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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Linear Approximation</h3>
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<h2 class="hd hd-2 unit-title">2. Linear approximation</h2>
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<p><b class="bfseries">Approximation of functions and derivatives</b></p><p>
The <span style="color:#99182C"><b class="bf">tangent line</b></span> at [mathjaxinline]x=a[/mathjaxinline] is a good approximation for the <span style="color:#99182C"><b class="bf">function</b></span> near [mathjaxinline]x=a[/mathjaxinline]. The <span style="color:#99182C"><b class="bf">slope of the secant line</b></span> between [mathjaxinline]x=a[/mathjaxinline] and [mathjaxinline]x=b[/mathjaxinline] is a good approximation for the <span style="color:#99182C"><b class="bf">derivative</b></span> between [mathjaxinline]x=a[/mathjaxinline] and [mathjaxinline]x=b[/mathjaxinline]. </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>
Understand where the <span style="color:#27408C"><b class="bf">tangent line</b></span> is a good <span style="color:#27408C"><b class="bf">linear approximation</b></span> to the function. </p></li><li><p>
Understand where the <span style="color:#27408C"><b class="bf">average rate of change</b></span> is a good <span style="color:#27408C"><b class="bf">linear approximation</b></span> to the <span style="color:#27408C"><b class="bf">instantaneous rate of change</b></span>. </p></li><li><p><span style="color:#27408C"><b class="bf">Approximate values of functions</b></span> given derivatives. </p></li><li><p><span style="color:#27408C"><b class="bf">Approximate values of derivatives</b></span> given function data. </p></li><li><p>
Understand how <span style="color:#27408C"><b class="bf">convexity/concavity</b></span> of the function affects the quality of the <span style="color:#27408C"><b class="bf">linear approximation</b></span>, both qualitatively and quantitatively. </p></li></ul><p><b class="bfseries">Contents: 9 pages</b></p><p>
4 videos (13 minutes 1x speed) 14 questions </p>
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<h2 class="hd hd-2 unit-title">3. Boat in a canal</h2>
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A boat in a canal
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Let [mathjaxinline]x=x(t)[/mathjaxinline] be the position of a boat along a canal as a function of time, where [mathjaxinline]t[/mathjaxinline] is measured in seconds and [mathjaxinline]x[/mathjaxinline] in meters. Suppose that at [mathjaxinline]t = 20[/mathjaxinline], we know that the boat is at the 150 meter point of the canal and is traveling in the positive direction at a velocity of [mathjaxinline]0.4[/mathjaxinline] meters per second. </p>
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<p style="display:inline">In terms of [mathjaxinline]\ x[/mathjaxinline], this is saying that [mathjaxinline]150 =[/mathjaxinline] </p>
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<option value="[mathjaxinline]x^{\prime }(20)[/mathjaxinline]"> [mathjaxinline]x^{\prime }(20)[/mathjaxinline]</option>
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From the information that we have, what can we determine about the graph of [mathjaxinline]x=x(t)[/mathjaxinline]? </p>
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<font size="2">Image of axes provided only to indicate that [mathjaxinline]t[/mathjaxinline] lies along the horizontal axis, and [mathjaxinline]x[/mathjaxinline] is the vertical axis. The scaling along these axes is not needed to answer the question below.</font>
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<text> Only the slope of the tangent line to the graph of [mathjaxinline]x(t)[/mathjaxinline] at [mathjaxinline]t=20[/mathjaxinline]</text>
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<text> The entire tangent line to the graph of [mathjaxinline]x(t)[/mathjaxinline] at [mathjaxinline]t=20[/mathjaxinline]</text>
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Underlying assumptions of estimate
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Recall the given information: at [mathjaxinline]t = 20[/mathjaxinline], the boat is at the 150 meter point of the canal and is traveling in the positive direction at a velocity of [mathjaxinline]0.4[/mathjaxinline] meters per second. </p>
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Suppose we want to estimate the position of the boat at [mathjaxinline]t = 30[/mathjaxinline]. </p>
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That is just an estimate; we don't know that the boat will be exactly at the extrapolated position. </p>
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<p style="display:inline">It would be exactly correct if we were guaranteed that the </p>
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<p style="display:inline">of the boat between [mathjaxinline]t=20[/mathjaxinline] and [mathjaxinline]t=30[/mathjaxinline] never changed.</p>
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<p style="display:inline">In other words, it would be guaranteed if the </p>
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<p>
Recall the given information: at [mathjaxinline]t = 20[/mathjaxinline], the boat is at the 150 meter point of the canal and is traveling in the positive direction at a velocity of [mathjaxinline]0.4[/mathjaxinline] meters per second. </p>
<p>
Extrapolate the position of the boat at [mathjaxinline]t = 30[/mathjaxinline]. </p>
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(Think about the units of the numbers 20, 30, 150, and 0.4. What arithmetic can you do with these numbers to get your estimate?) </p>
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<p>
We can make a similar estimate for the position of the boat at [mathjaxinline]t = 250[/mathjaxinline]. Would you have more confidence in your estimate for [mathjaxinline]t=30[/mathjaxinline] or for [mathjaxinline]t=250[/mathjaxinline]? </p>
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Use the technique described in the video to estimate [mathjaxinline]\sqrt {104}[/mathjaxinline] <b class="bf">without a calculator</b>, given that you can find [mathjaxinline]\sqrt {100}[/mathjaxinline] and also the derivative of the function [mathjaxinline]g(x) = \sqrt {x}[/mathjaxinline] at [mathjaxinline]x=100[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">6. Linear approximation</h2>
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The linear approximation for a function [mathjaxinline]\ f[/mathjaxinline] near a point [mathjaxinline]x=a[/mathjaxinline] is given by the following equivalent formulas: </p><table id="a0000000191" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000192"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta f[/mathjaxinline]
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[mathjaxinline]\displaystyle \approx \left. \frac{df}{dx} \right|_{x=a} \cdot \Delta x \qquad \ \mathrm{for} \ \Delta x \ \mathrm{near} \ 0[/mathjaxinline]
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[mathjaxinline]\displaystyle f(x)[/mathjaxinline]
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[mathjaxinline]\displaystyle \approx f'(a) (x-a) + f(a) \ \ \ \mathrm{for} \ x \ \mathrm{near} \ a[/mathjaxinline]
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Use linear approximation, <b class="bf">without a calculator</b>, to estimate [mathjaxinline]3.97^{2.5}[/mathjaxinline]. </p>
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Recall from the last video that [mathjaxinline]g(x) = \sqrt {x}[/mathjaxinline]. What is [mathjaxinline]g^{\prime \prime }(100)[/mathjaxinline]? </p>
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Concavity and the tangent line
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Still taking [mathjaxinline]g(x) = \sqrt {x}[/mathjaxinline] as in the previous video, compare the graph of [mathjaxinline]g[/mathjaxinline] to the tangent line to the graph at [mathjaxinline]x=100[/mathjaxinline]. Near [mathjaxinline]x=100[/mathjaxinline], is the graph of [mathjaxinline]g[/mathjaxinline] above or below the line? (Hint: Use your answer to the previous question.) </p>
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Overestimate or underestimate?
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In the previous video, for [mathjaxinline]g(x) = \sqrt {x}[/mathjaxinline] we estimated that [mathjaxinline]g(104) \approx 10.2[/mathjaxinline]. Is this estimate [mathjaxinline]10.2[/mathjaxinline] higher than the actual value or lower? (Hint: Use your answers to the previous two questions.) </p>
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Boat type
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<p>
Let's go back to the boat. </p>
<p>
Recall, at [mathjaxinline]t = 20[/mathjaxinline], the boat was at the 150 meter point of the canal and was traveling in the positive direction at a velocity of [mathjaxinline]0.4[/mathjaxinline] meters per second: </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000201" style="table-layout:auto" width="100%">
<tr id="a0000000202">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f(20)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 150[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.1)</td>
</tr>
<tr id="a0000000203">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(20)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 0.4.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.2)</td>
</tr>
</table>
<p>
We estimated that at [mathjaxinline]t=30[/mathjaxinline], the position of the boat would be [mathjaxinline]154[/mathjaxinline] meters; [mathjaxinline]f(30) \approx 154[/mathjaxinline]. </p>
<p>
Is this position estimate likely to be more accurate if the boat was an oil tanker or a canoe? </p>
<p>
(Think about what this question has to do with the previous ones! ) </p>
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<h2 class="hd hd-2 unit-title">8. Concavity</h2>
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<h3 class="hd hd-2">Concavity and Linear Approximation</h3>
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<h2 class="hd hd-2 unit-title">9. Some problems</h2>
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Change in volume
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<p>
A ball is supposed to be manufactured with radius 10cm. If it gets made with a radius of 9.97cm instead, it would take up approximately how much less volume (in cubic centimeters)? Do not use a calculator. </p>
<p>
(The volume of a ball is [mathjaxinline]4\pi r^3/3[/mathjaxinline], where [mathjaxinline]r[/mathjaxinline] is the radius. Keep your answer in terms of [mathjaxinline]\pi[/mathjaxinline] by typing pi and keep your answer as a positive number. Type [mathjaxinline]*[/mathjaxinline] for multiplication, / for division, and [mathjaxinline]\wedge[/mathjaxinline] for exponents. Do not use a calculator.) </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_diff_1-tab9-problem1_2_1" class="inputtype formulaequationinput">
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What is the percentage decrease? </p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
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<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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</tr>
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<font color="#0078b0">2/3</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<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>
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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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<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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<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>
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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 \( \sqrt{x^2-y} \)
</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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Going backwards!
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Suppose instead we told you that the sphere was manufactured with an excess volume of [mathjaxinline]20\pi[/mathjaxinline] cubic centimeters. In centimeters, approximately what radius would you expect this sphere to have? </p>
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(Do not use a calculator. Enter answer to 2 decimal places.) </p>
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<th class="formulainput" scope="col">Example Entries</th>
</tr>
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<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>
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<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>
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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>
</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>
</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>
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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 \( \sqrt{x^2-y} \)
</td>
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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>
</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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Big clock
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The minute hand extends 14 feet out from the center of the clock face on Elizabeth Tower. It is pointed at the first minute mark of the clock at 7:01.</p>
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<img alt="The clock on Elizabeth Tower" src="/assets/courseware/v1/2fd6efd317e86edfb51b370f0e172b02/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_bigben.png" width="495"/>
<br/>
<font size="2">Photo by John Morgan on Flickr, CC BY </font>
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<p>
Use linear approximation to estimate how much its tip of the minute hand moves horizontally between 7:01 and 7:02. Do not use a calculator. (Hint: approximate from the position of the minute hand at 7:00.) </p>
<p>
(Please enter your expression in terms of [mathjaxinline]\pi[/mathjaxinline]. Type <b class="bf">pi</b> for [mathjaxinline]\pi[/mathjaxinline]. Type [mathjaxinline]*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]\wedge[/mathjaxinline] for exponents.) </p>
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<span class="trailing_text" id="trailing_text_diff_1-tab9-problem3_2_1">feet</span>
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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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<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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<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>
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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>
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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>
</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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<h2 class="hd hd-2 unit-title">10. Summary</h2>
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<p><b class="bfseries">Linear approximation</b></p><p>
The linear approximation for a function [mathjaxinline]\ f[/mathjaxinline] near a point [mathjaxinline]x=a[/mathjaxinline] is given by the following equivalent formulas: </p><table id="a0000000208" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000209"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta f[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \approx \left. \frac{df}{dx} \right|_{x=a} \cdot \Delta x \qquad \ \mathrm{for} \ \Delta x \ \mathrm{near} \ 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000210"><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:left; border:none">
[mathjaxinline]\displaystyle \approx f'(a) (x-a) + f(a) \ \ \ \mathrm{for} \ x \ \mathrm{near} \ a[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table>
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