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

   (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

Problem 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

Problem 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.

Problem 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

Problem 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

Problem 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

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

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 $L(x)$ 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

Problem 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

Problem 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.