# Difference between revisions of "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.

## Limits

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

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)
$h(x)=4x\sin(x)+e(x^{2}+2)^{2}.$ ## Continuity and Differentiability

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) Consider the following 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.

## Implicit Differentiation

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

## Derivatives and Graphing

5. Consider the function   $h(x)={\frac {x^{3}}{3}}-2x^{2}-5x+{\frac {35}{3}}}.$ (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 $f(x)$ .

## Asymptotes

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

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

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

## Related Rates

9. A bug is crawling along the $x$ -axis at a constant speed of   ${\frac {dx}{dt}}=30$ . How fast is the distance between the bug and the point $(3,4)$ 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

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