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 or letters to go to a solution.

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

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.