Kayla Murray: /* Problem 10 */

2017-12-03T06:16:37Z

Problem 10

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2017-12-03T06:14:41Z

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'''This is a sample, and is meant to represent the material usually covered in Math 7A for the final. An actual test may or may not be similar.'''

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<div class="noautonum">TOC</div>

== [[007A_Sample Final 3,_Problem_1| Problem 1 ]] ==
Find each of the following limits if it exists. If you think the limit does not exist provide a reason.

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

(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>

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

== [[007A_Sample Final 3,_Problem_2| Problem 2 ]] ==
 Find the derivative of the following functions:

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

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

== [[007A_Sample Final 3,_Problem_3| Problem 3 ]] ==
Find the derivative of the following function using the limit definition of the derivative:

::<math>f(x)=3x-x^2</math>

== [[007A_Sample Final 3,_Problem_4| Problem 4 ]] ==
 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>

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?

== [[007A_Sample Final 3,_Problem_5| Problem 5 ]] ==
 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>

== [[007A_Sample Final 3,_Problem_6| Problem 6 ]] ==
 Let

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

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

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

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

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

== [[007A_Sample Final 3,_Problem_7| Problem 7 ]] ==

Compute

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

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

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

== [[007A_Sample Final 3,_Problem_8| Problem 8 ]] ==

If  <math style="vertical-align: 0px">W</math>  denotes the weight in pounds of an individual, and  <math style="vertical-align: 0px">t</math>  denotes the time in months, then  <math style="vertical-align: -13px">\frac{dW}{dt}</math>  is the rate of weight gain or loss in lbs/mo. The current speed record for weight loss is a drop in weight from 487 pounds to 130 pounds over an eight month period. Show that the rate of weight loss exceeded 44 lbs/mo at some time during the eight month period.

== [[007A_Sample Final 3,_Problem_9| Problem 9 ]] ==

Let

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

(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>

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

== [[007A_Sample Final 3,_Problem_10| Problem 10 ]] ==

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

(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>

(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>

← Older revision		Revision as of 06:16, 3 December 2017
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	== [[007A_Sample Final 3,_Problem_10\|<span class="biglink"><span style="font-size:80%"> Problem 10 </span>]] ==		== [[007A_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">When treating cancer, it is an advantage to know the rate of growth of the cancer tumor. A mathematical model for cancer growth shows that when the diameter of a particular spherical tumor is 16 mm, it is growing at a rate of 0.4 mm a day. How fast is the volume of the tumor changing at that time?
−
−	~~<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>~~