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:

$f(x)=\left\{{\begin{array}{lr}x+5&{\text{if }}x<3\\4{\sqrt {x+1}}&{\text{if }}x\geq 3\end{array}}\right.$

a) Show that $f(x)$ is continuous at $x=3$.

b) Using the limit definition of the derivative, and computing the limits from both sides, show that $f(x)$ is differentiable at $x=3$.

 Problem 3 

Find the derivatives of the following functions.

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

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

 Problem 4 

Find the interval of convergence of the following series.

$\sum _{n=0}^{\infty }(-1)^{n}{\frac {(x+2)^{n}}{n^{2}}}$

 Problem 5 

Let

$f(x)=\sum _{n=1}^{\infty }nx^{n}$

a) Find the radius of convergence of the power series.

b) Determine the interval of convergence of the power series.

c) Obtain an explicit formula for the function $f(x)$.

 Problem 6 

Find the Taylor polynomial of degree 4 of $f(x)=\cos ^{2}x$ at $a={\frac {\pi }{4}}$.

 Problem 7 

A curve is given in polar coordinates by

$r=1+\sin \theta$

a) Sketch the curve.

b) Compute $y'={\frac {dy}{dx}}$.

c) Compute $y''={\frac {d^{2}y}{dx^{2}}}$.

 Problem 8 

A curve is given in polar coordinates by

$r=1+\sin 2\theta$

$0\leq \theta \leq 2\pi$

a) Sketch the curve.

b) Find the area enclosed by the curve.

 Problem 9 

A curve is given in polar coordinates by

$r=\theta$

$0\leq \theta \leq 2\pi$

Find the length of the curve.

 Problem 10 

A curve is given in polar parametrically by

$x(t)=3\sin t$

$y(t)=4\cos t$

$0\leq t\leq 2\pi$

a) Sketch the curve.

b) Compute the equation of the tangent line at $t_{0}={\frac {\pi }{4}}$.