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<h2 class="hd hd-2 unit-title">2. Inverse functions and their derivatives</h2>
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<p><b class="bfseries">What functions can we differentiate so far?</b></p><p>
powers of [mathjaxinline]x[/mathjaxinline]<br/>polynomials<br/>trigonometric functions <br/>products of basic functions<br/>quotients of basic functions <br/>compositions of basic functions<br/>implicit functions<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>
Compute and evaluate inverse functions. </p></li><li><p>
Find graphs of inverse functions by reflecting the function graph across the diagonal. </p></li><li><p>
Find the derivative of <span style="color:#27408C"><b class="bf">inverse</b></span> function. </p></li></ul><p><b class="bfseries">Contents: 19 pages</b></p><p>
9 videos (38 minutes 1x speed) 21 questions </p>
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<h2 class="hd hd-2 unit-title">3. What is an inverse function?</h2>
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Cube roots
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If [mathjaxinline]\ f(x) = x^3[/mathjaxinline], then here are some values of [mathjaxinline]\ f[/mathjaxinline]: </p>
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[mathjaxinline]\displaystyle f(-11)[/mathjaxinline]
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[mathjaxinline]\displaystyle = -1331[/mathjaxinline]
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[mathjaxinline]\displaystyle f(12)[/mathjaxinline]
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[mathjaxinline]\displaystyle = 1728[/mathjaxinline]
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[mathjaxinline]\displaystyle f(-13)[/mathjaxinline]
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[mathjaxinline]\displaystyle = -2197.[/mathjaxinline]
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Let [mathjaxinline]g(y) = \displaystyle {\sqrt [3]{y}}[/mathjaxinline]. What is [mathjaxinline]g(-2197)[/mathjaxinline]? </p>
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<h3 class="hd hd-2">What is an inverse function?</h3>
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<h2 class="hd hd-2 unit-title">4. Definition of inverse function</h2>
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If a function [mathjaxinline]\ f[/mathjaxinline] has an inverse function [mathjaxinline]\ f^{-1}[/mathjaxinline], then [mathjaxinline]\ f^{-1}(b)= a[/mathjaxinline] if and only if [mathjaxinline]\ f(a) = b[/mathjaxinline]. </p>
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Find an inverse
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Suppose we know the following values of a function [mathjaxinline]g[/mathjaxinline]: </p>
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[mathjaxinline]\displaystyle g(0)[/mathjaxinline]
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[mathjaxinline]\displaystyle = 1/2[/mathjaxinline]
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<tr id="a0000000313">
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[mathjaxinline]\displaystyle g(1/2)[/mathjaxinline]
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[mathjaxinline]\displaystyle = 1[/mathjaxinline]
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<tr id="a0000000314">
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[mathjaxinline]\displaystyle g(3/2)[/mathjaxinline]
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[mathjaxinline]\displaystyle = 2[/mathjaxinline]
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[mathjaxinline]\displaystyle g(2)[/mathjaxinline]
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[mathjaxinline]\displaystyle = 5[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
If [mathjaxinline]g[/mathjaxinline] has an inverse function, find [mathjaxinline]g^{-1}(2)[/mathjaxinline]. </p>
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Find another inverse
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The function [mathjaxinline]\ f(x) = 6x - 16[/mathjaxinline] has an inverse function. Find [mathjaxinline]\ f^{-1}(4)[/mathjaxinline]. </p>
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(Enter answer either as a fraction or as a decimal number to 2 decimal places.) </p>
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<h2 class="hd hd-2 unit-title">6. Graphing inverses</h2>
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Points on a graph
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Suppose we know that [mathjaxinline]\ f[/mathjaxinline] has an inverse function and that [mathjaxinline](2,5)[/mathjaxinline] is on the graph of [mathjaxinline]\ f[/mathjaxinline]. What point do we know is on the graph of [mathjaxinline]\ f^{-1}[/mathjaxinline]? </p>
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(Enter your answer in the form [mathjaxinline](a,b)[/mathjaxinline] where [mathjaxinline]b[/mathjaxinline] is the value of [mathjaxinline]f^{-1}(a)[/mathjaxinline]. ) </p>
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Draw a picture
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Draw the two points. What is the relationship of the new point to the original point? </p>
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<text> Rotation by 90 degrees around the origin</text>
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<text> Reflection across the x-axis</text>
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<text> Reflection across the y-axis</text>
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<text> Reflection across the diagonal [mathjaxinline]y=x[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">7. Does every function have an inverse?</h2>
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<p><b class="bfseries">No inverse?</b></p><p>
You may have noticed that we've been constantly saying, “<span style="color:#99182C"><b class="bf">If</b></span> [mathjaxinline]\ f[/mathjaxinline] has an inverse..." These next few questions explain what could go wrong. </p>
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Square roots
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Let [mathjaxinline]\ \displaystyle {h(x) = x^2}[/mathjaxinline], and [mathjaxinline]\ j(x) = \displaystyle {\sqrt {x}}[/mathjaxinline]. </p>
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Note that [mathjaxinline]\ \displaystyle {h(-2) = 4}[/mathjaxinline]. What is [mathjaxinline]\ j(4)[/mathjaxinline]? <p style="display:inline">[mathjaxinline]j(4) =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_diff_6-tab7-problem1_2_1" class=" capa_inputtype inline textline">
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Is [mathjaxinline]\ j(h(-2)) = -2?[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">8. What functions have inverses?</h2>
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<h2 class="hd hd-2 unit-title">9. One-to-one definition and problems</h2>
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<p><b class="bfseries">Definition of one-to-one</b></p><p>
A function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#99182C"><b class="bf">one-to-one</b></span> if [mathjaxinline]\ f(a) \ne f(b)[/mathjaxinline] whenever [mathjaxinline]a\ne b[/mathjaxinline]. It is one-to-one if and only if its graph satisfies the horizontal line test (no horizontal line intersects its graph at more than one place). </p>
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Showing one-to-one
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Let [mathjaxinline]h(x) = \displaystyle {3-\frac{2}{x}}[/mathjaxinline]. Show that [mathjaxinline]h[/mathjaxinline] is one-to-one by finding a formula for [mathjaxinline]h^{-1}(y)[/mathjaxinline]. </p>
<p>
(Type [mathjaxinline]*[/mathjaxinline] for multiplication; e.g. 2[mathjaxinline]*[/mathjaxinline]y for [mathjaxinline]2y[/mathjaxinline]. Type / for division. Type [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses to denote order of operations.) </p>
<p>
<p style="display:inline">[mathjaxinline]h^{-1}(y) =[/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>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
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<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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<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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<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 \(\lambda \)
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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 \( \sqrt{x^2-y} \)
</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>
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<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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Which are one-to-one on the entire real line?
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<p>
Click the functions that are one-to-one on the entire real line. There may be more than one. </p>
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<text>[mathjaxinline]\ f(x)=5[/mathjaxinline]</text>
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<text>[mathjaxinline]g(x) = \sin x[/mathjaxinline]</text>
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<text>[mathjaxinline]h(x) = -3x^3 + x + 2[/mathjaxinline]</text>
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<text>[mathjaxinline]j(x) = -3x^3 - x + 2[/mathjaxinline]</text>
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<text>[mathjaxinline]k(x) = -4+5x + \cos (3x)[/mathjaxinline]</text>
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<p><b class="bfseries">Domain and range, interval notation</b></p><p>
Recall that the <span style="color:#99182C"><b class="bf">domain</b></span> of a function [mathjaxinline]\ f[/mathjaxinline] is the set of allowable input values. For instance, the domain of the function [mathjaxinline]\ f(x) = 1/x[/mathjaxinline] is the set of all non-zero real numbers. </p><p>
The <span style="color:#99182C"><b class="bf">range</b></span> of [mathjaxinline]\ f[/mathjaxinline] is the set of all possible output values. For instance, the range of the function [mathjaxinline]g(x) = x^2[/mathjaxinline] is the set of all real numbers that are non-negative. </p><p>
We often use interval notation to express sets of numbers like domains and ranges. A <span style="color:#99182C"><b class="bf">closed interval</b></span>, denoted [mathjaxinline][a,b][/mathjaxinline], is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]a \le x \le b[/mathjaxinline]. </p><p>
An <span style="color:#99182C"><b class="bf">open interval</b></span>, denoted [mathjaxinline](a,b)[/mathjaxinline], is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]a < x < b[/mathjaxinline]. </p><p>
One can have a half-open, half-closed interval. For instance, [mathjaxinline][-1, 3)[/mathjaxinline] is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]-1 \le x <3[/mathjaxinline]. One can also use [mathjaxinline]\pm \infty[/mathjaxinline] as endpoints: [mathjaxinline](-\infty , 0)[/mathjaxinline] is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]-\infty <x < 0[/mathjaxinline] (the set of negative numbers, in other words). </p><p>
This notation using round parentheses for open intervals is not universal; many mathematicians use reversed square brackets instead. For instance, they would denote the interval [mathjaxinline]3 < x < 7[/mathjaxinline] as [mathjaxinline]]3,7[[/mathjaxinline] rather than [mathjaxinline](3,7)[/mathjaxinline]. In this course, however, we will stick to round parentheses for open intervals. </p>
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<h2 class="hd hd-2 unit-title">10. Partial inverses</h2>
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<h3 class="hd hd-2">Partial Inverses and Trig Inverses</h3>
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Defining arcsin
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Here is the graph of [mathjaxinline]\sin \theta[/mathjaxinline]. </p>
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If we want to restrict [mathjaxinline]\theta[/mathjaxinline] to a certain interval in order to be able to define a partial inverse for this function, what interval would be best to choose? </p>
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<text> [mathjaxinline]\left(-\infty ,\infty \right)[/mathjaxinline]</text>
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<text> [mathjaxinline]\left[0,\infty \right)[/mathjaxinline]</text>
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<text> [mathjaxinline]\left[-\pi /2, \pi /2\right][/mathjaxinline]</text>
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<text> [mathjaxinline]\left(-\pi /2, \pi /2\right)[/mathjaxinline]</text>
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<text> [mathjaxinline]\left[0,\pi \right][/mathjaxinline]</text>
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<text> [mathjaxinline]\left[0,2\pi \right][/mathjaxinline]</text>
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<text> [mathjaxinline]\left[-1,1\right][/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">11. Inverse trig functions</h2>
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[mathjaxinline]\displaystyle {\arcsin x} =\theta[/mathjaxinline] </td><td style="text-align:left; border:none">
in [mathjaxinline]\displaystyle \left[-\pi /2,\pi /2\right][/mathjaxinline]</td><td style="text-align:left; border:none">
such that [mathjaxinline]\sin \theta = x.[/mathjaxinline] </td></tr><tr><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\arccos x} =\theta[/mathjaxinline] </td><td style="text-align:left; border:none">
in [mathjaxinline]\displaystyle \left[0,\pi \right][/mathjaxinline]</td><td style="text-align:left; border:none">
such that [mathjaxinline]\cos \theta = x.[/mathjaxinline] </td></tr><tr><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\arctan x} =\theta[/mathjaxinline] </td><td style="text-align:left; border:none">
in [mathjaxinline]\displaystyle \left(-\pi /2,\pi /2 \right)[/mathjaxinline] </td><td style="text-align:left; border:none">
such that [mathjaxinline]\tan \theta = x.[/mathjaxinline] </td></tr></table>
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Domains
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What is the domain of the function [mathjaxinline]\arcsin (x)[/mathjaxinline]? </p>
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<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline](-\infty , +\infty )[/mathjaxinline]</text>
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<input type="radio" name="input_diff_6-tab11-problem1_2_1" id="input_diff_6-tab11-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_6-tab11-problem1_2_1-choice_2-label" for="input_diff_6-tab11-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_2_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline][0,+\infty )[/mathjaxinline]</text>
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<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[-\pi /2, \pi /2\right][/mathjaxinline]</text>
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<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left(-\pi /2, \pi /2\right)[/mathjaxinline]</text>
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<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[0,\pi \right][/mathjaxinline]</text>
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<input type="radio" name="input_diff_6-tab11-problem1_2_1" id="input_diff_6-tab11-problem1_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="diff_6-tab11-problem1_2_1-choice_6-label" for="input_diff_6-tab11-problem1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_2_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[0,2\pi \right][/mathjaxinline]</text>
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<input type="radio" name="input_diff_6-tab11-problem1_2_1" id="input_diff_6-tab11-problem1_2_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="diff_6-tab11-problem1_2_1-choice_7-label" for="input_diff_6-tab11-problem1_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_2_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[-1,1\right][/mathjaxinline]</text>
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<text> None of the above</text>
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What is the domain of the function [mathjaxinline]\arccos (x)[/mathjaxinline]? </p>
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<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline](-\infty , +\infty )[/mathjaxinline]</text>
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<input type="radio" name="input_diff_6-tab11-problem1_3_1" id="input_diff_6-tab11-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_6-tab11-problem1_3_1-choice_2-label" for="input_diff_6-tab11-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_3_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline][0,+\infty )[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_3_1" id="input_diff_6-tab11-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_6-tab11-problem1_3_1-choice_3-label" for="input_diff_6-tab11-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_3_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[-\pi /2, \pi /2\right][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_3_1" id="input_diff_6-tab11-problem1_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_6-tab11-problem1_3_1-choice_4-label" for="input_diff_6-tab11-problem1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_3_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left(-\pi /2, \pi /2\right)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_3_1" id="input_diff_6-tab11-problem1_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diff_6-tab11-problem1_3_1-choice_5-label" for="input_diff_6-tab11-problem1_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_3_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[0,\pi \right][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_3_1" id="input_diff_6-tab11-problem1_3_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="diff_6-tab11-problem1_3_1-choice_6-label" for="input_diff_6-tab11-problem1_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_3_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[0,2\pi \right][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_3_1" id="input_diff_6-tab11-problem1_3_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="diff_6-tab11-problem1_3_1-choice_7-label" for="input_diff_6-tab11-problem1_3_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_3_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[-1,1\right][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_3_1" id="input_diff_6-tab11-problem1_3_1_choice_8" class="field-input input-radio" value="choice_8"/><label id="diff_6-tab11-problem1_3_1-choice_8-label" for="input_diff_6-tab11-problem1_3_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_3_1">
<text> None of the above</text>
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What is the domain of the function [mathjaxinline]\arctan (x)[/mathjaxinline]? </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_6-tab11-problem1_4_1">
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<input type="radio" name="input_diff_6-tab11-problem1_4_1" id="input_diff_6-tab11-problem1_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_6-tab11-problem1_4_1-choice_1-label" for="input_diff_6-tab11-problem1_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_4_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline](-\infty , +\infty )[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_4_1" id="input_diff_6-tab11-problem1_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_6-tab11-problem1_4_1-choice_2-label" for="input_diff_6-tab11-problem1_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_4_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline][0,+\infty )[/mathjaxinline]</text>
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<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_4_1" id="input_diff_6-tab11-problem1_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_6-tab11-problem1_4_1-choice_3-label" for="input_diff_6-tab11-problem1_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_4_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[-\pi /2, \pi /2\right][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_4_1" id="input_diff_6-tab11-problem1_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_6-tab11-problem1_4_1-choice_4-label" for="input_diff_6-tab11-problem1_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_4_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left(-\pi /2, \pi /2\right)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_4_1" id="input_diff_6-tab11-problem1_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diff_6-tab11-problem1_4_1-choice_5-label" for="input_diff_6-tab11-problem1_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_4_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[0,\pi \right][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_4_1" id="input_diff_6-tab11-problem1_4_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="diff_6-tab11-problem1_4_1-choice_6-label" for="input_diff_6-tab11-problem1_4_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_4_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[0,2\pi \right][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_4_1" id="input_diff_6-tab11-problem1_4_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="diff_6-tab11-problem1_4_1-choice_7-label" for="input_diff_6-tab11-problem1_4_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_4_1">
<text> [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]\left[-1,1\right][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_6-tab11-problem1_4_1" id="input_diff_6-tab11-problem1_4_1_choice_8" class="field-input input-radio" value="choice_8"/><label id="diff_6-tab11-problem1_4_1-choice_8-label" for="input_diff_6-tab11-problem1_4_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab11-problem1_4_1">
<text> None of the above</text>
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<span class="status unanswered" id="status_diff_6-tab11-problem1_4_1" data-tooltip="Not yet answered.">
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<span id="solution_diff_6-tab11-problem1_solution_1"/>
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Inverse trig functions
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<p>
What is [mathjaxinline]\arccos (-1)[/mathjaxinline]? </p>
<p>
(Give your answer in terms of [mathjaxinline]\pi[/mathjaxinline], typing pi for [mathjaxinline]\pi[/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_diff_6-tab11-problem2_2_1" class="inputtype formulaequationinput">
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<div id="input_diff_6-tab11-problem2_2_1_preview" class="equation">
\(\)
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<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
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<p>
What is [mathjaxinline]\arcsin (-1)[/mathjaxinline]? </p>
<p>
(Give your answer in terms of [mathjaxinline]\pi[/mathjaxinline], typing pi for [mathjaxinline]\pi[/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_diff_6-tab11-problem2_3_1" class="inputtype formulaequationinput">
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<div id="input_diff_6-tab11-problem2_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
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<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p>
What is [mathjaxinline]\arctan (-1)[/mathjaxinline]? </p>
<p>
(Give your answer in terms of [mathjaxinline]\pi[/mathjaxinline], typing pi for [mathjaxinline]\pi[/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_diff_6-tab11-problem2_4_1" class="inputtype formulaequationinput">
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<input type="text" name="input_diff_6-tab11-problem2_4_1" id="input_diff_6-tab11-problem2_4_1" data-input-id="diff_6-tab11-problem2_4_1" value="" aria-describedby="status_diff_6-tab11-problem2_4_1" size="20"/>
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<div id="input_diff_6-tab11-problem2_4_1_preview" class="equation">
\(\)
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<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<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 \( (2\pi+y)dy \)
</td>
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<h2 class="hd hd-2 unit-title">12. Trig functions of arc-trig functions</h2>
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Calculate [mathjaxinline]\sin \left(\arctan (3/4)\right)[/mathjaxinline] exactly. </p>
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<h2 class="hd hd-2 unit-title">13. Derivatives of inverse functions</h2>
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Let [mathjaxinline]y = \displaystyle {\frac{5}{2}}x - 3[/mathjaxinline]. This is a linear function with slope [mathjaxinline]\displaystyle {\frac{5}{2}}[/mathjaxinline]. Solve the equation for [mathjaxinline]x[/mathjaxinline] to get a formula for the inverse function; you should get a linear function. What is its slope? </p>
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Let [mathjaxinline]\ f[/mathjaxinline] be a function whose input is measured in meters and whose output is measured in liters. Let [mathjaxinline]g[/mathjaxinline] be an inverse or partial inverse for [mathjaxinline]\ f[/mathjaxinline]. </p>
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What are the units of [mathjaxinline]\ f'[/mathjaxinline]? </p>
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What are the units of [mathjaxinline]g'[/mathjaxinline]? </p>
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What derivatives are related?
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Suppose that [mathjaxinline]\ f(2) = 3[/mathjaxinline]. So the point [mathjaxinline](2,3)[/mathjaxinline] on the graph of [mathjaxinline]\ f[/mathjaxinline], and [mathjaxinline]\ f'(2)[/mathjaxinline] is the slope of the tangent line there. </p>
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Suppose that [mathjaxinline]\ f[/mathjaxinline] has an inverse function [mathjaxinline]g[/mathjaxinline]. Recall that the graph of [mathjaxinline]g[/mathjaxinline] is the graph of [mathjaxinline]\ f[/mathjaxinline] reflected across the [mathjaxinline]x=y[/mathjaxinline] line. </p>
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This suggests that [mathjaxinline]\ f'(2)[/mathjaxinline] is related to which of the following? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_6-tab13-problem3_2_1">
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<h2 class="hd hd-2 unit-title">14. The derivative of an inverse function</h2>
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If [mathjaxinline]g[/mathjaxinline] is a (full or partial) inverse of a function [mathjaxinline]\ f[/mathjaxinline], then </p><table id="a0000000323" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle {g'(x) = \frac{1}{f'\left(g(x)\right)}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all [mathjaxinline]x[/mathjaxinline] where [mathjaxinline]\displaystyle {f'\left(g(x)\right)}[/mathjaxinline] exists and is non-zero. </p>
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<h2 class="hd hd-2 unit-title">15. Inverse derivative practice</h2>
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The derivative of an inverse
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Suppose that [mathjaxinline]g = f^{-1}[/mathjaxinline]. If [mathjaxinline]g(3) = 2[/mathjaxinline], what is [mathjaxinline]g'(3)[/mathjaxinline]? Express your answer solely in terms of [mathjaxinline]\ f[/mathjaxinline], not [mathjaxinline]g[/mathjaxinline]. </p>
<p>
(Use an apostrophe ' to denote derivatives where needed. Type / for division.) </p>
<p>
<p style="display:inline">[mathjaxinline]g'(3) =[/mathjaxinline] </p>
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Find the derivative of an inverse
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<p>
Suppose that </p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]g(0)[/mathjaxinline] </td>
<td style="text-align:left; border:none">
= [mathjaxinline]1/2[/mathjaxinline] &#8195;&#8195;&#8195;&#8195;</td>
<td style="text-align:right; border:none">
[mathjaxinline]g'(0)[/mathjaxinline] </td>
<td style="text-align:left; border:none">
= [mathjaxinline]3[/mathjaxinline]</td>
</tr>
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]g(1/2)[/mathjaxinline] </td>
<td style="text-align:left; border:none">
= [mathjaxinline]1[/mathjaxinline] &#8195;&#8195;&#8195;&#8195;</td>
<td style="text-align:right; border:none">
[mathjaxinline]g'(1/2)[/mathjaxinline] </td>
<td style="text-align:left; border:none">
= [mathjaxinline]7[/mathjaxinline]</td>
</tr>
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]g(3/2)[/mathjaxinline] </td>
<td style="text-align:left; border:none">
= [mathjaxinline]2[/mathjaxinline]&#8195;&#8195;&#8195;&#8195;</td>
<td style="text-align:right; border:none">
[mathjaxinline]g'(3/2)[/mathjaxinline] </td>
<td style="text-align:left; border:none">
= [mathjaxinline]4[/mathjaxinline]</td>
</tr>
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]g(2)[/mathjaxinline] </td>
<td style="text-align:left; border:none">
= [mathjaxinline]5[/mathjaxinline]&#8195;&#8195;&#8195;&#8195;</td>
<td style="text-align:right; border:none">
[mathjaxinline]g'(2)[/mathjaxinline] </td>
<td style="text-align:left; border:none">
= [mathjaxinline]1/2[/mathjaxinline]. </td>
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<p>
Let [mathjaxinline]h[/mathjaxinline] be the inverse of [mathjaxinline]g[/mathjaxinline]. Find [mathjaxinline]h'(2)[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]h^{\prime }(2) =[/mathjaxinline] </p>
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Find the derivative of an inverse 2
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The function [mathjaxinline]\ f(x) = -2x^3 - 7x + 5[/mathjaxinline] is one-to-one and has an inverse function. (How would you verify this?) Let [mathjaxinline]g = f^{-1}[/mathjaxinline]. Find [mathjaxinline]g(5)[/mathjaxinline] and [mathjaxinline]g'(5)[/mathjaxinline]. </p>
<p>
(Enter as fractions or to 2 decimal places of accuracy.) </p>
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<p style="display:inline">[mathjaxinline]g(5) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]g^{\prime }(5) =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">16. Derivatives of inverse trig functions</h2>
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<p><b class="bfseries"><span style="color:#FF7F00">Note on video:</span></b> In the video, the quantity [mathjaxinline]\sin '(\arcsin x) = \cos (\arcsin x)[/mathjaxinline]. Observe that this is not the same as [mathjaxinline]\left(\sin (\arcsin x)\right)'[/mathjaxinline] which requires the chain rule to evaluate. </p>
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<p><b class="bfseries">Derivatives of inverse functions using implicit differentiation</b></p><p>
This is another way of finding the derivative of [mathjaxinline]\theta = \arcsin x[/mathjaxinline]. The relationship between [mathjaxinline]\theta[/mathjaxinline] and [mathjaxinline]x[/mathjaxinline] is given by </p><table id="a0000000326" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\sin \theta = x.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Differentiating both sides with respect to [mathjaxinline]x[/mathjaxinline] yields: </p><table id="a0000000327" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000328"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx}\sin \theta[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{d}{dx}x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000329"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos \theta \frac{d\theta }{dx}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 1[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000330"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d\theta }{dx}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{\cos \theta }[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
We know that </p><table id="a0000000331" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\cos ^2 \theta + x^2 = \cos ^2 \theta + \sin ^2 \theta = 1,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
so [mathjaxinline]\cos \theta = \pm \sqrt {1-x^2}[/mathjaxinline]. Since [mathjaxinline]\theta[/mathjaxinline] must lie in [mathjaxinline][-\pi /2, \pi /2][/mathjaxinline], [mathjaxinline]\cos \theta[/mathjaxinline] is positive. Hence </p><table id="a0000000332" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{d\theta }{dx} = \frac{1}{\cos \theta } = \frac{1}{\sqrt {1-x^2}}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
We can similarly find the derivative of [mathjaxinline]\theta = \arccos x[/mathjaxinline]. The relationship between [mathjaxinline]\theta[/mathjaxinline] and [mathjaxinline]x[/mathjaxinline] is given by </p><table id="a0000000333" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\cos \theta = x.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Differentiating both sides with respect to [mathjaxinline]x[/mathjaxinline] yields: </p><table id="a0000000334" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000335"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx}\cos \theta[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{d}{dx}x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000336"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle -\sin \theta \frac{d\theta }{dx}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 1[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000337"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d\theta }{dx}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\frac{1}{\sin \theta }[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
We know that </p><table id="a0000000338" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\sin ^2 \theta + x^2 = \sin ^2 \theta + \cos ^2 \theta = 1,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
so [mathjaxinline]\sin \theta = \pm \sqrt {1-x^2}[/mathjaxinline]. Since [mathjaxinline]\theta[/mathjaxinline] must lie in [mathjaxinline][0, \pi ][/mathjaxinline], [mathjaxinline]\sin \theta[/mathjaxinline] is positive. Hence </p><table id="a0000000339" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{d\theta }{dx} = -\frac{1}{\sin \theta } = -\frac{1}{\sqrt {1-x^2}}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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<h3 class="hd hd-2">Recitation video: arccos in more detail</h3>
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<h2 class="hd hd-2 unit-title">17. Arctan</h2>
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Where is arctan differentiable?
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At what points is [mathjaxinline]\arctan x[/mathjaxinline] differentiable? </p>
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<text> All real numbers [mathjaxinline]x[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_diff_6-tab17-problem1_2_1" id="input_diff_6-tab17-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_6-tab17-problem1_2_1-choice_2-label" for="input_diff_6-tab17-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab17-problem1_2_1">
<text> All [mathjaxinline]x&gt;0[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_diff_6-tab17-problem1_2_1" id="input_diff_6-tab17-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_6-tab17-problem1_2_1-choice_3-label" for="input_diff_6-tab17-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab17-problem1_2_1">
<text> All [mathjaxinline]x\neq 0[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_diff_6-tab17-problem1_2_1" id="input_diff_6-tab17-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_6-tab17-problem1_2_1-choice_4-label" for="input_diff_6-tab17-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_6-tab17-problem1_2_1">
<text> [mathjaxinline]-1 \leq x \leq 1[/mathjaxinline]</text>
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<text> [mathjaxinline]-\pi /2 &lt; x &lt; \pi /2[/mathjaxinline]</text>
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Find the derivative
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Calculate [mathjaxinline]\displaystyle \frac{d}{dx} \arctan x.[/mathjaxinline] </p>
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(You can type sin, cos, or tan for the trig functions, sqrt for [mathjaxinline]\sqrt {}[/mathjaxinline], and pi for [mathjaxinline]\pi[/mathjaxinline]. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents.) </p>
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<h2 class="hd hd-2 unit-title">18. Final practice</h2>
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<p><b class="bfseries">Derivatives of the inverse trig functions</b></p><p>
We now have more basic functions that we can differentiate. </p><table id="a0000000347" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000348"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx} \arcsin x[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{\sqrt {1-x^2}}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000349"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx} \arccos x[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac{-1}{\sqrt {1-x^2}}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000350"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx} \arctan x[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac{1}{1+x^2}[/mathjaxinline]
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Practice derivative 1
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Let [mathjaxinline]\ f(x) = \arctan (3x)[/mathjaxinline]. What is [mathjaxinline]\ f'(-1)[/mathjaxinline]? </p>
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(You can type pi for [mathjaxinline]\pi[/mathjaxinline], e for [mathjaxinline]e[/mathjaxinline], and sqrt for [mathjaxinline]\sqrt {}[/mathjaxinline]. ) </p>
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Practice derivative 2
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Let [mathjaxinline]g(x) = x^2 \arccos x[/mathjaxinline]. What is [mathjaxinline]\displaystyle g'\left(\frac{1}{2} \right)[/mathjaxinline]? </p>
<p>
(You can type pi for [mathjaxinline]\pi[/mathjaxinline], e for [mathjaxinline]e[/mathjaxinline], and sqrt for [mathjaxinline]\sqrt {}[/mathjaxinline]. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents. <i class="itshape">Use an exact expression rather than a decimal.</i>) </p>
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<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 \(\lambda \)
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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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<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>
</tr>
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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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A constant function
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Let [mathjaxinline]h(x) = \arcsin x + \arccos x.[/mathjaxinline] Note that </p>
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<td class="equation" style="width:80%; border:none">[mathjax]h'(x) = \frac{1}{\sqrt {1-x^2}}+\frac{-1}{\sqrt {1-x^2}} = 0.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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This tells us that [mathjaxinline]h(x)[/mathjaxinline] is actually a constant function. What is its value? </p>
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(You can type pi for [mathjaxinline]\pi[/mathjaxinline], e for [mathjaxinline]e[/mathjaxinline], and sqrt for [mathjaxinline]\sqrt {}[/mathjaxinline]. <i class="itshape">Use an exact expression rather than a decimal.</i> ) </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>
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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>
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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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<p><b class="bfseries">Definition of inverse function</b></p><p>
If functions [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] satisfy [mathjaxinline]g\left(f(x)\right) = x[/mathjaxinline] and [mathjaxinline]\ f\left(g(y)\right) = y[/mathjaxinline], then we say [mathjaxinline]g[/mathjaxinline] is the inverse of [mathjaxinline]\ f[/mathjaxinline], and denote it by [mathjaxinline]\ f^{-1}[/mathjaxinline]. (Similarly, [mathjaxinline]\ f = g^{-1}[/mathjaxinline].) </p><p>
If a function [mathjaxinline]\ f[/mathjaxinline] has an inverse function [mathjaxinline]\ f^{-1}[/mathjaxinline], then [mathjaxinline]\ f^{-1}(b)= a[/mathjaxinline] if and only if [mathjaxinline]\ f(a) = b[/mathjaxinline]. </p><p><b class="bfseries">Definition of one-to-one</b></p><p>
A function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#99182C"><b class="bf">one-to-one</b></span> if [mathjaxinline]\ f(a) \ne f(b)[/mathjaxinline] whenever [mathjaxinline]a\ne b[/mathjaxinline]. It is one-to-one if and only if its graph satisfies the horizontal line test (no horizontal line intersects its graph at more than one place). </p><p><b class="bfseries">Domain and range, interval notation</b></p><p>
Recall that the <span style="color:#99182C"><b class="bf">domain</b></span> of a function [mathjaxinline]\ f[/mathjaxinline] is the set of allowable input values. For instance, the domain of the function [mathjaxinline]\ f(x) = 1/x[/mathjaxinline] is the set of all non-zero real numbers. </p><p>
The <span style="color:#99182C"><b class="bf">range</b></span> of [mathjaxinline]\ f[/mathjaxinline] is the set of all possible output values. For instance, the range of the function [mathjaxinline]g(x) = x^2[/mathjaxinline] is the set of all real numbers that are non-negative. </p><p>
We often use interval notation to express sets of numbers like domains and ranges. A <span style="color:#99182C"><b class="bf">closed interval</b></span>, denoted [mathjaxinline][a,b][/mathjaxinline], is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]a \le x \le b[/mathjaxinline]. </p><p>
An <span style="color:#99182C"><b class="bf">open interval</b></span>, denoted [mathjaxinline](a,b)[/mathjaxinline], is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]a < x < b[/mathjaxinline]. </p><p>
One can have a half-open, half-closed interval. For instance, [mathjaxinline][-1, 3)[/mathjaxinline] is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]-1 \le x <3[/mathjaxinline]. One can also use [mathjaxinline]\pm \infty[/mathjaxinline] as endpoints: [mathjaxinline](-\infty , 0)[/mathjaxinline] is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]-\infty <x < 0[/mathjaxinline] (the set of negative numbers, in other words). </p><p>
This notation using round parentheses for open intervals is not universal; many mathematicians use reversed square brackets instead. For instance, they would denote the interval [mathjaxinline]3 < x < 7[/mathjaxinline] as [mathjaxinline]]3,7[[/mathjaxinline] rather than [mathjaxinline](3,7)[/mathjaxinline]. In this course, however, we will stick to round parentheses for open intervals. </p><p><b class="bfseries">The inverse trig functions</b></p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\arcsin x} =[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\theta[/mathjaxinline] in [mathjaxinline]\displaystyle \left[-\pi /2,\pi /2\right][/mathjaxinline]</td><td style="text-align:left; border:none">
such that [mathjaxinline]\sin \theta = x.[/mathjaxinline] </td></tr><tr><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\arccos x} =[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\theta[/mathjaxinline] in [mathjaxinline]\displaystyle \left[0,\pi \right][/mathjaxinline]</td><td style="text-align:left; border:none">
such that [mathjaxinline]\cos \theta = x.[/mathjaxinline] </td></tr><tr><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\arctan x} =[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\theta[/mathjaxinline] in [mathjaxinline]\displaystyle \left(-\pi /2,\pi /2 \right)[/mathjaxinline] </td><td style="text-align:left; border:none">
such that [mathjaxinline]\tan \theta = x.[/mathjaxinline] </td></tr></table><p><b class="bfseries">Derivatives of inverse functions</b></p><p>
If [mathjaxinline]g[/mathjaxinline] is a (full or partial) inverse of a function [mathjaxinline]\ f[/mathjaxinline], then </p><table id="a0000000356" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle {g'(x) = \frac{1}{f'\left(g(x)\right)}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all [mathjaxinline]x[/mathjaxinline] where [mathjaxinline]\displaystyle {f'\left(g(x)\right)}[/mathjaxinline] exists and is non-zero. </p><p><b class="bfseries">Derivatives of arctrig functions</b></p><p>
Where defined, </p><table id="a0000000357" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000358"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d}{dx}\arcsin (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 \frac{1}{\sqrt {1-x^2}}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000359"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx}\arccos (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 \frac{-1}{\sqrt {1-x^2}}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000360"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx}\arctan (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 \frac{1}{1+x^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table>
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