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 

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure  $P$  and volume  $V$  satisfy the equation  $PV=C$  where  $C$  is a constant. Suppose that at a certain instant, the volume is  $600{\text{ cm}}^{3},$  the pressure is  $150{\text{ kPa}},$  and the pressure is increasing at a rate of  $20{\text{ kPa/min}}.$  At what rate is the volume decreasing at this instant?

 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:  ${\frac {\pi }{4}}\approx 0.785.$