009A Sample Final 3

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 

Find each of the following limits if it exists. If you think the limit does not exist provide a reason.

a) $\lim _{x\rightarrow 0}{\frac {\sin(5x)}{1-{\sqrt {1-x}}}}$

b) $\lim _{x\rightarrow 8}f(x),$ given that $\lim _{x\rightarrow 8}{\frac {xf(x)}{3}}=-2$

c) $\lim _{x\rightarrow -\infty }{\frac {\sqrt {9x^{6}-x}}{3x^{3}+4x}}$

 Problem 2 

Find the derivative of the following functions:

a) $g(\theta )={\frac {\pi ^{2}}{(\sec \theta -\sin 2\theta )^{2}}}$

b) $y=\cos(3\pi )+\tan ^{-1}({\sqrt {x}})$

 Problem 3 

Find the derivative of the following function using the limit definition of the derivative:

$f(x)=3x-x^{2}$

 Problem 4 

Discuss, without graphing, if the following function is continuous at $x=0.$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \left\{ \begin{array}{lr} \frac{x}{|x|} & \text{if }x < 0\\ 0 & \text{if }x = 0\\ x-\cos x & \text{if }x > 0 \end{array} \right. }

If you think $f$ is not continuous at $x=0,$ what kind of discontinuity is it?

 Problem 5 

Calculate the equation of the tangent line to the curve defined by $x^{3}+y^{3}=2xy$ at the point, $(1,1).$

 Problem 6 

Let

$f(x)=4+8x^{3}-x^{4}$

a) Over what $x$-intervals is $f$ increasing/decreasing?

b) Find all critical points of $f$ and test each for local maximum and local minimum.

c) Over what $x$-intervals is $f$ concave up/down?

d) Sketch the shape of the graph of $f.$

 Problem 7 

Compute

a) $\lim _{x\rightarrow 0}{\frac {x}{3-{\sqrt {9-x}}}}$

b) $\lim _{x\rightarrow \pi }{\frac {\sin x}{\pi -x}}$

c) $\lim _{x\rightarrow -2}{\frac {x^{2}-x-6}{x^{3}+8}}$

 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 $[-8,8]$.