# 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. Click on the blue problem numbers to go to a solution.

## Limits

Problem 1. Find the following limits:
(a)   ${\displaystyle \lim _{x\rightarrow 0}{\frac {\tan(3x)}{x^{3}}}.}$

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

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

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

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

## Derivatives

Problem 2. Find the derivatives of the following functions:
(a)  ${\displaystyle f(x)={\frac {3x^{2}-5}{x^{3}-9}}.}$

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

(c)
${\displaystyle h(x)=4x\sin(x)+e(x^{2}+2)^{2}.}$

## Continuity and Differentiability

Problem 3. (Version I) Consider the following function:  ${\displaystyle f(x)={\begin{cases}{\sqrt {x}},&{\mbox{if }}x\geq 1,\\4x^{2}+C,&{\mbox{if }}x<1.\end{cases}}}$
(a) Find a value of  ${\displaystyle C}$ which makes ${\displaystyle f}$ continuous at ${\displaystyle x=1.}$
(b) With your choice of  ${\displaystyle C}$, is ${\displaystyle f}$ differentiable at ${\displaystyle x=1}$?  Use the definition of the derivative to motivate your answer.

Problem 3. (Version II) Consider the following function:  ${\displaystyle g(x)={\begin{cases}{\sqrt {x^{2}+3}},&\quad {\mbox{if }}x\geq 1\\{\frac {1}{4}}x^{2}+C,&\quad {\mbox{if }}x<1.\end{cases}}}$
(a) Find a value of  ${\displaystyle C}$ which makes ${\displaystyle f}$ continuous at ${\displaystyle x=1.}$
(b) With your choice of  ${\displaystyle C}$, is ${\displaystyle f}$ differentiable at ${\displaystyle x=1}$?  Use the definition of the derivative to motivate your answer.

## Implicit Differentiation

Problem 4. Find an equation for the tangent line to the function  ${\displaystyle -x^{3}-2xy+y^{3}=-1}$ at the point ${\displaystyle (1,1)}$.

## Derivatives and Graphing

Problem 5. Consider the function   ${\displaystyle h(x)={\displaystyle {\frac {x^{3}}{3}}-2x^{2}-5x+{\frac {35}{3}}}.}$
(a) Find the intervals where the function is increasing and decreasing.
(b) Find the local maxima and minima.
(c) Find the intervals on which ${\displaystyle f(x)}$ is concave upward and concave downward.
(d) Find all inflection points.
(e) Use the information in the above to sketch the graph of ${\displaystyle f(x)}$.

## Asymptotes

Problem 6. Find the vertical and horizontal asymptotes of the function  ${\displaystyle f(x)={\frac {\sqrt {4x^{2}+3}}{10x-20}}.}$

## Optimization

Problem 7. 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?

## Linear Approximation

Problem 8. (a) Find the linear approximation ${\displaystyle L(x)}$ to the function ${\displaystyle f(x)=\sec x}$ at the point ${\displaystyle x=\pi /3}$.
(b) Use ${\displaystyle L(x)}$ to estimate the value of ${\displaystyle \sec \,(3\pi /7)}$.

## Related Rates

Problem 9. A bug is crawling along the ${\displaystyle x}$-axis at a constant speed of   ${\displaystyle {\frac {dx}{dt}}=30}$. How fast is the distance between the bug and the point ${\displaystyle (3,4)}$ changing when the bug is at the origin? (Note that if the distance is decreasing, then you should have a negative answer).

## Two Important Theorems

Problem 10. Consider the function   ${\displaystyle f(x)=2x^{3}+4x+{\sqrt {2}}.}$
(a) Use the Intermediate Value Theorem to show that ${\displaystyle f(x)}$ has at least one zero.
(b) Use Rolle's Theorem to show that ${\displaystyle f(x)}$ has exactly one zero.