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family="Arial Cyr" charset="204" size="12">Задачи из Ефимова [24]. Ряды Фурье. </f> <f family="Arial Cyr" charset="204" size="12"> Задача №12.484 </f> </text> </region> <region region-id="39" left="42" top="36" width="417" height="36" align-x="42" align-y="48" 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 задаче требуется разложить функцию в ряд Фурье. Зададим функцию </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">) </f> <f family="Arial Cyr" charset="204" size="12">следу</f> <f family="Arial Cyr" charset="204" size="12">ющим образом </f> <region region-id="212" left="157.5" top="51" width="45" height="21" align-x="183" align-y="67.5" 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">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </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"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> (см. рис.1):</f> </text> </region> <region region-id="218" left="36" top="90.75" width="158.25" height="13.5" align-x="36" align-y="102" 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">Корректнее будет задать так</f> </text> </region> <region region-id="219" left="216" top="85.5" width="149.25" height="21" align-x="248.25" align-y="102" 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">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="223" left="396" top="85.5" width="47.25" height="21" align-x="422.25" align-y="102" 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:boundVars> </ml:function> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="220" left="492" top="90.75" width="15.75" height="15.75" align-x="498.75" align-y="102" 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="5"/> </region> <region region-id="40" left="36" top="126" width="471" height="27.75" align-x="36" align-y="138" 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 периодической функции </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">) задан п</f> <f family="Arial Cyr" charset="204" size="12">олупериод </f> <f family="Arial Cyr" charset="204" size="12"> L </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> который можно выбрать, исходя </f> <f family="Arial Cyr" charset="204" size="12">из удобства вычисления (см. [13], стр. 328) коэффициентов ряда Фурье, равный:</f> </text> </region> <region region-id="58" left="48" top="168.75" width="30.75" height="15.75" align-x="62.25" 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:id xml:space="preserve">q</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="57" left="108" top="168" width="43.5" height="16.5" align-x="123" 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 warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">q</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="53" left="42" top="198" width="465.75" height="28.5" align-x="42" align-y="210" 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">Введём параметр анимации </f> <f family="Arial Cyr" charset="204" size="12"> FRAME </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">так, чтобы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> 20<</f> <f family="Arial Cyr" charset="204" size="12"> FRAME </f> <f family="Arial Cyr" charset="204" size="12">, через который выразим число </f> <f family="Arial Cyr" charset="204" size="12"> N </f> <f family="Arial Cyr" charset="204" size="12"> членов ряда Фурье, например, так:<sp count="2"/> </f> </text> </region> <region region-id="59" left="180" top="234.75" width="87" height="15.75" align-x="197.25" align-y="246" 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">N</ml:id> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="61" left="36" top="266.25" width="435" height="12" align-x="36" align-y="276" 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="10"> Примечание </f> <f family="Arial Cyr" charset="204" size="10">. Такое ограничение 20<</f> <f family="Arial Cyr" charset="204" size="10"> FRAME </f> <f family="Arial Cyr" charset="204" size="10"> связано со временем записи видеоклипа и только. </f> </text> </region> <region region-id="25" left="36" top="288" width="475.5" height="27.75" align-x="36" align-y="300" 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">Сумма ряда </f> <f family="Arial Cyr" charset="204" size="12"> S </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">) приближает функцию </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12">. Покажем это в режиме </f> <f family="Arial Cyr" charset="204" size="12">анимации на рис.1, предварительно вычислив коэффициенты </f> <f family="Arial Cyr" charset="204" size="12">ряда Фурье</f> </text> </region> <region region-id="107" left="36" top="336" width="171.75" height="56.25" align-x="64.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">a</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="124" left="264" top="336" width="168.75" height="56.25" align-x="293.25" 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:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">b</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="125" left="492" top="354.75" width="15.75" height="15.75" align-x="498.75" 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>2</ml:real> </ml:parens> </math> <rendering item-idref="11"/> </region> <region region-id="70" left="36" top="408.75" width="464.25" height="41.25" align-x="36" align-y="420" 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">Проведём визуализацию численных значений коэффициентов </f> <f family="Arial Cyr" charset="204" size="12">ряда Фурье и запишем</f> <f family="Arial Cyr" charset="204" size="12">его </f> <f family="Arial Cyr" charset="204" size="12">сумму<sp count="2"/>S(x), оставив в ней только коэффициенты </f> <f family="Arial Cyr" charset="204" size="12"> a, </f> <f family="Arial Cyr" charset="204" size="12"> так как </f> <f family="Arial Cyr" charset="204" size="12">коэффициенты </f> <f family="Arial Cyr" charset="204" size="12"> b </f> <f family="Arial Cyr" charset="204" size="12"> по</f> <f family="Arial Cyr" charset="204" size="12">определению для чётной функции равны нулю, что подтверждается ниже</f> </text> </region> <region region-id="204" left="42" top="480.75" width="44.25" height="15.75" align-x="54" align-y="492" 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:real>3</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="205" left="108" top="480.75" width="50.25" height="81" align-x="117" align-y="492" 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">a</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>6.5797362673929056</ml:real> <ml:real>-3.9999999999999991</ml:real> <ml:real>0.99999999999999956</ml:real> <ml:real>-0.44444444444444514</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="13"/> </resultFormat> </math> <rendering item-idref="14"/> </region> <region region-id="206" left="180" top="480.75" width="36.75" height="81" align-x="189" align-y="492" 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">b</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="15"/> </resultFormat> </math> <rendering item-idref="16"/> </region> <region region-id="207" left="258" top="465.75" width="205.5" height="50.25" align-x="288" align-y="492" 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:function> <ml:id xml:space="preserve">S</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="208" left="492" top="480.75" width="15.75" height="15.75" align-x="498.75" align-y="492" 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="18"/> </region> <region region-id="195" left="384" top="580.5" width="66" height="33.75" align-x="413.25" align-y="600" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="19"/> </region> <region region-id="189" left="42" top="580.5" width="280.5" height="45" align-x="70.5" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="20"/> </region> <region region-id="190" left="36" top="648.75" width="470.25" height="35.25" align-x="36" align-y="666" 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">Так как для целых </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">=0, 1, 2, ...</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> справедливо </f> <region region-id="191" left="285" top="654" width="64.5" height="16.5" align-x="334.5" align-y="666" 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">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="21"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> и </inlineAttr> </f> <region region-id="192" left="363.75" top="649.5" width="87.75" height="21" align-x="415.5" align-y="666" 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">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="22"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, то коэф-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">фициент </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">a</inlineAttr> </f> <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"> <inlineAttr font-weight="normal" underline="false" line-through="false">n</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">) можно упростить до вида</inlineAttr> </f> </text> </region> <region region-id="194" left="48" top="723.75" width="222.75" height="45.75" align-x="216" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:command> <ml:sequence> <ml:id xml:space="preserve">substitute</ml:id> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:command> <ml:command> <ml:sequence> <ml:id xml:space="preserve">substitute</ml:id> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:sequence> </ml:command> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="23"/> </region> <region region-id="224" left="312" top="730.5" width="83.25" height="39" align-x="340.5" align-y="750" 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:function> <ml:id xml:space="preserve">a</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="226" left="48" top="786.75" width="457.5" height="27" align-x="48" align-y="798" 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">Исходя из найденных коэффициентов ряда Фурье, запишем сумму (3) и представим</f> <f family="Arial Cyr" charset="204" size="12">её график в режиме анимации на рис.1.</f> </text> </region> <region region-id="227" left="54" top="831.75" width="199.5" height="50.25" align-x="84" align-y="858" 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:function> <ml:id xml:space="preserve">S</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="88" left="30" top="912" width="308.25" height="227.25" align-x="30" align-y="912" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="26"/> <rendering item-idref="27"/> </region> <region region-id="229" left="30" top="1176.75" width="478.5" height="68.25" align-x="30" align-y="1188" 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">Надо сказать, что при нахождении коэффициентов a(n) и b(n) под знак интеграла была</f> <f family="Arial Cyr" charset="204" size="12">помещена функция f(x), а не f1(x), так как последняя не позволяла вести далее сим- вольные вычисления. </f> <f family="Arial Cyr" charset="204" size="12">Рассмотрим функцию Хевисайда </f> <f family="Arial Cyr" charset="204" size="12"> Ф </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">) и то как с помощью этой функции можно задать<sp count="2"/>функцию f(x) на примере данной задачи (см. рис.2):</f> </text> </region> <region region-id="112" left="66" top="1284" width="177.75" height="16.5" align-x="97.5" align-y="1296" 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:function> <ml:id xml:space="preserve">H</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Φ</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Φ</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="151" left="30" top="1336.5" width="470.25" height="33.75" align-x="30" align-y="1350" 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">В условиях задачи </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">№12.484 функция задана как f</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">)=x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> 2 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">, а на рис.2 функцию f</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">) имитируют</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">функция<sp count="2"/>H(x) на отрезке</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <region region-id="153" left="240.75" top="1353.75" width="37.5" height="16.5" align-x="258.75" align-y="1365.75" 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:sequence> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:sequence> </ml:parens> </math> <rendering item-idref="29"/> </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="50" left="48" top="1416" width="336.75" height="167.25" align-x="48" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="30"/> <rendering item-idref="31"/> </region> <region region-id="173" left="42" top="1607.25" width="441" height="44.25" 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">Итак, разложим теперь функцию H(x) в ряд Фурье, с полуперидом </f> <region region-id="157" left="417" top="1608" width="30" height="16.5" align-x="431.25" align-y="1620" 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 style="equal"/> <ml:id xml:space="preserve">L</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </math> <rendering item-idref="32"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Тогда</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">будем иметь </inlineAttr> </f> </text> </region> <region region-id="233" left="60" top="1650" width="185.25" height="56.25" align-x="96.75" align-y="1680" 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">Ha</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">H</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="244" left="348" top="1660.5" width="74.25" height="33.75" align-x="385.5" align-y="1680" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Ha</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="34"/> </region> <region region-id="234" left="48" top="1720.5" width="288.75" height="45" align-x="84.75" align-y="1746" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Ha</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="35"/> </region> <region region-id="237" left="48" top="1770.75" width="243.75" height="13.5" align-x="48" align-y="1782" 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">Последнее выражение можно упростить так:</f> </text> </region> <region region-id="238" left="54" top="1809" width="231" height="63.75" align-x="230.25" align-y="1842" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Ha</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:command> <ml:sequence> <ml:id xml:space="preserve">substitute</ml:id> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:command> <ml:command> <ml:sequence> <ml:id xml:space="preserve">substitute</ml:id> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:sequence> </ml:command> <ml:command> <ml:id xml:space="preserve">simplify</ml:id> </ml:command> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="36"/> </region> <region region-id="239" left="360" top="1822.5" width="91.5" height="39" align-x="396.75" align-y="1842" 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:function> <ml:id xml:space="preserve">Ha</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="245" left="48" top="1890.75" width="426.75" height="27" align-x="48" align-y="1902" 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">Запишем сумму ряда Фурье для функции H(x) или<sp count="2"/>f(x) в окончательном виде и </f> <f family="Arial Cyr" charset="204" size="12">снимем видеоклип на рис. 3</f> </text> </region> <region region-id="242" left="54" top="1947.75" width="204.75" height="50.25" align-x="89.25" align-y="1974" 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">Sh</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="246" left="36" top="2028.75" width="473.25" height="27" align-x="36" align-y="2040" 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">Построенный график функции H(x)<sp count="2"/>и приближающий её ряд<sp count="2"/>Sh(x) рис.3 </f> <f family="Arial Cyr" charset="204" size="12">в режиме анимации</f> <f family="Arial Cyr" charset="204" size="12"> дают оценку точности.</f> </text> </region> <region region-id="172" left="48" top="2112" width="338.25" height="191.25" align-x="48" align-y="2112" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="39"/> <rendering item-idref="40"/> </region> <region region-id="187" left="36" top="2340.75" width="475.5" height="27" align-x="36" align-y="2352" 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">Полученный ряд Фурье является неполным, так как один из коэффициентов ряда b(n) равен нулю, что характерно для четных функций. </f> </text> </region> </regions> <binaryContent> <item item-id="1" 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family="Arial Cyr" charset="204" size="12">Задачи из Ефимова [24]. Ряды Фурье. </f> <f family="Arial Cyr" charset="204" size="12"> Задача №12.484 </f> </text> </region> <region region-id="39" left="42" top="36" width="417" height="36" align-x="42" align-y="48" 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 задаче требуется разложить функцию в ряд Фурье. Зададим функцию </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">) </f> <f family="Arial Cyr" charset="204" size="12">следу</f> <f family="Arial Cyr" charset="204" size="12">ющим образом </f> <region region-id="212" left="157.5" top="51" width="45" height="21" align-x="183" align-y="67.5" 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">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </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"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> (см. рис.1):</f> </text> </region> <region region-id="218" left="36" top="90.75" width="158.25" height="13.5" align-x="36" align-y="102" 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">Корректнее будет задать так</f> </text> </region> <region region-id="219" left="216" top="85.5" width="149.25" height="21" align-x="248.25" align-y="102" 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">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="223" left="396" top="85.5" width="47.25" height="21" align-x="422.25" align-y="102" 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:boundVars> </ml:function> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="220" left="492" top="90.75" width="15.75" height="15.75" align-x="498.75" align-y="102" 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="5"/> </region> <region region-id="40" left="36" top="126" width="471" height="27.75" align-x="36" align-y="138" 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 периодической функции </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">) задан п</f> <f family="Arial Cyr" charset="204" size="12">олупериод </f> <f family="Arial Cyr" charset="204" size="12"> L </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> который можно выбрать, исходя </f> <f family="Arial Cyr" charset="204" size="12">из удобства вычисления (см. [13], стр. 328) коэффициентов ряда Фурье, равный:</f> </text> </region> <region region-id="58" left="48" top="168.75" width="30.75" height="15.75" align-x="62.25" 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:id xml:space="preserve">q</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="57" left="108" top="168" width="43.5" height="16.5" align-x="123" 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 warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">q</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="53" left="42" top="198" width="465.75" height="28.5" align-x="42" align-y="210" 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">Введём параметр анимации </f> <f family="Arial Cyr" charset="204" size="12"> FRAME </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">так, чтобы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> 20<</f> <f family="Arial Cyr" charset="204" size="12"> FRAME </f> <f family="Arial Cyr" charset="204" size="12">, через который выразим число </f> <f family="Arial Cyr" charset="204" size="12"> N </f> <f family="Arial Cyr" charset="204" size="12"> членов ряда Фурье, например, так:<sp count="2"/> </f> </text> </region> <region region-id="59" left="180" top="234.75" width="87" height="15.75" align-x="197.25" align-y="246" 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">N</ml:id> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="61" left="36" top="266.25" width="435" height="12" align-x="36" align-y="276" 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="10"> Примечание </f> <f family="Arial Cyr" charset="204" size="10">. Такое ограничение 20<</f> <f family="Arial Cyr" charset="204" size="10"> FRAME </f> <f family="Arial Cyr" charset="204" size="10"> связано со временем записи видеоклипа и только. </f> </text> </region> <region region-id="25" left="36" top="288" width="475.5" height="27.75" align-x="36" align-y="300" 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">Сумма ряда </f> <f family="Arial Cyr" charset="204" size="12"> S </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">) приближает функцию </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12">. Покажем это в режиме </f> <f family="Arial Cyr" charset="204" size="12">анимации на рис.1, предварительно вычислив коэффициенты </f> <f family="Arial Cyr" charset="204" size="12">ряда Фурье</f> </text> </region> <region region-id="107" left="36" top="336" width="171.75" height="56.25" align-x="64.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">a</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="124" left="264" top="336" width="168.75" height="56.25" align-x="293.25" 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:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">b</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="125" left="492" top="354.75" width="15.75" height="15.75" align-x="498.75" 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>2</ml:real> </ml:parens> </math> <rendering item-idref="11"/> </region> <region region-id="70" left="36" top="408.75" width="464.25" height="41.25" align-x="36" align-y="420" 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">Проведём визуализацию численных значений коэффициентов </f> <f family="Arial Cyr" charset="204" size="12">ряда Фурье и запишем</f> <f family="Arial Cyr" charset="204" size="12">его </f> <f family="Arial Cyr" charset="204" size="12">сумму<sp count="2"/>S(x), оставив в ней только коэффициенты </f> <f family="Arial Cyr" charset="204" size="12"> a, </f> <f family="Arial Cyr" charset="204" size="12"> так как </f> <f family="Arial Cyr" charset="204" size="12">коэффициенты </f> <f family="Arial Cyr" charset="204" size="12"> b </f> <f family="Arial Cyr" charset="204" size="12"> по</f> <f family="Arial Cyr" charset="204" size="12">определению для чётной функции равны нулю, что подтверждается ниже</f> </text> </region> <region region-id="204" left="42" top="480.75" width="44.25" height="15.75" align-x="54" align-y="492" 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:real>3</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="205" left="108" top="480.75" width="50.25" height="81" align-x="117" align-y="492" 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">a</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>6.5797362673929056</ml:real> <ml:real>-3.9999999999999991</ml:real> <ml:real>0.99999999999999956</ml:real> <ml:real>-0.44444444444444514</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="13"/> </resultFormat> </math> <rendering item-idref="14"/> </region> <region region-id="206" left="180" top="480.75" width="36.75" height="81" align-x="189" align-y="492" 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">b</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="15"/> </resultFormat> </math> <rendering item-idref="16"/> </region> <region region-id="207" left="258" top="465.75" width="205.5" height="50.25" align-x="288" align-y="492" 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:function> <ml:id xml:space="preserve">S</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="208" left="492" top="480.75" width="15.75" height="15.75" align-x="498.75" align-y="492" 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="18"/> </region> <region region-id="195" left="384" top="580.5" width="66" height="33.75" align-x="413.25" align-y="600" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="19"/> </region> <region region-id="189" left="42" top="580.5" width="280.5" height="45" align-x="70.5" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="20"/> </region> <region region-id="190" left="36" top="648.75" width="470.25" height="35.25" align-x="36" align-y="666" 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">Так как для целых </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">=0, 1, 2, ...</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> справедливо </f> <region region-id="191" left="285" top="654" width="64.5" height="16.5" align-x="334.5" align-y="666" 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">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="21"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> и </inlineAttr> </f> <region region-id="192" left="363.75" top="649.5" width="87.75" height="21" align-x="415.5" align-y="666" 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">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="22"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, то коэф-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">фициент </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">a</inlineAttr> </f> <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"> <inlineAttr font-weight="normal" underline="false" line-through="false">n</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">) можно упростить до вида</inlineAttr> </f> </text> </region> <region region-id="194" left="48" top="723.75" width="222.75" height="45.75" align-x="216" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:command> <ml:sequence> <ml:id xml:space="preserve">substitute</ml:id> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:command> <ml:command> <ml:sequence> <ml:id xml:space="preserve">substitute</ml:id> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:sequence> </ml:command> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="23"/> </region> <region region-id="224" left="312" top="730.5" width="83.25" height="39" align-x="340.5" align-y="750" 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:function> <ml:id xml:space="preserve">a</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="226" left="48" top="786.75" width="457.5" height="27" align-x="48" align-y="798" 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">Исходя из найденных коэффициентов ряда Фурье, запишем сумму (3) и представим</f> <f family="Arial Cyr" charset="204" size="12">её график в режиме анимации на рис.1.</f> </text> </region> <region region-id="227" left="54" top="831.75" width="199.5" height="50.25" align-x="84" align-y="858" 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:function> <ml:id xml:space="preserve">S</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="88" left="30" top="912" width="308.25" height="227.25" align-x="30" align-y="912" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="26"/> <rendering item-idref="27"/> </region> <region region-id="229" left="30" top="1176.75" width="478.5" height="68.25" align-x="30" align-y="1188" 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">Надо сказать, что при нахождении коэффициентов a(n) и b(n) под знак интеграла была</f> <f family="Arial Cyr" charset="204" size="12">помещена функция f(x), а не f1(x), так как последняя не позволяла вести далее сим- вольные вычисления. </f> <f family="Arial Cyr" charset="204" size="12">Рассмотрим функцию Хевисайда </f> <f family="Arial Cyr" charset="204" size="12"> Ф </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">) и то как с помощью этой функции можно задать<sp count="2"/>функцию f(x) на примере данной задачи (см. рис.2):</f> </text> </region> <region region-id="112" left="66" top="1284" width="177.75" height="16.5" align-x="97.5" align-y="1296" 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:function> <ml:id xml:space="preserve">H</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Φ</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Φ</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="151" left="30" top="1336.5" width="470.25" height="33.75" align-x="30" align-y="1350" 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">В условиях задачи </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">№12.484 функция задана как f</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">)=x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> 2 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">, а на рис.2 функцию f</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">) имитируют</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">функция<sp count="2"/>H(x) на отрезке</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <region region-id="153" left="240.75" top="1353.75" width="37.5" height="16.5" align-x="258.75" align-y="1365.75" 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:sequence> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:sequence> </ml:parens> </math> <rendering item-idref="29"/> </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="50" left="48" top="1416" width="336.75" height="167.25" align-x="48" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="30"/> <rendering item-idref="31"/> </region> <region region-id="173" left="42" top="1607.25" width="441" height="44.25" 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">Итак, разложим теперь функцию H(x) в ряд Фурье, с полуперидом </f> <region region-id="157" left="417" top="1608" width="30" height="16.5" align-x="431.25" align-y="1620" 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 style="equal"/> <ml:id xml:space="preserve">L</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </math> <rendering item-idref="32"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Тогда</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">будем иметь </inlineAttr> </f> </text> </region> <region region-id="233" left="60" top="1650" width="185.25" height="56.25" align-x="96.75" align-y="1680" 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">Ha</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">H</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="244" left="348" top="1660.5" width="74.25" height="33.75" align-x="385.5" align-y="1680" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Ha</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="34"/> </region> <region region-id="234" left="48" top="1720.5" width="288.75" height="45" align-x="84.75" align-y="1746" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Ha</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="35"/> </region> <region region-id="237" left="48" top="1770.75" width="243.75" height="13.5" align-x="48" align-y="1782" 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">Последнее выражение можно упростить так:</f> </text> </region> <region region-id="238" left="54" top="1809" width="231" height="63.75" align-x="230.25" align-y="1842" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Ha</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:command> <ml:sequence> <ml:id xml:space="preserve">substitute</ml:id> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:command> <ml:command> <ml:sequence> <ml:id xml:space="preserve">substitute</ml:id> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:sequence> </ml:command> <ml:command> <ml:id xml:space="preserve">simplify</ml:id> </ml:command> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="36"/> </region> <region region-id="239" left="360" top="1822.5" width="91.5" height="39" align-x="396.75" align-y="1842" 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:function> <ml:id xml:space="preserve">Ha</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="245" left="48" top="1890.75" width="426.75" height="27" align-x="48" align-y="1902" 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">Запишем сумму ряда Фурье для функции H(x) или<sp count="2"/>f(x) в окончательном виде и </f> <f family="Arial Cyr" charset="204" size="12">снимем видеоклип на рис. 3</f> </text> </region> <region region-id="242" left="54" top="1947.75" width="204.75" height="50.25" align-x="89.25" align-y="1974" 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">Sh</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="246" left="36" top="2028.75" width="473.25" height="27" align-x="36" align-y="2040" 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">Построенный график функции H(x)<sp count="2"/>и приближающий её ряд<sp count="2"/>Sh(x) рис.3 </f> <f family="Arial Cyr" charset="204" size="12">в режиме анимации</f> <f family="Arial Cyr" charset="204" size="12"> дают оценку точности.</f> </text> </region> <region region-id="172" left="48" top="2112" width="338.25" height="191.25" align-x="48" align-y="2112" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="39"/> <rendering item-idref="40"/> </region> <region region-id="187" left="36" top="2340.75" width="475.5" height="27" align-x="36" align-y="2352" 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">Полученный ряд Фурье является неполным, так как один из коэффициентов ряда b(n) равен нулю, что характерно для четных функций. </f> </text> </region> </regions> <binaryContent> <item item-id="1" 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