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<h3 class="hd hd-2">Numerical methods</h3>
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<h2 class="hd hd-2 unit-title">2. Numerical methods</h2>
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<p><h3>Objectives</h3></p><ul class="itemize"><li><p>
Use <span style="color:#27408C"><b class="bf">Riemann sums</b></span>, the <span style="color:#27408C"><b class="bf">Midpoint Rule</b></span>, the <span style="color:#27408C"><b class="bf">Trapezoidal Rule</b></span>, and <span style="color:#27408C"><b class="bf">Simpson's Rule</b></span> to approximate integrals. </p></li><li><p>
Recognize <span style="color:#27408C"><b class="bf">numerical methods</b></span> as weighted averages. </p></li><li><p>
Derive Simpson's Rule. </p></li><li><p>
Use <span style="color:#27408C"><b class="bf">error bounds</b></span> of these methods to understand the accuracy of your approximations. </p></li></ul><p><h3>Contents: 11 pages</h3></p><p>
9 videos (34 minutes at 1x speed) 12 problems </p>
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<h2 class="hd hd-2 unit-title">3. Riemann sums</h2>
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<h3 class="hd hd-2">Introduction to numerical methods</h3>
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<p>
To approximate the integral [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline]: </p><ul class="itemize"><li><p>
Subdivide the interval into [mathjaxinline]n[/mathjaxinline] pieces [mathjaxinline]a = x_0 < x_1 < \dotsb < x_{n-1} < x_ n = b[/mathjaxinline]. </p></li><li><p>
Let [mathjaxinline]y_0 = f(x_0)[/mathjaxinline], [mathjaxinline]y_1 = f(x_1)[/mathjaxinline], ... , [mathjaxinline]y_{n-1}= f(x_{n-1})[/mathjaxinline], and [mathjaxinline]y_ n = f(x_ n)[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\Delta x = \displaystyle \frac{b-a}{n}[/mathjaxinline]. </p></li><li><p><b class="bf">Left Riemann sum</b> [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx \approx \Delta x\left( y_0 + y_1 + \dotsb + y_{n-1}\right)[/mathjaxinline] </p></li><li><p><b class="bf">Right Riemann sum</b> [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx \approx \Delta x\left( y_1 + y_2 + \dotsb + y_ n\right)[/mathjaxinline] </p></li></ul>
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Use the left Riemann sum to approximate [mathjaxinline]\displaystyle \int _0^1 x\, dx[/mathjaxinline] with [mathjaxinline]n=3[/mathjaxinline]. </p>
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Use the right Riemann sum to approximate [mathjaxinline]\displaystyle \int _0^1 x\, dx[/mathjaxinline] with [mathjaxinline]n=3[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">4. Riemann sums: Midpoint Rule</h2>
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Review: Underestimates and overestimates
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Consider a function [mathjaxinline]f(x)&gt;0[/mathjaxinline] on the interval [mathjaxinline]a &lt; x &lt; b[/mathjaxinline]. </p>
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If the function is everywhere decreasing on this interval, then the approximation from the <b class="bf">Left</b> Riemann sum is an </p>
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<text> overestimate.</text>
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If the function is everywhere decreasing on this interval, then the approximation from the <b class="bf">Right</b> Riemann sum is an </p>
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<p><b class="bfseries">Riemann sums: Midpoint Rule</b></p><p>
To approximate the integral [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline], instead of taking the left or right endpoints, we can instead take the midpoint! </p><ul class="itemize"><li><p>
Subdivide the interval into [mathjaxinline]n[/mathjaxinline] pieces [mathjaxinline]a = x_0 < x_1 < \dotsb < x_{n-1} < x_ n = b[/mathjaxinline]. </p></li><li><p>
Find the value of the function on the midpoint of each interval </p><p>
[mathjaxinline]\displaystyle \tilde{y}_1 = f\left(\frac{x_0+x_1}{2}\right)[/mathjaxinline], [mathjaxinline]\displaystyle \tilde{y}_2 = f\left(\frac{x_1+x_2}{2}\right)[/mathjaxinline], ... , [mathjaxinline]\displaystyle \tilde{y}_{n}= f\left(\frac{x_{n-1}+x_ n}{2}\right)[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\Delta x = \displaystyle \frac{b-a}{n}[/mathjaxinline]. </p></li><li><p><b class="bf">Midpoint Rule</b> [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx \approx \Delta x\left( \tilde{y}_1 + \tilde{y}_2 + \dotsb + \tilde{y}_{n}\right)[/mathjaxinline] </p></li></ul>
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<h2 class="hd hd-2 unit-title">5. Trapezoidal Rule</h2>
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<p><b class="bfseries">Trapezoidal Rule</b></p><p>
To approximate the integral [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline]: </p><ul class="itemize"><li><p>
Subdivide the interval into [mathjaxinline]n[/mathjaxinline] pieces [mathjaxinline]a = x_0 < x_1 < \dotsb < x_{n-1} < x_ n = b[/mathjaxinline]. </p></li><li><p>
Let [mathjaxinline]y_0 = f(x_0)[/mathjaxinline], [mathjaxinline]y_1 = f(x_1)[/mathjaxinline], ... , [mathjaxinline]y_{n-1}= f(x_{n-1})[/mathjaxinline], and [mathjaxinline]y_ n = f(x_ n)[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\Delta x = \displaystyle \frac{b-a}{n}[/mathjaxinline]. </p></li><li><p><b class="bf">Trapezoidal Rule</b> [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx \approx \Delta x\left( \frac12 y_0 + y_1 + \dotsb + y_{n-1} + \frac12 y_ n \right).[/mathjaxinline] </p></li></ul>
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Use the Trapezoidal Rule to approximate [mathjaxinline]\displaystyle \int _0^{\pi } \sin (x) \, dx[/mathjaxinline] with [mathjaxinline]n=2[/mathjaxinline]. </p>
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Use the Trapezoidal Rule to approximate [mathjaxinline]\displaystyle \int _0^{\pi } \sin (x) \, dx[/mathjaxinline] with [mathjaxinline]n=4[/mathjaxinline]. </p>
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Consider a function [mathjaxinline]f(x)&gt;0[/mathjaxinline] on the interval [mathjaxinline]a &lt; x &lt; b[/mathjaxinline]. If the function is everywhere concave up on this interval, then the approximation from the Trapezoidal Rule is an </p>
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Under and over estimates
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Consider a function [mathjaxinline]f(x)&gt;0[/mathjaxinline] on the interval [mathjaxinline]a &lt; x &lt; b[/mathjaxinline]. If the function is linear on this interval, then the approximation from the Trapezoidal Rule is an </p>
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<h2 class="hd hd-2 unit-title">6. Simpson's Rule</h2>
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<p>
To approximate the integral [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline]: </p><ul class="itemize"><li><p>
Subdivide the interval into [mathjaxinline]n[/mathjaxinline] pieces ([mathjaxinline]n[/mathjaxinline] must be even) [mathjaxinline]a = x_0 < x_1 < \dotsb < x_{n-1} < x_ n = b[/mathjaxinline]. </p></li><li><p>
Let [mathjaxinline]y_0 = f(x_0)[/mathjaxinline], [mathjaxinline]y_1 = f(x_1)[/mathjaxinline], ... , [mathjaxinline]y_{n-1}= f(x_{n-1})[/mathjaxinline], and [mathjaxinline]y_ n = f(x_ n)[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\Delta x = \displaystyle \frac{b-a}{n}[/mathjaxinline]. </p></li><li><p><b class="bf">Simpson's Rule</b> [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx \approx \frac{\Delta x}{3} \left( y_0 + 4y_1 + 2y_2 + 4 y_3 + \dotsb +2y_{n-2} + 4y_{n-1} + y_ n \right)[/mathjaxinline] </p></li></ul>
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Practice Simpson&#39;s rule
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Use Simpson's rule to approximate [mathjaxinline]\displaystyle \int _0^{\pi } \sin (x)\, dx[/mathjaxinline] with [mathjaxinline]n=2[/mathjaxinline]. </p>
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Use Simpson's rule to approximate [mathjaxinline]\displaystyle \int _0^{\pi } \sin (x) \, dx[/mathjaxinline] with [mathjaxinline]n=4[/mathjaxinline]. </p>
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<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">7. Compare the rules</h2>
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Compare the rules
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<p>
Use the Trapezoidal Rule to approximate [mathjaxinline]\displaystyle \int _1^2 \frac{1}{x} \, dx[/mathjaxinline] &#8201; with [mathjaxinline]n=2[/mathjaxinline]. </p>
<p>
(Enter as decimal to 3 decimal places.) </p>
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Use Simpson's rule to approximate [mathjaxinline]\displaystyle \int _1^2 \frac{1}{x} \, dx[/mathjaxinline] with [mathjaxinline]n=2[/mathjaxinline]. </p>
<p>
(Enter as decimal to 3 decimal places.) </p>
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<p>
Compute the exact integral [mathjaxinline]\displaystyle \int _1^2 \frac{1}{x} \, dx[/mathjaxinline]. </p>
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(Enter as decimal to 3 decimal places.) </p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<td class="formulainput">Fractions</td>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h3 class="hd hd-2">Worked example: using the Trapezoid and Simpson's Rules</h3>
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<h2 class="hd hd-2 unit-title">8. Finding Simpson's Rule</h2>
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<h3 class="hd hd-2">Simpson's Rule setup</h3>
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Finding Simpson&#39;s rule
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<center>
<img alt="A function f of x is shown on the interval negative h to h. The values y sub 0, y sub 1, and y sub 2 correspond to the heights of the function at x equals negative h, x equals 0, and x equals h, respectively. The parabola intersecting these points is given by A x squared plus B x plus C." src="/assets/courseware/v1/226defeb814b732a123917cd9ff39193/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_numerical-simpsonsetup.svg" style="margin: 10px 25px 25px 25px" width="300px"/>
</center>
<p>
Start by computing the signed area bounded by the parabola [mathjaxinline]Ax^2 + Bx +C[/mathjaxinline] over the interval [mathjaxinline]-h \leq x \leq h[/mathjaxinline]. </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000001597" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{-h}^ h \left(Ax^2 + Bx +C\right) \, dx[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
(Express your answer in terms of [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], and [mathjaxinline]h[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \int _{-h}^ h (Ax^2 + Bx +C)\, dx =[/mathjaxinline]</p>
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\(\)
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<p>
Determine the values of [mathjaxinline]C[/mathjaxinline] and [mathjaxinline]2Ah^2[/mathjaxinline] if the parabola passes through the points [mathjaxinline](-h, y_0)[/mathjaxinline], [mathjaxinline](0, y_1)[/mathjaxinline], and [mathjaxinline](h, y_2)[/mathjaxinline]. </p>
<p>
(Express your answer in terms of [mathjaxinline]y_0[/mathjaxinline], [mathjaxinline]y_1[/mathjaxinline], [mathjaxinline]y_2[/mathjaxinline], and [mathjaxinline]h[/mathjaxinline].) </p>
<p>
(Type y_0 for [mathjaxinline]y_0[/mathjaxinline], etc.) </p>
<p>
<p style="display:inline">[mathjaxinline]C=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]2Ah^2=[/mathjaxinline]</p>
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<p>
Put it all together to find the formula for the signed area bounded by the parabola in Simpson's rule. </p>
<p>
(Express your answer in terms of [mathjaxinline]y_0[/mathjaxinline], [mathjaxinline]y_1[/mathjaxinline], [mathjaxinline]y_2[/mathjaxinline], and [mathjaxinline]h[/mathjaxinline].) </p>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
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<h2 class="hd hd-2 unit-title">9. Error bounds</h2>
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Estimates and error bounds
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Using the mathlet below, compare the Riemann Sum evaluated at the midpoint (the Midpoint Rule), the Trapezoidal Rule, and Simpson's Rule for the function [mathjaxinline]\ f(x) = e^{x/2}[/mathjaxinline] on the interval [mathjaxinline][0, 1][/mathjaxinline]. </p>
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(Hint, use the geometric picture as well as the graph to the left which shows the result of the approximation. Use the slider to vary the size of [mathjaxinline]n[/mathjaxinline] and observe how the approximations behave as [mathjaxinline]n[/mathjaxinline] gets large. Choose undetermined if you cannot tell if it is an underestimate or an overestimate from the picture.) </p>
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The Midpoint Rule is: </td>
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The Trapezoidal Rule is: </td>
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Simpson's Rule is:</td>
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<text> an underestimate</text>
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<p><b class="bfseries">Error bounds</b></p><p><h3>Error bound of Trapezoidal Rule</h3> Suppose [mathjaxinline]\displaystyle \left| f^{\prime \prime }(x) \right| \leq M[/mathjaxinline] for [mathjaxinline]a \leq x \leq b[/mathjaxinline]. Then the error of the Trapezoidal Rule over [mathjaxinline]n[/mathjaxinline] subintervals is given by </p><table id="a0000001603" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \left| E_ T \right| \leq \frac{M(b-a)^3}{12n^2}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><h3>Error bound of Midpoint Rule</h3> Suppose [mathjaxinline]\displaystyle \left| f^{\prime \prime }(x) \right| \leq M[/mathjaxinline] for [mathjaxinline]a \leq x \leq b[/mathjaxinline]. Then the error of the Midpoint Rule over [mathjaxinline]n[/mathjaxinline] subintervals is given by </p><table id="a0000001604" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \left| E_ M \right| \leq \frac{M(b-a)^3}{24n^2}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><h3>Error bound of Simpson's Rule</h3> Suppose [mathjaxinline]\displaystyle \left| f^{(4)}(x) \right| \leq M[/mathjaxinline] for [mathjaxinline]a \leq x \leq b[/mathjaxinline]. Then the error of the Simpson's Rule over [mathjaxinline]n[/mathjaxinline] subintervals is given by </p><table id="a0000001605" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \left| E_ S \right| \leq \frac{M(b-a)^5}{180n^4}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Accuracy
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For this problem, consider [mathjaxinline]\displaystyle \int _1^2 \frac{1}{x} \, dx[/mathjaxinline]. </p>
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What is the minimum number of subintervals [mathjaxinline]n[/mathjaxinline] we can divide the interval [mathjaxinline]1 \leq x \leq 2[/mathjaxinline] into to guarantee that the Midpoint Rule is accurate to within [mathjaxinline]10^{-6}[/mathjaxinline]? </p>
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(Enter a whole number.) </p>
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<p style="display:inline">[mathjaxinline]n=[/mathjaxinline]</p>
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What is the minimum number of subintervals [mathjaxinline]n[/mathjaxinline] we can divide the interval [mathjaxinline]1 \leq x \leq 2[/mathjaxinline] into to guarantee that the Trapezoidal Rule is accurate to within [mathjaxinline]10^{-6}[/mathjaxinline]? </p>
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(Enter a whole number.) </p>
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<p style="display:inline">[mathjaxinline]n=[/mathjaxinline]</p>
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What is the minimum number of subintervals [mathjaxinline]n[/mathjaxinline] we can divide the interval [mathjaxinline]1 \leq x \leq 2[/mathjaxinline] into to guarantee that the Simpson's Rule is accurate to within [mathjaxinline]10^{-6}[/mathjaxinline]? </p>
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(Enter a whole number.) </p>
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<p style="display:inline">[mathjaxinline]n=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">10. Practice with probability</h2>
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Practice with probability
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In the United States, the average height of a woman is 64 inches with a standard deviation of 3 inches. Thus the probability density function is given by </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle f(x) = \frac{1}{3 \sqrt {2\pi }} e^{-\frac{(x-64)^2}{18}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(3.77)</td>
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Use Simpson's rule to estimate the probability that a woman is between 64 and 68 inches tall with [mathjaxinline]n=4[/mathjaxinline]. </p>
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(Enter answer as a decimal accurate to 2 decimal places.) </p>
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<h2 class="hd hd-2 unit-title">11. Summary</h2>
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<p><b class="bfseries">Riemann sums: left and right</b></p><p>
To approximate the integral [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline]: </p><ul class="itemize"><li><p>
Subdivide the interval into [mathjaxinline]n[/mathjaxinline] pieces [mathjaxinline]a = x_0 < x_1 < \dotsb < x_{n-1} < x_ n = b[/mathjaxinline]. </p></li><li><p>
Let [mathjaxinline]y_0 = f(x_0)[/mathjaxinline], [mathjaxinline]y_1 = f(x_1)[/mathjaxinline], ... , [mathjaxinline]y_{n-1}= f(x_{n-1})[/mathjaxinline], and [mathjaxinline]y_ n = f(x_ n)[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\Delta x = \displaystyle \frac{b-a}{n}[/mathjaxinline]. </p></li><li><p><b class="bf">Left Riemann sum</b> [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx \approx \Delta x\left( y_0 + y_1 + \dotsb + y_{n-1}\right)[/mathjaxinline] </p></li><li><p><b class="bf">Right Riemann sum</b> [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx \approx \Delta x\left( y_1 + y_2 + \dotsb + y_ n\right)[/mathjaxinline] </p></li></ul><p><b class="bfseries">Riemann sums: Midpoint Rule</b></p><p>
To approximate the integral [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline]: </p><ul class="itemize"><li><p>
Subdivide the interval into [mathjaxinline]n[/mathjaxinline] pieces [mathjaxinline]a = x_0 < x_1 < \dotsb < x_{n-1} < x_ n = b[/mathjaxinline]. </p></li><li><p>
Find the value of the function on the midpoint of each interval </p><p>
[mathjaxinline]\displaystyle \tilde{y}_1 = f\left(\frac{x_0+x_1}{2}\right)[/mathjaxinline], [mathjaxinline]\displaystyle \tilde{y}_2 = f\left(\frac{x_1+x_2}{2}\right)[/mathjaxinline], ... , [mathjaxinline]\displaystyle \tilde{y}_{n}= f\left(\frac{x_{n-1}+x_ n}{2}\right)[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\Delta x = \displaystyle \frac{b-a}{n}[/mathjaxinline]. </p></li><li><p><b class="bf">Midpoint Rule</b> [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx \approx \Delta x\left( \tilde{y}_1 + \tilde{y}_2 + \dotsb + \tilde{y}_{n}\right)[/mathjaxinline] </p></li></ul><p><b class="bfseries">Trapezoidal Rule</b></p><p>
To approximate the integral [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline]: </p><ul class="itemize"><li><p>
Subdivide the interval into [mathjaxinline]n[/mathjaxinline] pieces [mathjaxinline]a = x_0 < x_1 < \dotsb < x_{n-1} < x_ n = b[/mathjaxinline]. </p></li><li><p>
Let [mathjaxinline]y_0 = f(x_0)[/mathjaxinline], [mathjaxinline]y_1 = f(x_1)[/mathjaxinline], ... , [mathjaxinline]y_{n-1}= f(x_{n-1})[/mathjaxinline], and [mathjaxinline]y_ n = f(x_ n)[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\Delta x = \displaystyle \frac{b-a}{n}[/mathjaxinline]. </p></li><li><p><b class="bf">Trapezoidal Rule</b> [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx \approx \Delta x\left( \frac12 y_0 + y_1 + \dotsb + y_{n-1} + \frac12 y_ n \right).[/mathjaxinline] </p></li></ul><p><b class="bfseries">Simpson's Rule</b></p><p>
To approximate the integral [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline]: </p><ul class="itemize"><li><p>
Subdivide the interval into [mathjaxinline]n[/mathjaxinline] pieces ([mathjaxinline]n[/mathjaxinline] must be even) [mathjaxinline]a = x_0 < x_1 < \dotsb < x_{n-1} < x_ n = b[/mathjaxinline]. </p></li><li><p>
Let [mathjaxinline]y_0 = f(x_0)[/mathjaxinline], [mathjaxinline]y_1 = f(x_1)[/mathjaxinline], ... , [mathjaxinline]y_{n-1}= f(x_{n-1})[/mathjaxinline], and [mathjaxinline]y_ n = f(x_ n)[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\Delta x = \displaystyle \frac{b-a}{n}[/mathjaxinline]. </p></li><li><p><b class="bf">Simpson's Rule</b> [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx \approx \frac{\Delta x}{3} \left( y_0 + 4y_1 + 2y_2 + 4 y_3 + \dotsb +2y_{n-2} + 4y_{n-1} + y_ n \right)[/mathjaxinline] </p></li></ul><p><b class="bfseries">Error bounds</b></p><p><h3>Error bound of Trapezoidal Rule</h3> Suppose [mathjaxinline]\displaystyle \left| f^{\prime \prime }(x) \right| \leq M[/mathjaxinline] for [mathjaxinline]a \leq x \leq b[/mathjaxinline]. Then the error of the Trapezoidal Rule over [mathjaxinline]n[/mathjaxinline] subintervals is given by </p><table id="a0000001621" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \left| E_ T \right| \leq \frac{M(b-a)^3}{12n^2}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><h3>Error bound of Midpoint Rule</h3> Suppose [mathjaxinline]\displaystyle \left| f^{\prime \prime }(x) \right| \leq M[/mathjaxinline] for [mathjaxinline]a \leq x \leq b[/mathjaxinline]. Then the error of the Midpoint Rule over [mathjaxinline]n[/mathjaxinline] subintervals is given by </p><table id="a0000001622" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \left| E_ M \right| \leq \frac{M(b-a)^3}{24n^2}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><h3>Error bound of Simpson's Rule</h3> Suppose [mathjaxinline]\displaystyle \left| f^{(4)}(x) \right| \leq M[/mathjaxinline] for [mathjaxinline]a \leq x \leq b[/mathjaxinline]. Then the error of the Simpson's Rule over [mathjaxinline]n[/mathjaxinline] subintervals is given by </p><table id="a0000001623" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \left| E_ S \right| \leq \frac{M(b-a)^5}{180n^4}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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