009A Sample Final 2

This is a sample, and is meant to represent the material usually covered in Math 9A for the final. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

Compute

a) $\lim _{x\rightarrow 4}{\frac {{\sqrt {x+5}}-3}{x-4}}$

b) $\lim _{x\rightarrow 0}{\frac {\sin ^{2}x}{3x}}$

c) $\lim _{x\rightarrow -\infty }{\frac {\sqrt {x^{2}+2}}{2x-1}}$

 Problem 2 

Let

$f(x)=\left\{{\begin{array}{lr}{\frac {x^{2}-2x-3}{x-3}}&{\text{if }}x\neq 3\\5&{\text{if }}x=3\end{array}}\right.$

For what values of $x$ is $f$ continuous?

 Problem 3 

Compute ${\frac {dy}{dx}}.$

a) $y={\bigg (}{\frac {x^{2}+3}{x^{2}-1}}{\bigg )}^{3}$

b) $y=x\cos({\sqrt {x+1}})$

c) $y=\sin ^{-1}x$

 Problem 4 

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

$3x^{2}+xy+y^{2}=5$ at the point $(1,-2)$

 Problem 5 

A lighthouse is located on a small island 3km away from the nearest point $P$ on a straight shoreline and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline on a point 1km away from $P?$

 Problem 6 

Consider the following function:

$f(x)=3x-2\sin x+7$

a) Use the Intermediate Value Theorem to show that $f(x)$ has at least one zero.

b) Use the Mean Value Theorem to show that $f(x)$ has at most one zero.

 Problem 7 

A curve is defined implicitly by the equation

$x^{3}+y^{3}=6xy.$

a) Using implicit differentiation, compute  ${\frac {dy}{dx}}$.

b) Find an equation of the tangent line to the curve $x^{3}+y^{3}=6xy$ at the point $(3,3)$.

 Problem 8 

Let

$y=x^{3}.$

a) Find the differential $dy$ of $y=x^{3}$ at $x=2$.

b) Use differentials to find an approximate value for $1.9^{3}$.

 Problem 9 

Given the function $f(x)=x^{3}-6x^{2}+5$,

a) Find the intervals in which the function increases or decreases.

b) Find the local maximum and local minimum values.

c) Find the intervals in which the function concaves upward or concaves downward.

d) Find the inflection point(s).

e) Use the above information (a) to (d) to sketch the graph of $y=f(x)$.

 Problem 10 

Consider the following continuous function:

$f(x)=x^{1/3}(x-8)$

defined on the closed, bounded interval $[-8,8]$.

a) Find all the critical points for $f(x)$.

b) Determine the absolute maximum and absolute minimum values for $f(x)$ on the interval $[-8,8]$.