009A Sample Final A

This is a sample final, and is meant to represent the material usually covered in Math 9A. Moreover, it contains enough questions to represent a three hour test. An actual test may or may not be similar.

1. Find the following limits:

(a)   $\lim _{x\rightarrow 0}{\frac {\tan(3x)}{x^{3}}}.$

(b)   $\lim _{x\rightarrow -\infty }{\frac {\sqrt {x^{6}+6x^{2}+2}}{x^{3}+x-1}}.$

(c)   $\lim _{x\rightarrow 3}{\frac {x-3}{{\sqrt {x+1}}-2}}.$

(d)   $\lim _{x\rightarrow 3}{\frac {x-1}{{\sqrt {x+1}}-1}}.$

(e)   $\lim _{x\rightarrow \infty }{\frac {5x^{2}-2x+3}{1-3x^{2}}}.$

2. Find the derivatives of the following functions:

(a)   $f(x)={\frac {3x^{2}-5}{x^{3}-9}}.$

(b)   $g(x)=\pi +2\cos({\sqrt {x-2}}).$

(c)   $h(x)=4x\sin(x)+e(x^{2}+2)^{2}.$
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3. (Version I) Consider the following function:

$f(x)={\begin{cases}{\sqrt {x}},&{\mbox{if }}x\geq 1,\\4x^{2}+C,&{\mbox{if }}x<1.\end{cases}}$

(a) Find a value of  $C$ which makes $f$ continuous at $x=1.$

(b) With your choice of  $C$, is $f$ differentiable at $x=1$?  Use the definition of the derivative to motivate your answer.

3. (Version II) Repeat the above for the function:

$g(x)={\begin{cases}{\sqrt {x^{2}+3}},&\quad {\mbox{if }}x\geq 1\\{\frac {1}{4}}x^{2}+C,&\quad {\mbox{if }}x<1.\end{cases}}$

(a) Find a value of  $C$ which makes $f$ continuous at $x=1.$

(b) With your choice of  $C$, is $f$ differentiable at $x=1$?  Use the definition of the derivative to motivate your answer.

4. Use implicit differentiation to find:

an equation for the tangent line to the function  $-x^{3}-2xy+y^{3}=-1$ at the point $(1,1)$.

5. Consider the function:

$h(x)={\displaystyle {\frac {x^{3}}{3}}-2x^{2}-5x+{\frac {35}{3}}}.$
(a) Find the intervals where the function is increasing and decreasing.

(b) Find the local maxima and minima.

(c) Find the intervals on which $f(x)$ is concave upward and concave downward.

(d) Find all inflection points.

(e) Use the information in the above to sketch the graph of $f(x)$.

6. Find the vertical and horizontal asymptotes of:

$f(x)={\frac {\sqrt {4x^{2}+3}}{10x-20}}.$

7. An optimization problem:

A farmer wishes to make 4 identical rectangular pens, each with 500 sq. ft. of area. What dimensions for each pen will use the least amount of total fencing?

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8. Linear Approximation:

(a) Find the linear approximation $L(x)$ to the function $f(x)=\sec x$ at the point $x=\pi /3$.