>3.1874849202</ml:real> </result> </ml:eval> </math> <rendering item-idref="18"/> </region> <region region-id="710" left="234" top="396.75" width="60.75" height="17.25" align-x="252" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>5</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.72102</ml:real> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <region region-id="45" left="24" top="420.75" width="302.25" height="13.5" align-x="24" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Точное решение этого дифференциального уравнения:</f> </text> </region> <region region-id="47" left="342" top="415.5" width="93" height="21" align-x="372" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Q</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="789" left="24" top="450.75" width="207" height="13.5" align-x="24" align-y="462" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Значение функции на правой границе:</f> </text> </region> <region region-id="790" left="252" top="450.75" width="81" height="15.75" align-x="284.25" align-y="462" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>3.43656365691809</ml:real> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <region region-id="52" left="36" top="498" width="242.25" height="135.75" align-x="36" align-y="498" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="22"/> <rendering item-idref="23"/> </region> <region region-id="864" left="36" top="660.75" width="400.5" height="13.5" align-x="36" align-y="672" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Рис. 1.<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Возможна анимация графика </f> <f family="Arial Cyr" charset="204" size="12">по параметру "n".</f> </text> </region> <region region-id="56" left="30" top="720.75" width="469.5" height="135" align-x="30" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Разностная схема (метод) не всегда работает.<sp count="2"/>Если f(x,y)=f(y) и на каком-то шаге f(y</f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">)=0, то далее </f> <f family="Arial Cyr" charset="204" size="12">в силу (1) последующие значения y</f> <f family="Arial Cyr" charset="204" size="12"> i+1 </f> <f family="Arial Cyr" charset="204" size="12"> будут равны y</f> <f family="Arial Cyr" charset="204" size="12">i </f> <f family="Arial Cyr" charset="204" size="12">, что не всегда так</f> <f family="Arial Cyr" charset="204" size="12"> . Примером тому<sp count="2"/>может служить процесс, описываемый уравнением<sp count="3"/> </f> <region region-id="67" left="30" top="768" width="111.75" height="44.25" align-x="126.75" align-y="794.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="24"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">или </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">уравнением (2)</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> с заданными начальными условиями: z(0)=zo, z'(0)=q. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Независимо от значения "q" , z(t) это изменяемая во времени функция.<sp count="2"/>Однако, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">при q=0 разностная схема<sp count="2"/>дает другой результат - z(t)=const. При </inlineAttr> </f> <region region-id="74" left="30" top="840" width="28.5" height="15.75" align-x="43.5" align-y="851.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:notEqual/> <ml:id xml:space="preserve">q</ml:id> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="25"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> картина получается достоверной.</inlineAttr> </f> </text> </region> <region region-id="795" left="36" top="873" width="213" height="42.75" align-x="72" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">q</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </math> <rendering item-idref="26"/> </region> <region region-id="794" left="282" top="888.75" width="30.75" height="15.75" align-x="296.25" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">q</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="793" left="336" top="879.75" width="39.75" height="34.5" align-x="354" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="792" left="444" top="888.75" width="15.75" height="15.75" align-x="450.75" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="29"/> </region> <region region-id="130" left="36" top="930.75" width="423.75" height="27" align-x="36" align-y="942" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Если не ввести отдельно обозначение константы, то возникнут трудности при </f> <f family="Arial Cyr" charset="204" size="12">записи<sp count="2"/>f1(z).<sp count="2"/>Поэтому </f> </text> </region> <region region-id="423" left="168" top="972.75" width="81.75" height="17.25" align-x="202.5" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">CON</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="424" left="456" top="972.75" width="15.75" height="15.75" align-x="462.75" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="31"/> </region> <region region-id="87" left="30" top="1008.75" width="485.25" height="27" align-x="30" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Пусть "z" и "t" принадлежат соответственно интервалам [-zo,zo] и [0,T]. Запишем переменную t в дискретном виде (значение Т подобрано как полупериод) </f> </text> </region> <region region-id="801" left="36" top="1056.75" width="91.5" height="15.75" align-x="52.5" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <ml:apply> <ml:minus/> <ml:real>50</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="800" left="144" top="1056.75" width="65.25" height="15.75" align-x="156" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="799" left="228" top="1056.75" width="40.5" height="15.75" align-x="243" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">T</ml:id> <ml:real>3.3</ml:real> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="798" left="276" top="1048.5" width="37.5" height="33.75" align-x="291.75" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">δ</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="797" left="330" top="1056" width="39.75" height="18" align-x="344.25" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">δ</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="796" left="396" top="1056.75" width="52.5" height="15.75" align-x="416.25" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">q1</ml:id> <ml:real>-0.2</ml:real> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="809" left="30" top="1110" width="204.75" height="16.5" align-x="30" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Введем обозначение (</f> <region region-id="185" left="153" top="1110.75" width="70.5" height="15.75" align-x="208.5" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">CON</ml:id> <ml:id xml:space="preserve">q</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="38"/> </region> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="808" left="264" top="1095" width="173.25" height="41.25" align-x="295.5" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:id xml:space="preserve">CON</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">q</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="807" left="480" top="1110.75" width="15.75" height="15.75" align-x="486.75" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="40"/> </region> <region region-id="825" left="24" top="1164.75" width="220.5" height="27" align-x="24" align-y="1176" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Здесь (1.1) может быть записано в виде</f> <f family="Arial Cyr" charset="204" size="12">первых 2-х слагаемых ряда Тейлора:</f> </text> </region> <region region-id="822" left="276" top="1176" width="97.5" height="18" align-x="303" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">δ</ml:id> <ml:apply> <ml:id xml:space="preserve">f1</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="823" left="396" top="1176.75" width="66" height="17.25" align-x="413.25" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>4</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.5707963267948966</ml:real> </result> </ml:eval> </math> <rendering item-idref="42"/> </region> <region region-id="824" left="480" top="1176.75" width="15.75" height="15.75" align-x="486.75" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </ml:parens> </math> <rendering item-idref="43"/> </region> <region region-id="867" left="30" top="1200.75" width="239.25" height="16.5" align-x="30" align-y="1212" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Где f1(z</f> <f family="Arial Cyr" charset="204" size="12"> j </f> <f family="Arial Cyr" charset="204" size="12">) - первая производная на<sp count="2"/>j-м шаге.</f> </text> </region> <region region-id="717" left="36" top="1230.75" width="438.75" height="30" align-x="36" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">В результате получаем ошибочное решение Z</f> <f family="Arial Cyr" charset="204" size="12"> j </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> представляемое прямой линией </f> <f family="Arial Cyr" charset="204" size="12">в виде графика рис. 2</f> </text> </region> <region region-id="156" left="36" top="1290" width="258" height="16.5" align-x="36" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">С другой стороны при </f> <region region-id="152" left="158.25" top="1290.75" width="56.25" height="15.75" align-x="199.5" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:notEqual/> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">q</ml:id> <ml:id xml:space="preserve">q1</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="44"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>будем иметь:</inlineAttr> </f> </text> </region> <region region-id="175" left="318" top="1275" width="179.25" height="41.25" align-x="349.5" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:id xml:space="preserve">CON</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">q1</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="162" left="42" top="1326.75" width="44.25" height="17.25" align-x="66" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z2</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="163" left="114" top="1326" width="115.5" height="18" align-x="147" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z2</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z2</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">δ</ml:id> <ml:apply> <ml:id xml:space="preserve">f2</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z2</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="718" left="42" top="1362.75" width="444.75" height="13.5" align-x="42" align-y="1374" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Это решение также представлено на рис. 2 и является единственно правильным.</f> </text> </region> <region region-id="865" left="48" top="1446" width="165" height="135.75" align-x="48" align-y="1446" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="48"/> <rendering item-idref="49"/> </region> <region region-id="826" left="42" top="1608.75" width="330" height="13.5" align-x="42" align-y="1620" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Рис. 2.<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Возможна анимация графика.</f> </text> </region> <region region-id="719" left="48" top="1632" width="335.25" height="16.5" align-x="48" align-y="1644" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> Упражнение: </f> <f family="Arial Cyr" charset="204" size="12"> 1) снять клип по параметру "q1" :<sp count="2"/> </f> <region region-id="722" left="309" top="1632.75" width="73.5" height="15.75" align-x="366.75" align-y="1644" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>-0.2</ml:real> <ml:id xml:space="preserve">q1</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="50"/> </region> </text> </region> <region region-id="724" left="48" top="1656.75" width="465" height="40.5" align-x="48" align-y="1668" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">2) привести другие примеры неустойчивой работы этого метода.</f> <f family="Arial Cyr" charset="204" size="12">3) решить дифференциаальное уравнение y'(x)=cos(y(x)) при заданных начальных</f> <f family="Arial Cyr" charset="204" size="12">условиях х=0, у(0)=0.</f> </text> </region> <region region-id="188" left="48" top="1715.25" width="432.75" height="197.25" align-x="48" align-y="1728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="14">1.2 Метод Адамса.</f> <f family="Arial Cyr" charset="204" size="12">Решить уравнение вида: </f> <region region-id="237" left="184.5" top="1731.75" width="79.5" height="36.75" align-x="221.25" align-y="1752.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="51"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> для заданных начальных</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> условий </inlineAttr> </f> <region region-id="238" left="96.75" top="1769.25" width="9" height="17.25" align-x="100.5" align-y="1780.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="52"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <region region-id="239" left="112.5" top="1769.25" width="51.75" height="17.25" align-x="144.75" align-y="1780.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="53"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">,<sp count="2"/>численно методом конечных разностей. </f> </inlineAttr> <region region-id="194" left="393.75" top="1769.25" width="51" height="17.25" align-x="429" align-y="1780.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </math> <rendering item-idref="54"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">Запишем формулу Адамса с четырьмя членами [2]: </f> </inlineAttr> <region region-id="189" left="48" top="1800.75" width="231" height="33.75" align-x="80.25" align-y="1820.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:real>12</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">Δ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="55"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">,<sp count="44"/>(6)</f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">где </f> </inlineAttr> <region region-id="190" left="69" top="1841.25" width="72.75" height="17.25" align-x="81.75" align-y="1852.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="56"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">, </f> </inlineAttr> <region region-id="191" left="148.5" top="1840.5" width="106.5" height="18" align-x="190.5" align-y="1852.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="57"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <region region-id="192" left="261.75" top="1835.25" width="142.5" height="23.25" align-x="313.5" align-y="1852.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">Δ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX'</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="58"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <region region-id="193" left="48" top="1859.25" width="125.25" height="17.25" align-x="107.25" align-y="1870.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X'</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:sequence> </math> <rendering item-idref="59"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">Формула Адамса дает возможность, зная </f> </inlineAttr> <region region-id="195" left="279" top="1877.25" width="75.75" height="17.25" align-x="325.5" align-y="1888.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:sequence> </math> <rendering item-idref="60"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">,<sp count="2"/>определить </f> </inlineAttr> <region region-id="196" left="431.25" top="1877.25" width="25.5" height="17.25" align-x="443.25" align-y="1888.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="61"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">.</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Таким образом, зная </inlineAttr> </f> <region region-id="197" left="165" top="1895.25" width="53.25" height="17.25" align-x="200.25" align-y="1906.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real font="0">1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real font="0">2</ml:real> </ml:apply> </ml:sequence> </math> <rendering item-idref="62"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,<sp count="2"/>мы можем определить </inlineAttr> </f> <region region-id="198" left="355.5" top="1895.25" width="14.25" height="17.25" align-x="361.5" align-y="1906.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>3</ml:real> </ml:apply> </math> <rendering item-idref="63"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> и далее </f> </inlineAttr> <region region-id="199" left="418.5" top="1895.25" width="46.5" height="17.25" align-x="434.25" align-y="1906.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:sequence> <ml:real>5</ml:real> <ml:id xml:space="preserve" font="1">...</ml:id> </ml:sequence> </ml:apply> </ml:sequence> </math> <rendering item-idref="64"/> </region> </text> </region> <region region-id="869" left="48" top="1920" width="430.5" height="27.75" align-x="48" align-y="1932" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Моделирование алгоритма Адамса в системе MathCAD. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">На примере задачи, решенной выше.)</inlineAttr> </f> </text> </region> <region region-id="226" left="42" top="1968.75" width="30.75" height="13.5" align-x="42" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Дано:</f> </text> </region> <region region-id="225" left="96" top="1968" width="80.25" height="16.5" align-x="141" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">ψ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="65"/> </region> <region region-id="332" left="210" top="1968.75" width="33" height="17.25" align-x="226.5" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="66"/> </region> <region region-id="331" left="258" top="1968.75" width="38.25" height="17.25" align-x="279.75" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="67"/> </region> <region region-id="903" left="312" top="1960.5" width="110.25" height="33.75" align-x="325.5" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:minus/> <ml:real>0.5</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>50</ml:real> </ml:apply> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="68"/> </region> <region region-id="904" left="444" top="1968.75" width="39" height="15.75" align-x="457.5" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="69"/> </region> <region region-id="905" left="42" top="1998.75" width="413.25" height="13.5" align-x="42" align-y="2010" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Для проведения анимации необходимо убрать h:=0.1, заменив его на<sp count="2"/>h=0.1</f> </text> </region> <region region-id="898" left="54" top="2037" width="93" height="35.25" align-x="71.25" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">K</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="70"/> </region> <region region-id="877" left="156" top="2046.75" width="49.5" height="15.75" align-x="170.25" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">K</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="71"/> </region> <region region-id="876" left="234" top="2046.75" width="63.75" height="17.25" align-x="250.5" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="72"/> </region> <region region-id="733" left="48" top="2076.75" width="174" height="13.5" align-x="48" align-y="2088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Просмотр численного значения:</f> </text> </region> <region region-id="879" left="234" top="2076.75" width="44.25" height="15.75" align-x="250.5" align-y="2088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">K</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>10</ml:real> </result> </ml:eval> </math> <rendering item-idref="73"/> </region> <region region-id="731" left="294" top="2076.75" width="35.25" height="15.75" align-x="307.5" align-y="2088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">b</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="74"/> </region> <region region-id="737" left="48" top="2109" width="450" height="72" align-x="48" align-y="2136" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Вычисляем значение n-й производной: </f> <region region-id="251" left="261" top="2109.75" width="113.25" height="44.25" align-x="309" align-y="2136" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">Xn</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:id xml:space="preserve">n</ml:id> </ml:degree> </ml:apply> </ml:apply> </math> <rendering item-idref="75"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> . В системе Maple достаточно предыдущей символьной записи, а в MathCAD - каждую производную надо записывать отдельно:</inlineAttr> </f> </text> </region> <region region-id="880" left="42" top="2202.75" width="189.75" height="13.5" align-x="42" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Первая производная уже имеется:</f> </text> </region> <region region-id="886" left="258" top="2202" width="96.75" height="16.5" align-x="307.5" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">X1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="76"/> </region> <region region-id="739" left="42" top="2256.75" width="123" height="13.5" align-x="42" align-y="2268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Вторая производная:<sp count="2"/> </f> </text> </region> <region region-id="741" left="180" top="2247" width="228.75" height="36.75" align-x="229.5" align-y="2268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">X2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="77"/> </region> <region region-id="742" left="42" top="2292.75" width="121.5" height="13.5" align-x="42" align-y="2304" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Третья производная:<sp count="2"/> </f> </text> </region> <region region-id="242" left="48" top="2337.75" width="378" height="87" align-x="97.5" align-y="2364" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">X3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> </ml:function> <ml:parens> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">X</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">X</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:define> </math> <rendering item-idref="78"/> </region> <region region-id="743" left="36" top="2454.75" width="452.25" height="54" align-x="36" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Запись последующих производных в MathCAD становится обременительной. В </f> <f family="Arial Cyr" charset="204" size="12">Maple эту процедуру символьного дифференцирования выполняет сама система, </f> <f family="Arial Cyr" charset="204" size="12">что позволяет записывать производные практически любого порядка и вести </f> <f family="Arial Cyr" charset="204" size="12">далее с ними работу.</f> </text> </region> <region region-id="346" left="36" top="2520.75" width="454.5" height="27" align-x="36" align-y="2532" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Можно осуществить проверку<sp count="2"/>правильности </f> <f family="Arial Cyr" charset="204" size="12">вычислен</f> <f family="Arial Cyr" charset="204" size="12">ия производных для выбранных значений функции в отдельных точках:</f> </text> </region> <region region-id="744" left="36" top="2556.75" width="82.5" height="17.25" align-x="96.75" align-y="2568" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">X3</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.9999999999997282</ml:real> </result> </ml:eval> </math> <rendering item-idref="79"/> </region> <region region-id="745" left="120" top="2556.75" width="82.5" height="17.25" align-x="180.75" align-y="2568" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">X2</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0000000000000089</ml:real> </result> </ml:eval> </math> <rendering item-idref="80"/> </region> <region region-id="748" left="36" top="2580.75" width="352.5" height="13.5" align-x="36" align-y="2592" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Вывод: вычисляет правильно, в том числе и для других функций.</f> </text> </region> <region region-id="262" left="36" top="2610.75" width="122.25" height="13.5" align-x="36" align-y="2622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Введем обозначения </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="274" left="168" top="2610" width="87" height="18" align-x="195.75" align-y="2622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X1</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="81"/> </region> <region region-id="275" left="264" top="2610.75" width="91.5" height="17.25" align-x="291.75" align-y="2622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X2</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X2</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="82"/> </region> <region region-id="276" left="360" top="2610.75" width="91.5" height="17.25" align-x="387.75" align-y="2622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X3</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X3</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="83"/> </region> <region region-id="755" left="36" top="2646" width="307.5" height="14.25" align-x="36" align-y="2658" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Просмотр численных значений и проверка: </f> </text> </region> <region region-id="756" left="36" top="2670.75" width="48.75" height="17.25" align-x="63" align-y="2682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X1</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="84"/> </region> <region region-id="757" left="102" top="2670.75" width="48.75" height="17.25" align-x="129" align-y="2682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X2</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0000000000000089</ml:real> </result> </ml:eval> </math> <rendering item-idref="85"/> </region> <region region-id="758" left="174" top="2670.75" width="48.75" height="17.25" align-x="201" align-y="2682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X3</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.9999999999997282</ml:real> </result> </ml:eval> </math> <rendering item-idref="86"/> </region> <region region-id="370" left="36" top="2694" width="174" height="14.25" align-x="36" align-y="2706" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> Проверка завершена успешно. </f> </text> </region> <region region-id="280" left="36" top="2718" width="462" height="36" align-x="36" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Используя разложение </f> <region region-id="281" left="164.25" top="2718.75" width="21.75" height="15.75" align-x="172.5" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </math> <rendering item-idref="87"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> в ряд Тейлора в окрестности точки </inlineAttr> </f> <region region-id="282" left="383.25" top="2718.75" width="9" height="17.25" align-x="387" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="88"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> и беря его первые</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">четыре члена, определим таким образом </inlineAttr> </f> <region region-id="283" left="264.75" top="2736.75" width="57.75" height="17.25" align-x="285.75" align-y="2748" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="89"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="3"/>и<sp count="2"/> </inlineAttr> </f> <region region-id="284" left="344.25" top="2736.75" width="57.75" height="17.25" align-x="365.25" align-y="2748" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="90"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> как </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">:</inlineAttr> </f> </text> </region> <region region-id="293" left="42" top="2759.25" width="188.25" height="39" align-x="63.75" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X1</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X2</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:real>6</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X3</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="91"/> </region> <region region-id="292" left="252" top="2759.25" width="235.5" height="39" align-x="273.75" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X1</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X2</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> <ml:real>6</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X3</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="92"/> </region> <region region-id="889" left="36" top="2808.75" width="454.5" height="50.25" align-x="36" align-y="2820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Значения X</f> <f family="Arial Cyr" charset="204" size="12"> o </f> <f family="Arial Cyr" charset="204" size="12">, X</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12">, X</f> <f family="Arial Cyr" charset="204" size="12"> 2 </f> <f family="Arial Cyr" charset="204" size="12"> необходимы только вначале для запуска алгоритма Адамса.</f> <f family="Arial Cyr" charset="204" size="12">Вычисления последующих X</f> <f family="Arial Cyr" charset="204" size="12">k</f> <f family="Arial Cyr" charset="204" size="12">,<sp count="2"/> </f> <region region-id="888" left="202.5" top="2826" width="29.25" height="15.75" align-x="216" align-y="2837.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:greaterOrEqual/> <ml:id xml:space="preserve">k</ml:id> <ml:real>3</ml:real> </ml:apply> </math> <rendering item-idref="93"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> идет автоматически по рекуррентной формуле</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">через разности.</inlineAttr> </f> </text> </region> <region region-id="760" left="42" top="2874" width="176.25" height="14.25" align-x="42" align-y="2886" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> Проверка численных значений: </f> </text> </region> <region region-id="761" left="42" top="2904.75" width="69.75" height="17.25" align-x="63" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.1103333333333334</ml:real> </result> </ml:eval> </math> <rendering item-idref="94"/> </region> <region region-id="762" left="114" top="2904.75" width="69.75" height="17.25" align-x="135" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>2</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.2426666666666666</ml:real> </result> </ml:eval> </math> <rendering item-idref="95"/> </region> <region region-id="763" left="186" top="2904" width="105" height="18" align-x="242.25" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.2103333333333335</ml:real> </result> </ml:eval> </math> <rendering item-idref="96"/> </region> <region region-id="764" left="306" top="2904" width="105" height="18" align-x="362.25" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.4426666666666665</ml:real> </result> </ml:eval> </math> <rendering item-idref="97"/> </region> <region region-id="766" left="42" top="2928.75" width="99" height="13.5" align-x="42" align-y="2940" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Первые разности:</f> </text> </region> <region region-id="364" left="42" top="2958.75" width="46.5" height="15.75" align-x="56.25" align-y="2970" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="98"/> </region> <region region-id="357" left="108" top="2958" width="189.75" height="18" align-x="154.5" align-y="2970" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX1</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="99"/> </region> <region region-id="361" left="312" top="2958" width="52.5" height="56.25" align-x="330.75" align-y="2970" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX1</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="2" cols="1"> <ml:real>0.21033333333333348</ml:real> <ml:real>0.23233333333333306</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="100"/> </resultFormat> </math> <rendering item-idref="101"/> </region> <region region-id="767" left="48" top="3024.75" width="97.5" height="13.5" align-x="48" align-y="3036" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Вторые разности:</f> </text> </region> <region region-id="385" left="48" top="3054.75" width="30.75" height="15.75" align-x="62.25" align-y="3066" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="102"/> </region> <region region-id="374" left="114" top="3054" width="146.25" height="18" align-x="160.5" align-y="3066" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Δ2X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX1</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ΔX1</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="103"/> </region> <region region-id="375" left="282" top="3054" width="88.5" height="18" align-x="327.75" align-y="3066" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Δ2X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.021999999999999575</ml:real> </result> </ml:eval> </math> <rendering item-idref="104"/> </region> <region region-id="371" left="42" top="3084" width="174" height="14.25" align-x="42" align-y="3096" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Проверка завершена успешно. </f> </text> </region> <region region-id="294" left="42" top="3114.75" width="418.5" height="27" align-x="42" align-y="3126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Далее все последующие значения "х" будем вычислять по формуле Адамса,</f> <f family="Arial Cyr" charset="204" size="12">имеющей вид:</f> </text> </region> <region region-id="426" left="48" top="3162.75" width="49.5" height="15.75" align-x="62.25" align-y="3174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>2</ml:real> <ml:id xml:space="preserve">K</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="105"/> </region> <region region-id="425" left="126" top="3154.5" width="359.25" height="33.75" align-x="159" align-y="3174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>23</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:real>12</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:real>12</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="106"/> </region> <region region-id="892" left="48" top="3198.75" width="423.75" height="13.5" align-x="48" align-y="3210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Результаты вычислений представим в виде вектора X и графически на рис. 3</f> </text> </region> <region region-id="908" left="48" top="3228.75" width="408.75" height="31.5" align-x="70.5" align-y="3246" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:transpose/> <ml:id xml:space="preserve">X</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="1" cols="12"> <ml:real>1</ml:real> <ml:real>1.1103333333333334</ml:real> <ml:real>1.2426666666666666</ml:real> <ml:real>1.3994666666666666</ml:real> <ml:real>1.5832727777777778</ml:real> <ml:real>1.7969156157407404</ml:real> <ml:real>2.043532516165123</ml:real> <ml:real>2.32659053207208</ml:real> <ml:real>2.64992086588641</ml:real> <ml:real>3.0177574824119078</ml:real> <ml:real>3.4347794899256723</ml:real> <ml:real>3.9061579305851053</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="107"/> </resultFormat> </math> <rendering item-idref="108"/> </region> <region region-id="893" left="120" top="3294.75" width="47.25" height="15.75" align-x="132" align-y="3306" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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align-x="372" align-y="3312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">QQ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">tt</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:id xml:space="preserve">tt</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">tt</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="111"/> </region> <region region-id="829" left="66" top="3354" width="201.75" height="153.75" align-x="66" align-y="3354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="112"/> <rendering item-idref="113"/> </region> <region region-id="830" left="318" top="3414.75" width="42.75" height="13.5" align-x="318" align-y="3426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Рис. 3</f> </text> </region> <region region-id="524" left="48" top="3522.75" width="435" height="31.5" align-x="48" align-y="3534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Сравнительный анализ, полученных численных результатов разными методами</f> <f family="Arial Cyr" charset="204" size="12">(метод Эйлера - </f> <region region-id="435" left="139.5" top="3537" width="11.25" height="17.25" align-x="144" align-y="3548.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </math> <rendering item-idref="114"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> , </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">метод</f> <sp/> <f family="Arial Cyr" charset="204" size="12">Адамса -</f> <f family="Arial Cyr" charset="204"> <sp/> </f> <region region-id="436" left="249.75" top="3537" width="14.25" height="17.25" align-x="255.75" align-y="3548.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </math> <rendering item-idref="115"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,точное решение - </inlineAttr> </f> <region region-id="437" left="368.25" top="3537" width="24.75" height="15.75" align-x="377.25" align-y="3548.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Q</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </math> <rendering item-idref="116"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">).</inlineAttr> </f> </text> </region> <region region-id="834" left="36" top="3564.75" width="117" height="13.5" align-x="36" align-y="3576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Значение в точке 0.4:</f> </text> </region> <region region-id="839" left="168" top="3564.75" width="66.75" height="17.25" align-x="186" align-y="3576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>4</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.5282</ml:real> </result> </ml:eval> </math> <rendering item-idref="117"/> </region> <region region-id="842" left="312" top="3564.75" width="69.75" height="17.25" align-x="333" align-y="3576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>4</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.5832727777777778</ml:real> </result> </ml:eval> </math> <rendering item-idref="118"/> </region> <region region-id="831" left="420" top="3564.75" width="90" height="15.75" align-x="461.25" align-y="3576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.4</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.5836493952825408</ml:real> </result> </ml:eval> </math> <rendering item-idref="119"/> </region> <region region-id="837" left="36" top="3600.75" width="83.25" height="27" align-x="36" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Относительная</f> <f family="Arial Cyr" charset="204" size="12">ошибка в %:</f> </text> </region> <region region-id="838" left="168" top="3591" width="138" height="35.25" align-x="257.25" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.4</ml:real> 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<ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.4</ml:real> </ml:apply> </ml:apply> <ml:real>100</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.023781621480418645</ml:real> </result> </ml:eval> </math> <rendering item-idref="121"/> </region> <region region-id="539" left="36" top="3666.75" width="117" height="13.5" align-x="36" align-y="3678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Значение в точке 0.6:</f> </text> </region> <region region-id="531" left="168" top="3666.75" width="66.75" height="17.25" align-x="186" align-y="3678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>6</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.943122</ml:real> </result> </ml:eval> </math> <rendering item-idref="122"/> </region> <region region-id="534" left="318" top="3666.75" width="69.75" height="17.25" align-x="339" align-y="3678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>6</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.043532516165123</ml:real> </result> </ml:eval> </math> <rendering item-idref="123"/> </region> <region region-id="507" left="426" top="3666.75" width="90" height="15.75" align-x="467.25" align-y="3678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:real>0.6</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0442376007810177</ml:real> </result> </ml:eval> </math> <rendering item-idref="124"/> </region> <region region-id="540" left="36" top="3720.75" width="83.25" height="27" align-x="36" align-y="3732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Относительная</f> <f family="Arial Cyr" charset="204" size="12">ошибка в %:</f> </text> </region> <region region-id="532" left="168" top="3711" width="138" height="35.25" align-x="257.25" align-y="3732" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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1) Решить уравнение x''(t)=-t</f> <f family="Arial Cyr" charset="204" size="12"> 2 </f> <f family="Arial Cyr" charset="204" size="12">*x(t), преобразовав его к виду </f> <region region-id="897" left="30" top="3777" width="130.5" height="44.25" align-x="98.25" align-y="3803.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">t</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="127"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>методом припасовывания при условии х(0)=1, x'(0)=0.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">2) Решить уравнение x'(t)=t*x(t) при условии х(0)=1 методом Адамса.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">3) </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Решить уравнение x'=cos(x) при условии х(0)=0 методом Адамса.</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> </text> </region> <region region-id="543" left="36" top="3875.25" width="459" height="191.25" align-x="36" align-y="3888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="14">1.3 Метод разложения в ряд Тейлора<sp count="2"/>(метод припасовывания).</f> <f family="Arial Cyr" charset="204" size="12">В основе метода лежит представление решения рядом Тейлора. Решение д.б. гладким и иметь непрерывные производные до </f> <f family="Arial Cyr" charset="204" size="12">"n+1"-го порядка включительно. Ряд Тейлора заменяется его первыми </f> <f family="Arial Cyr" charset="204" size="12">"n+1" членами разложения в окрестности точки </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <region region-id="542" left="71.25" top="3944.25" width="9" height="17.25" align-x="75" align-y="3955.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </math> <rendering item-idref="128"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">на каждом участке "k". Конечные значения аргументов на этом участке становятся начальными на следующем участке "k+1" (подробнее см.[3]).</f> <f family="Arial Cyr" charset="204" size="12">На примере ранее рассмотренного уравнения </f> <region region-id="544" left="292.5" top="3975.75" width="86.25" height="36.75" align-x="332.25" align-y="3996.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="129"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>с начальными условиями </inlineAttr> </f> <region region-id="546" left="96.75" top="4013.25" width="42.75" height="17.25" align-x="107.25" align-y="4024.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:sequence> </math> <rendering item-idref="130"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> покажем как работает этот метод.</inlineAttr> </f> <inlineAttr font-weight="normal" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> Дано (см. выше)<sp count="2"/> </f> </inlineAttr> <region region-id="563" left="132.75" top="4031.25" width="207.75" height="35.25" align-x="325.5" align-y="4052.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:sequence> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">K</ml:id> <ml:real>6</ml:real> </ml:apply> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">k</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> <ml:id xml:space="preserve">K</ml:id> </ml:range> </math> <rendering item-idref="131"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10"> <sp/> </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">:</inlineAttr> </f> </text> </region> <region region-id="567" left="36" top="4086" width="78" height="16.5" align-x="78.75" align-y="4098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">φ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">V</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="132"/> </region> <region region-id="568" left="150" top="4086.75" width="33" height="17.25" align-x="166.5" align-y="4098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="133"/> </region> <region region-id="569" left="198" top="4086.75" width="38.25" height="17.25" align-x="219.75" align-y="4098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="134"/> </region> <region region-id="580" left="42" top="4122.75" width="78" height="13.5" align-x="42" align-y="4134" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Вычисляем </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="578" left="150" top="4113" width="222" height="36.75" align-x="199.5" align-y="4134" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">V2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="135"/> </region> <region region-id="575" left="48" top="4161.75" width="369" height="87" align-x="97.5" align-y="4188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">V3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> </ml:function> <ml:parens> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">V</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:define> </math> <rendering item-idref="136"/> </region> <region region-id="845" left="48" top="4260" width="57.75" height="14.25" align-x="48" align-y="4272" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Проверка: </f> </text> </region> <region region-id="614" left="54" top="4290.75" width="30.75" height="15.75" align-x="68.25" align-y="4302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="137"/> </region> <region region-id="602" left="102" top="4290.75" width="91.5" height="17.25" align-x="129.75" align-y="4302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V2</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">V2</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="138"/> </region> <region region-id="603" left="210" top="4290.75" width="91.5" height="17.25" align-x="237.75" align-y="4302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V3</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">V3</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="139"/> </region> <region region-id="604" left="330" top="4290.75" width="48.75" height="17.25" align-x="357" align-y="4302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V2</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0000000000000089</ml:real> </result> </ml:eval> </math> <rendering item-idref="140"/> </region> <region region-id="605" left="390" top="4290.75" width="48.75" height="17.25" align-x="417" align-y="4302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">V3</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.9999999999997282</ml:real> </result> </ml:eval> </math> <rendering item-idref="141"/> </region> <region region-id="774" left="42" top="4326" width="427.5" height="28.5" align-x="42" align-y="4338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Численные значения производных в заданных точках подтверждены. Проверка завершена успешно. </f> </text> </region> <region region-id="775" left="48" top="4374" width="356.25" height="28.5" align-x="48" align-y="4386" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Алгоритм вычислений. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">(некоторые параметры следует переопределить еще раз) :</inlineAttr> </f> </text> </region> <region region-id="617" left="48" top="4428.75" width="33" height="17.25" align-x="64.5" align-y="4440" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="dd62d7d7-9936-4965-89ae-cf836ae1798f" version-id="e99ad359-0854-47c7-96be-02c370bb29e5" branch-id="00000000-0000-0000-0000-000000000000" revision-num="3" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЧислМетоды\МетПрипасовывания.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="dd62d7d7-9936-4965-89ae-cf836ae1798f" version-id="e99ad359-0854-47c7-96be-02c370bb29e5" branch-id="00000000-0000-0000-0000-000000000000" revision-num="3" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЧислМетоды\МетПрипасовывания.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">d20f025e5ae4d43040e9208e58ddda37</contentHash> <ml:define> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:define> </ml:provenance> </math> <rendering item-idref="142"> <element-image-map> <box left="1.5" top="0.75" width="31.5" height="15.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="618" left="96" top="4428.75" width="38.25" height="17.25" align-x="117.75" align-y="4440" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="dd62d7d7-9936-4965-89ae-cf836ae1798f" version-id="e99ad359-0854-47c7-96be-02c370bb29e5" branch-id="00000000-0000-0000-0000-000000000000" revision-num="3" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЧислМетоды\МетПрипасовывания.xmcd" 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width="83.25" height="27" align-x="96" align-y="5100" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Относительная</f> <f family="Arial Cyr" charset="204" size="12">ошибка в %:</f> </text> </region> <region region-id="683" left="354" top="5079" width="177.75" height="36.75" align-x="451.5" align-y="5100" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> 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<inlineAttr font-weight="normal" underline="false">недопустимо определение </inlineAttr> </f> <region region-id="674" left="213" top="5136.75" width="61.5" height="17.25" align-x="235.5" align-y="5148" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Q</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="162"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false" line-through="false">!<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Формула в тексте идет как </inlineAttr> </f> <inlineAttr font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">исполняемая формула далее!</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Перемести выше формулу и убедись в этом!</inlineAttr> </f> </text> </region> <region region-id="780" left="36" top="5190.75" width="419.25" height="27" align-x="36" align-y="5202" show-border="false" show-highlight="false" is-protected="true" 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