009A Sample Final 1

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 

In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.

a) $\lim _{x\rightarrow -3}{\frac {x^{3}-9x}{6+2x}}$

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

c) $\lim _{x\rightarrow -\infty }{\frac {3x}{\sqrt {4x^{2}+x+5}}}$

 Problem 2 

Consider the following piecewise defined function:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=\left\{{\begin{array}{lr}x+5&{\text{if }}x<3\\4{\sqrt {x+1}}&{\text{if }}x\geq 3\end{array}}\right.}

a) Show that $f(x)$ is continuous at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=3} .

b) Using the limit definition of the derivative, and computing the limits from both sides, show that $f(x)$ is differentiable at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=3} .

 Problem 3 

Find the derivatives of the following functions.

a) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=\ln {\bigg (}{\frac {x^{2}-1}{x^{2}+1}}{\bigg )}}

b) $g(x)=2\sin(4x)+4\tan({\sqrt {1+x^{3}}})$

 Problem 4 

If

$y=x^{2}+\cos(\pi (x^{2}+1))$

compute ${\frac {dy}{dx}}$ and find the equation for the tangent line at $x_{0}=1$. You may leave your answers in point-slope form.

 Problem 5 

A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing

when 50 (meters) of the string has been let out?

 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 implicityly by the equation

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=f(x)} .

 Problem 10 

Consider the following continuous function:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=x^{1/3}(x-8)}

defined on the closed, bounded interval Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [-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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [-8,8]} .