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

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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.