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MATH 121 Lecture Notes - Lecture 27: Trigonometric Functions

7. 1-7. 2 finding inverse trig functions ( first graph is sine domain and range restricted) Restrict the domain of f(x)= sinx to make it 1-1 [- /2, /2] 

United States

Precalculus Mathematics

Mathematics

Kashcheyeva Olga S.

University of Illinois, Chicago

<p>Think About It</p>
<p><strong>In Exercises 73 and 74, the function <em>f</em> is differentiable on the indicated interval. The table shows</strong> <picture><source srcset="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/944873089973522/question/0.webp?img=4950bc7763c8a6a561eff4b531ba8d88" type="image/webp"><img src="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/944873089973522/question/0.png?img=6fbf256fedb19bd5fdf72b20530061e7" data-mlang="latex" data-equation="f%5E%7B%27%7D%28x%29"></picture></p>
<p><strong>for selected values of <em>x</em>. (a) Sketch the graph of <em>f</em>, (b) approximate the critical numbers, and (c) identify the relative extrema.</strong></p>
<p><em>f</em> is differentiable on <picture><source srcset="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/944873089973522/question/1.webp?img=73983c85c26e08aa5ba988f0cf76f206" type="image/webp"><img src="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/944873089973522/question/1.png?img=ba235a3b2b6dfa7904957ed70d9d9b08" data-mlang="latex" data-equation="lbrack0%2Cpirbrack"></picture>.</p>
<div class="scrolledContentWrap" data-scroll-wrapped-indicator="true">
<table border="1" rules="all">
<tbody>
<tr>
<td><em>x</em></td>
<td>0</td>
<td>π / 6</td>
<td>π / 4</td>
<td>π / 3</td>
<td>π / 2</td>
</tr>
<tr>
<td>f ' ( x )</td>
<td>3.14</td>
<td>-0.23</td>
<td>-2.45</td>
<td>-3.11</td>
<td>0.69</td>
</tr>
</tbody>
</table>
</div>
<div class="scrolledContentWrap" data-scroll-wrapped-indicator="true">
<table border="1" rules="all">
<tbody>
<tr>
<td><em>x</em></td>
<td>2 π / 3</td>
<td>3 π / 4</td>
<td>5 π / 6</td>
<td>π</td>
</tr>
<tr>
<td>f ' ( x )</td>
<td>3.00</td>
<td>1.37</td>
<td>-1.14</td>
<td>-2.84</td>
</tr>
</tbody>
</table>
<p> </p>
</div>

<p><strong>Numerical, Graphical, and Analytic Analysis</strong></p>
<p>Consider the functions <picture><source srcset="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/0.webp?img=4555015a6cedfd65e800e602f4b950dc" type="image/webp"><img src="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/0.png?img=70c7f8011a32eda433582e5b59c637c3" data-mlang="latex" data-equation="f%28x%29%3Dx%2C%5Ctext%7Band%7D%2Cg%28x%29%3Dsinx"></picture> on the interval <picture><source srcset="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/1.webp?img=6cd15005d8974ee83fc0cf90cdcff8d3" type="image/webp"><img src="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/1.png?img=c49099ccae2a6886cb82d9c3806dda10" data-mlang="latex" data-equation="%280%2Cpi%29"></picture>.</p>
<p>(a) Complete the table and make a conjecture about which is the greater function on the interval (0, π).</p>
<div class="scrolledContentWrap" data-scroll-wrapped-indicator="true">
<table border="1" rules="all">
<tbody>
<tr>
<td><em>x</em></td>
<td>0.5</td>
<td>1</td>
<td>1.5</td>
<td>2</td>
<td>2.5</td>
<td>3</td>
</tr>
<tr>
<td><em>f(x)</em></td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td><em>g(x)</em></td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
</div>
<p>(b) Use a graphing utility to graph the functions and use the graphs to make a conjecture about which is the greater function on the interval ( 0 , π ) .</p>
<p>(c) Prove that <picture><source srcset="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/2.webp?img=c845c84ec4fde32c8c019ffaca013aa7" type="image/webp"><img src="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/2.png?img=534c0c8ee9bb94c8a532d1002e19e2a5" data-mlang="latex" data-equation="f%28x%29%3E%2Cg%28x%29"></picture> on the interval <picture><source srcset="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/3.webp?img=a8a690a2d21eee0d670d6f87817c534a" type="image/webp"><img src="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/3.png?img=b23626da2107dcfb068863eca5f123ba" data-mlang="latex" data-equation="%280%2Cpi%29"></picture>. [<em>Hint</em>: Show that <picture><source srcset="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/4.webp?img=0d5d5838d9380a771c7e7ba015263cf2" type="image/webp"><img src="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/4.png?img=8efb85dd2d9639d8d2d943da00b9af3f" data-mlang="latex" data-equation="h%5E%7B%27%7D%28x%29%3E%2C0"></picture>, where <picture><source srcset="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/5.webp?img=a9767f0e9fb2a95a6cf47fa41819cf6b" type="image/webp"><img src="https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/057298178982163/question/5.png?img=65118a6237cdaab4e2f9d4218a2c3670" data-mlang="latex" data-equation="h%3Df-g"></picture>.]</p>

<p><picture><source srcset="https://prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/21/2116396.webp" type="image/webp"></source><img src="https://prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/21/2116396.png" alt=""></picture></p> <div style="margin:0px;padding:0px;border:0px;width:707px;height:0px;"><div style="height:145px;"></div></div> <div id="dr5" class="hidden">Let f : R2 rightarrow R be given by f(x, y) = 5x2 + 2y2 - 3. What is the domain and range of f? Restrict the domain so that f is one-to-one on the new domain. Restrict the codomain so that f is onto the new codomain. What is the domain and range of the following functions f. Sketch the domain. f(x, y)= 1/ </div> <span class="transcribed-text">Show transcribed image text</span> <span class="transcribed-image-text hidden">Let f : R2 rightarrow R be given by f(x, y) = 5x2 + 2y2 - 3. What is the domain and range of f? Restrict the domain so that f is one-to-one on the new domain. Restrict the codomain so that f is onto the new codomain. What is the domain and range of the following functions f. Sketch the domain. f(x, y)= 1/ </span>

MATH 121 Lecture Notes - Lecture 27: Trigonometric Functions

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Related Questions

x	0	π / 6	π / 4	π / 3	π / 2
f ' ( x )	3.14	-0.23	-2.45	-3.11	0.69

x	2 π / 3	3 π / 4	5 π / 6	π
f ' ( x )	3.00	1.37	-1.14	-2.84

x	0.5	1	1.5	2	2.5	3
f(x)
g(x)