009A Sample Final 3

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  boxed problem numbers  to go to a solution.

 Problem 1 

Find each of the following limits if it exists. If you think the limit does not exist provide a reason.

(a)  $\lim _{x\rightarrow 0}{\frac {\sin(5x)}{1-{\sqrt {1-x}}}}$

(b)  $\lim _{x\rightarrow 8}f(x),$  given that  $\lim _{x\rightarrow 8}{\frac {xf(x)}{3}}=-2$

(c)  $\lim _{x\rightarrow -\infty }{\frac {\sqrt {9x^{6}-x}}{3x^{3}+4x}}$

 Problem 2 

Find the derivative of the following functions:

(a)  $g(\theta )={\frac {\pi ^{2}}{(\sec \theta -\sin 2\theta )^{2}}}$

(b)  $y=\cos(3\pi )+\tan ^{-1}({\sqrt {x}})$

 Problem 3 

Find the derivative of the following function using the limit definition of the derivative:

$f(x)=3x-x^{2}$

 Problem 4 

Discuss, without graphing, if the following function is continuous at  $x=0.$

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

If you think  $f$  is not continuous at  $x=0,$  what kind of discontinuity is it?

 Problem 5 

Calculate the equation of the tangent line to the curve defined by  $x^{3}+y^{3}=2xy$  at the point,  $(1,1).$

 Problem 6 

Let

$f(x)=4+8x^{3}-x^{4}$

(a) Over what  $x$-intervals is  $f$  increasing/decreasing?

(b) Find all critical points of  $f$  and test each for local maximum and local minimum.

(c) Over what  $x$-intervals is  $f$  concave up/down?

(d) Sketch the shape of the graph of  $f.$

 Problem 7 

Compute

(a)  $\lim _{x\rightarrow 0}{\frac {x}{3-{\sqrt {9-x}}}}$

(b)  $\lim _{x\rightarrow \pi }{\frac {\sin x}{\pi -x}}$

(c)  $\lim _{x\rightarrow -2}{\frac {x^{2}-x-6}{x^{3}+8}}$

 Problem 8 

If  $W$  denotes the weight in pounds of an individual, and  $t$  denotes the time in months, then  ${\frac {dW}{dt}}$  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.

 Problem 9 

Let

$g(x)=(2x^{2}-8x)^{\frac {2}{3}}$

(a) Find all critical points of  $g$  over the  $x$-interval  $[0,8].$

(b) Find absolute maximum and absolute minimum of  $g$  over  $[0,8].$

 Problem 10 

Let  $y=\tan(x).$

(a) Find the differential  $dy$  of  $y=\tan(x)$  at  $x={\frac {\pi }{4}}.$

(b) Use differentials to find an approximate value for  $\tan(0.885).$  Hint:  ${\frac {\pi }{4}}\approx 0.785.$