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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h2 class="hd hd-2 unit-title">2. Linear Approximation and Measurement Error</h2>
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<p><span style="color:#99182C"><b class="bf">products</b></span><br/><span style="color:#99182C"><b class="bf">compositions</b></span><br/><span style="color:#99182C"><b class="bf">error</b></span><br/></p><p><b class="bfseries">Objectives</b></p><p>
At the end of this sequence, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Find <span style="color:#27408C"><b class="bf">linear approximations</b></span> of any function. </p></li><li><p>
Recognize linearizations of basic functions at 0. </p></li><li><p>
Find linear approximations of <span style="color:#27408C"><b class="bf">compositions and products</b></span> using basic approximations. </p></li><li><p>
Use the linear approximation to <span style="color:#27408C"><b class="bf">estimate error</b></span>. </p></li></ul><p><b class="bfseries">Contents: 14 pages</b></p><p>
8 videos (32 minutes 1x speed) 23 questions </p>
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<h2 class="hd hd-2 unit-title">3. A problem with a zipline</h2>
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<p><b class="bfseries">Linear approximations revisited</b></p><p>
You already know how to take linear approximations. This was the main tool for finding the derivatives of all of the basic functions in the previous unit. </p><p>
We return again to linear approximations because they are the main tool that we use in finding approximations in science and engineering. Here we develop some more tools for using linear approximations. We will also change our perspective and use linear approximations to estimate how measurement error propogates through a system, and understand the system's sensitivity to small perturbations. </p>
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<h3 class="hd hd-2">Zipline design problem setup</h3>
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Find the maximum velocity
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Find an expression for the maximum velocity of a rider on the zipline in terms of the total system energy [mathjaxinline]E[/mathjaxinline] and the rider mass [mathjaxinline]m[/mathjaxinline]. </p>
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(Enter expression in terms of E and m. Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
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Recall that the linear approximation of a function [mathjaxinline]\ f(x)[/mathjaxinline] for [mathjaxinline]x[/mathjaxinline] near [mathjaxinline]x_0[/mathjaxinline] is given by the formula [mathjaxinline]\ f(x) \approx f(x_0) + f'(x_0)(x-x_0)[/mathjaxinline]. </p>
<p>
Find the linear approximation of the function [mathjaxinline](1+u)^ r[/mathjaxinline] near [mathjaxinline]u=0[/mathjaxinline]. </p>
<p>
(Note that this answer will involve both [mathjaxinline]u[/mathjaxinline] and [mathjaxinline]r[/mathjaxinline]. Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline](1+u)^ r \approx[/mathjaxinline]</p>
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Let's make another simplifying assumption. Suppose that rider starts completely at rest. This means that initially, the kinetic energy is zero, so the total energy is given by [mathjaxinline]E = mgh[/mathjaxinline]. </p>
<p>
With this added assumption, suppose we change the height displaced during the ride by [mathjaxinline]2\%[/mathjaxinline], i.e. [mathjaxinline]\displaystyle \frac{\Delta h}{h} = .02[/mathjaxinline]. What percentage did [mathjaxinline]v[/mathjaxinline] change by? </p>
<p>
(Find an approximation for [mathjaxinline]\displaystyle \frac{\Delta v}{v}[/mathjaxinline]. Enter your answer as a decimal with 2 decimal digits rather than as a percentage.) </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{\Delta v}{v}=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">5. Linear approximations near zero</h2>
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Recall that the linear approximation of a function [mathjaxinline]\ f(x)[/mathjaxinline] for [mathjaxinline]x[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline] is given by the formula [mathjaxinline]\ f(x) \approx f(0) + f'(0)x[/mathjaxinline]. Find linear approximations of all of the basic functions near [mathjaxinline]x=0[/mathjaxinline]. We are going to be using these formulas a lot! So you may even want to memorize them. </p><p>
We call the linear approximation the <span style="color:#27408C"><b class="bf">linearization</b></span> of a function. </p>
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Linear approximations of basic functions
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Find the linear approximation of [mathjaxinline]\ \sin (x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab5-problem1_2_1" class="inputtype formulaequationinput">
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Find the linear approximation of [mathjaxinline]\ \cos (x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_approx_1-tab5-problem1_3_1" class="inputtype formulaequationinput">
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Find the linear approximation of [mathjaxinline]\ e^ x[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_approx_1-tab5-problem1_4_1" class="inputtype formulaequationinput">
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Find the linear approximation of [mathjaxinline]\ \ln (1+x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div id="formulaequationinput_approx_1-tab5-problem1_5_1" class="inputtype formulaequationinput">
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<h2 class="hd hd-2 unit-title">6. Combining approximations</h2>
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<p><b class="bfseries">Approximations (likely nonlinear) of compositions</b></p><p>
Suppose that [mathjaxinline]g(x)[/mathjaxinline] is a function such that [mathjaxinline]g(0) =0[/mathjaxinline]. To find an approximation of a function [mathjaxinline]f(g(x))[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline], we can take a linear approximation for [mathjaxinline]f(u)[/mathjaxinline] and then substitute [mathjaxinline]g(x)[/mathjaxinline] in for [mathjaxinline]u[/mathjaxinline]. The resulting approximation is likely nonlinear, but it is still an approximation! <b class="bfseries"><span style="color:#FF7F00">Warning:</span></b> this only works if [mathjaxinline]g(0) = 0[/mathjaxinline]. </p><table id="a0000000470" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000471"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f(u) \approx f(0) + f'(0)u \qquad \Longrightarrow \qquad f\left(g(x)\right) \approx f(g(0)) + f'(g(0))g(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.6)</td></tr></table>
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Exponential
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What is the linear approximation of [mathjaxinline]e^{-x}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline].? </p>
<p>
(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </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>
</tr>
<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>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<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>
<tr class="formulainput">
<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>
</tr>
<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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Powers
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What is the linear approximation to [mathjaxinline](1+3x)^{-1/2}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline].? </p>
<p>
(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<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>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<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>
<tr class="formulainput">
<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>
</tr>
<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 \( (2\pi+y)dy \)
</td>
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Approximation and compositions
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Use a linear approximation for [mathjaxinline]e^ u[/mathjaxinline] to find a (nonlinear in [mathjaxinline]x[/mathjaxinline] but still polynomial) approximation to the function [mathjaxinline]e^{x^3}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. </p>
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(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline]e^{x^3}\approx[/mathjaxinline]</p>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<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>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<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>
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<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 \( 2\pi \)
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<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 \( \sqrt{x^2-y} \)
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<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>
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<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>
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<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>
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<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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Approximation and compositions
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Combine linear approximations for the natural logarithm and square root functions to find a (nonlinear in [mathjaxinline]x[/mathjaxinline] but still polynomial) approximation of the function [mathjaxinline]\ln {\sqrt {1+x^2}}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. </p>
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(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline]\ln {\sqrt {1+x^2}} \approx[/mathjaxinline]</p>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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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>
<tr class="formulainput">
<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>
</tr>
<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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That last video mentioned quadratic approximations. We haven't quite gotten there yet, but that is coming next! Don't worry, the students in this class didn't know about quadratic approximations yet either. </p>
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<h2 class="hd hd-2 unit-title">8. Products of functions</h2>
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Find the linear approximation of the function [mathjaxinline]\ \displaystyle \frac{e^{-3x}}{\sqrt {1+x}}[/mathjaxinline] at [mathjaxinline]x=0[/mathjaxinline]. </p>
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(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
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<p>
To find the linear approximation of a function [mathjaxinline]h(x) = f(x)g(x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline], it suffices to find a linear approximation for [mathjaxinline]f(x)[/mathjaxinline], find a linear approximation for [mathjaxinline]g(x)[/mathjaxinline], and then the linear approximation for [mathjaxinline]h(x)[/mathjaxinline] is the product of these two approximations where we cancel all of the terms that are quadratic (or higher for more products). </p><table id="a0000000475" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000476"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle h(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(x)g(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.7)</td></tr><tr id="a0000000477"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0) + f'(0)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.8)</td></tr><tr id="a0000000478"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle g(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle g(0) + g'(0)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.9)</td></tr><tr id="a0000000479"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle h(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(f(0) + f'(0)x\right)\left(g(0) + g'(0)x\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.10)</td></tr><tr id="a0000000480"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0)g(0) + \left(f'(0)g(0) +f(0)g'(0)\right)x + f'(0)g'(0)x^2[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.11)</td></tr><tr id="a0000000481"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0)g(0) + \left(f'(0)g(0) +f(0)g'(0)\right)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.12)</td></tr></table>
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<p><b class="bfseries">Estimate of the height</b></p><p>
Here is a diagram of the measurements we just carried out. </p><center><img src="/assets/courseware/v1/4fd3055c881f14b6ee4b57d09c758fb4/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_greenbld_diagram.svg" width="350px" alt="See table below." style="margin: 10px 25px 25px 25px"/></center><p>
The height [mathjaxinline]y[/mathjaxinline] of the Green building is given by </p><table id="a0000000489" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]y = h+x\tan (\theta ).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Below is a table of the measurements made, as well as the error in each term. </p><table style="table-layout:auto"><tr><th> Quantity </th><th> Variable </th><th> Measurement </th><th> Error </th></tr><tr><td style="border:none"> Height to Jen's eyeball </td><td style="border:none"> h </td><td style="border:none"> 4.9 feet </td><td style="border:none"> [mathjaxinline]\pm [/mathjaxinline] .01 feet </td></tr><tr><td style="border:none"> Distance to Green Building </td><td style="border:none"> x</td><td style="border:none"> 175 feet </td><td style="border:none"> [mathjaxinline]\pm [/mathjaxinline] 3 feet </td></tr><tr><td style="border:none"> Angle </td><td style="border:none"> [mathjaxinline]\theta [/mathjaxinline]</td><td style="border:none"> 57 degrees </td><td style="border:none"> [mathjaxinline]\pm [/mathjaxinline] 3 degrees </td></tr></table><p>
Where is the error coming from? </p><ul class="itemize"><li><p>
The error in the measurement of Jen's height comes from rounding error and inherent error in using a tape measure to enough accuracy. </p></li><li><p>
The error in the distance to the Green building is coming from a few places: error in the actual length of the rope which may differ by a few inches, error in our walking in a perfectly straight line as we measure, and error coming from the fact that the rope is stretching when we pull on it, lengthening the rope. So in fact, we expect our measurement is likely an overestimate. </p></li><li><p>
The error in the angle is coming from wind and the string not hanging straight down, error in the construction of the device, and error in holding the device to read the measurement. </p></li></ul>
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Height estimate
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What is the estimated height of the Green Building? </p>
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(Enter an integer with 3 significant figures.) </p>
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<p><b class="bfseries">Error in estimate</b></p><p>
What is [mathjaxinline]\Delta y[/mathjaxinline], which is an error term representing the difference between the height that we calculated for the building from our measurements and its true height? </p><p>
First note that what we really have is the following. </p><center><img src="/assets/courseware/v1/d37cb727913f5ab0ae7c0a4fb3dc4b91/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_greenbld_diagram2.svg" width="355px" alt="Each variable includes a small error." style="margin: 10px 25px 25px 25px"/></center><p>
And </p><table id="a0000000490" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000491"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \tan (\theta +\Delta \theta )[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac{(y+\Delta y)-(h+\Delta h)}{x+\Delta x}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.15)</td></tr><tr id="a0000000492"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac{\Delta y -\Delta h + (y-h)}{x+\Delta x}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.16)</td></tr><tr id="a0000000493"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac{\Delta y -\Delta h + x \tan \theta }{x+\Delta x}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.17)</td></tr></table><p>
The last line uses the fact that [mathjaxinline]\displaystyle \tan (\theta ) = \frac{y-h}{x}[/mathjaxinline]. </p><p>
To determine the error, we solve for [mathjaxinline]\Delta y[/mathjaxinline]: </p><table id="a0000000494" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\Delta y = \Delta h + (x+\Delta x)\left(\tan (\theta + \Delta \theta )\right) - x\tan (\theta ).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
To get a sense of the order of this error term, let's find a linear approximation of [mathjaxinline]\tan (\theta + \Delta \theta )[/mathjaxinline] near [mathjaxinline]\theta = 57[/mathjaxinline] degrees. </p>
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<h2 class="hd hd-2 unit-title">10. Continuing the error estimate</h2>
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To estimate the error term, we find: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \Delta y = \Delta h + (x+\Delta x)\left(\tan (\theta + \Delta \theta )\right) - x\tan (\theta )[/mathjaxinline] </td><td style="text-align:left; border:none">
This is the term from the previous page.</td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \quad \approx \Delta h + (x+\Delta x)\left(\tan (\theta ) + \sec ^2(\theta )\Delta \theta \right)-x\tan (\theta )[/mathjaxinline] </td><td style="text-align:left; border:none">
Here we use the linear approximation for [mathjaxinline]\tan (\theta + \Delta \theta )[/mathjaxinline]. </td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\quad \displaystyle = \Delta h + \Delta x\tan (\theta ) + x\sec ^2(\theta )\Delta \theta + \Delta x \Delta \theta \sec ^2(\theta )[/mathjaxinline] </td><td style="text-align:left; border:none">
Note this is equal to the previous line. We multiplied out the middle term in the previous line, and canceled the [mathjaxinline]x\tan (\theta )[/mathjaxinline] terms.</td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \quad \approx \Delta h + \Delta x\tan (\theta ) + x\sec ^2(\theta )\Delta \theta .[/mathjaxinline] </td><td style="text-align:left; border:none">
This last is an approximate equality. We canceled the quadratic term as it is not linear. </td></tr></table>
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Identify largest error term
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In the error term [mathjaxinline]\Delta y \approx \Delta h + \Delta x\tan (\theta ) + x\sec ^2(\theta )\Delta \theta[/mathjaxinline], which term has the largest contribution to the error? </p>
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The error
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What is the magnitude of the error of our height measurement? </p>
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<span class="trailing_text" id="trailing_text_approx_1-tab10-problem2_2_1">feet</span>
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Compare this to the actual error given that the Green Building is actually 295 feet tall. Did our estimate fall within the error bounds? </p>
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<p>
Build your own altitude measuring equipment and find the heights of objects, including the heights of your friends, and other known quantities. Make sure to answer the following questions and test your hypothesis! </p><ol class="enumerate"><li value="1"><p>
What do you predict to be the measurement error in each measurement that you make? </p></li><li value="2"><p>
What is the error bound for the height? </p></li><li value="3"><p>
Does your height measurement fall within the bounds of your expected error? Why or why not? </p></li></ol>
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<h2 class="hd hd-2 unit-title">12. Error approximation practice</h2>
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Velocity measurement
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Suppose that you measure the velocity of an object by measuring that it takes 1 second to travel 1.2 meters. The measurement error is .001 meters in distance, and the error in time is .01 second. What is the absolute value of the error in the linear approximation for the velocity? </p>
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(Enter decimal answer to three decimal digits. Enter the units of the error using [mathjaxinline]m[/mathjaxinline] to denote meters and [mathjaxinline]s[/mathjaxinline] to denote seconds.) </p>
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<p style="display:inline">[mathjaxinline]\left| \Delta v \right| \leq[/mathjaxinline]</p>
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Kinetic energy
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Suppose you measure the velocity of a mosquito to be [mathjaxinline]1.3[/mathjaxinline] m/s with an error of [mathjaxinline]0.2[/mathjaxinline] m/s. We approximate the mosquito to weigh approximately equal to a grain of rice, which is on the order of magnitude of [mathjaxinline]10^{-6}[/mathjaxinline] kg. What is the error in the kinetic energy? Assume there is no rotational velocity. In the second answer box, enter units involving m, s, and kg to denote meters, seconds, and kilograms respectively. (Do <b class="bf">not</b> use J for Joules or N for Newtons.) </p>
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(Enter answer with 2 significant figures of accuracy, i.e.with 2 nonzero numbers.) </p>
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<p style="display:inline">[mathjaxinline]\left| \Delta K \right| \leq[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">13. Practice</h2>
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Practice problem 2
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Use a linear approximation for [mathjaxinline]\sin[/mathjaxinline] to find an approximation (this will not be linear) to the function [mathjaxinline]\sin (x^3)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\sin (x^3)\approx[/mathjaxinline]</p>
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Implicit differentiation and linear approximation (*)
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Consider the implicit function [mathjaxinline](w(x) + 1)e^{w(x)} = x[/mathjaxinline]. Approximate [mathjaxinline]w(1.1)[/mathjaxinline] using that [mathjaxinline]w(1) =0[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">14. Summary</h2>
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<p><b class="bfseries">Linear approximations revisited</b></p><p>
You already know how to take linear approximations. This was the main tool for finding the derivatives of all of the basic functions in the previous unit. </p><p>
We return again to linear approximations because they are the main tool that we use in finding approximations in science and engineering. Here we develop some more tools for using linear approximations, as well as changing our perspective to use linear approximations to estimate how measurement error perpetuates through a system, and understand the system's sensitivity to small perturbations. </p><p><b class="bfseries">Linear Approximations near [mathjaxinline]x=0[/mathjaxinline]</b></p><p>
Recall that the linear approximation of a function [mathjaxinline]\ f(x)[/mathjaxinline] for [mathjaxinline]x[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline] is given by the formula [mathjaxinline]\ f(x) \approx f(0) + f'(0)x[/mathjaxinline]. Find linear approximations of all of the basic functions near [mathjaxinline]x=0[/mathjaxinline]. We are going to be using these formulas a lot! So you may even want to memorize them. </p><p>
We call the linear approximation the <span style="color:#27408C"><p><b class="bf">linearization</b> of a function. </p><p><b class="bfseries">Approximations (likely nonlinear) of compositions</b></p><p>
Suppose that [mathjaxinline]g(x)[/mathjaxinline] is a function such that [mathjaxinline]g(0) =0[/mathjaxinline]. To find an approximation of a function [mathjaxinline]f(g(x))[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline], we can take a linear approximation for [mathjaxinline]f(u)[/mathjaxinline] and then substitute [mathjaxinline]g(x)[/mathjaxinline] in for [mathjaxinline]u[/mathjaxinline]. The resulting approximation is likely nonlinear, but it is still an approximation! <span style="color:#99182C"><p>
Warning: this only works if [mathjaxinline]g(0) = 0[/mathjaxinline]. </p><table id="a0000000529" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000530"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f(u) \approx f(0) + f'(0)u \qquad \Longrightarrow \qquad f\left(g(x)\right) \approx f(g(0)) + f'(g(0))g(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.28)</td></tr></table><p><b class="bfseries">Linear approximations of products</b></p><p>
To find the linear approximation of a function [mathjaxinline]h(x) = f(x)g(x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline], it suffices to find a linear approximation for [mathjaxinline]f(x)[/mathjaxinline], find a linear approximation for [mathjaxinline]g(x)[/mathjaxinline], and then the linear approximation for [mathjaxinline]h(x)[/mathjaxinline] is the product of these two approximations where we cancel all of the terms that are quadratic (or higher for more products). </p><table id="a0000000531" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000532"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle h(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(x)g(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.29)</td></tr><tr id="a0000000533"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0) + f'(0)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.30)</td></tr><tr id="a0000000534"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle g(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle g(0) + g'(0)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.31)</td></tr><tr id="a0000000535"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle h(x)[/mathjaxinline]
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[mathjaxinline]\displaystyle \approx[/mathjaxinline]
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[mathjaxinline]\displaystyle \left(f(0) + f'(0)x\right)\left(g(0) + g'(0)x\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.32)</td></tr><tr id="a0000000536"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle f(0)g(0) + \left(f'(0)g(0) +f(0)g'(0)\right)x + f'(0)g'(0)x^2[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.33)</td></tr><tr id="a0000000537"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0)g(0) + \left(f'(0)g(0) +f(0)g'(0)\right)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.34)</td></tr></table></span></p></span></p>
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