009A Sample Final 3 - Revision history

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2017-04-10T16:27:29Z

Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 9A for the final. An actual test may or may not be similar.''' '''Click on the''' '''<span..."

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'''This is a sample, and is meant to represent the material usually covered in Math 9A for the final. An actual test may or may not be similar.'''

'''Click on the''' '''<span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''
<div class="noautonum">TOC</div>

== [[009A_Sample Final 3,_Problem_1|<span class="biglink"><span style="font-size:80%"> Problem 1 </span></span>]] ==
<span class="exam">Find each of the following limits if it exists. If you think the limit does not exist provide a reason.

<span class="exam">(a)  <math style="vertical-align: -14px">\lim_{x\rightarrow 0} \frac{\sin(5x)}{1-\sqrt{1-x}}</math>

<span class="exam">(b)  <math style="vertical-align: -12px">\lim_{x\rightarrow 8} f(x),</math>  given that  <math style="vertical-align: -14px">\lim_{x\rightarrow 8}\frac{xf(x)}{3}=-2</math>

<span class="exam">(c)  <math style="vertical-align: -14px">\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}</math>

== [[009A_Sample Final 3,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] ==
<span class="exam"> Find the derivative of the following functions:

<span class="exam">(a)  <math style="vertical-align: -18px">g(\theta)=\frac{\pi^2}{(\sec\theta -\sin 2\theta)^2}</math>

<span class="exam">(b)  <math style="vertical-align: -5px">y=\cos(3\pi)+\tan^{-1}(\sqrt{x})</math>

== [[009A_Sample Final 3,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] ==
<span class="exam">Find the derivative of the following function using the limit definition of the derivative:

::<span class="exam"><math>f(x)=3x-x^2</math>

== [[009A_Sample Final 3,_Problem_4|<span class="biglink"><span style="font-size:80%"> Problem 4 </span>]] ==
<span class="exam"> Discuss, without graphing, if the following function is continuous at  <math style="vertical-align: 0px">x=0.</math>

::<math>f(x) = \left\{
\begin{array}{lr}
\frac{x}{|x|} & \text{if }x < 0\\
0 & \text{if }x = 0\\
x-\cos x & \text{if }x > 0
\end{array}
\right.
</math>

<span class="exam">If you think  <math style="vertical-align: -4px">f</math>  is not continuous at  <math style="vertical-align: -4px">x=0,</math>  what kind of discontinuity is it?

== [[009A_Sample Final 3,_Problem_5|<span class="biglink"><span style="font-size:80%"> Problem 5 </span>]] ==
<span class="exam"> Calculate the equation of the tangent line to the curve defined by  <math style="vertical-align: -4px">x^3+y^3=2xy</math>  at the point,  <math style="vertical-align: -5px">(1,1).</math>

== [[009A_Sample Final 3,_Problem_6|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] ==
<span class="exam"> Let

::<math>f(x)=4+8x^3-x^4</math>

<span class="exam">(a) Over what  <math style="vertical-align: 0px">x</math>-intervals is  <math style="vertical-align: -4px">f</math>  increasing/decreasing?

<span class="exam">(b) Find all critical points of  <math style="vertical-align: -4px">f</math>  and test each for local maximum and local minimum.

<span class="exam">(c) Over what  <math style="vertical-align: 0px">x</math>-intervals is  <math style="vertical-align: -4px">f</math>  concave up/down?

<span class="exam">(d) Sketch the shape of the graph of  <math style="vertical-align: -4px">f.</math>

== [[009A_Sample Final 3,_Problem_7|<span class="biglink"><span style="font-size:80%"> Problem 7 </span>]] ==

<span class="exam">Compute

<span class="exam">(a)  <math style="vertical-align: -18px">\lim_{x\rightarrow 0} \frac{x}{3-\sqrt{9-x}}</math>

<span class="exam">(b)  <math style="vertical-align: -16px">\lim_{x\rightarrow \pi} \frac{\sin x}{\pi-x}</math>

<span class="exam">(c)  <math style="vertical-align: -16px">\lim_{x\rightarrow -2} \frac{x^2-x-6}{x^3+8}</math>

== [[009A_Sample Final 3,_Problem_8|<span class="biglink"><span style="font-size:80%"> Problem 8 </span>]] ==

<span class="exam">Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure  <math style="vertical-align: 0px">P</math>  and volume  <math style="vertical-align: 0px">V</math>  satisfy the equation  <math style="vertical-align: 0px">PV=C</math>  where  <math style="vertical-align: 0px">C</math>  is a constant. Suppose that at a certain instant, the volume is  <math style="vertical-align: -4px">600 \text{ cm}^3,</math>  the pressure is  <math style="vertical-align: -4px">150 \text{ kPa},</math>  and the pressure is increasing at a rate of  <math style="vertical-align: -4px">20 \text{ kPa/min}.</math>  At what rate is the volume decreasing at this instant?

== [[009A_Sample Final 3,_Problem_9|<span class="biglink"><span style="font-size:80%"> Problem 9 </span>]] ==

<span class="exam">Let

::<math>g(x)=(2x^2-8x)^{\frac{2}{3}}</math>

<span class="exam">(a) Find all critical points of  <math style="vertical-align: -4px">g</math>  over the  <math style="vertical-align: 0px">x</math>-interval  <math style="vertical-align: -5px">[0,8].</math>

<span class="exam">(b) Find absolute maximum and absolute minimum of  <math style="vertical-align: -4px">g</math>  over  <math style="vertical-align: -5px">[0,8].</math>

== [[009A_Sample Final 3,_Problem_10|<span class="biglink"><span style="font-size:80%"> Problem 10 </span>]] ==

<span class="exam">Let  <math style="vertical-align: -5px">y=\tan(x).</math>

<span class="exam">(a) Find the differential  <math style="vertical-align: -4px">dy</math>  of  <math style="vertical-align: -5px">y=\tan (x)</math>  at  <math style="vertical-align: -15px">x=\frac{\pi}{4}.</math>

<span class="exam">(b) Use differentials to find an approximate value for  <math style="vertical-align: -5px">\tan(0.885).</math>  Hint:  <math style="vertical-align: -15px">\frac{\pi}{4}\approx 0.785.</math>