009C Sample Final 3

This is a sample, and is meant to represent the material usually covered in Math 9C 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 

Which of the following sequences $(a_{n})_{n\geq 1}$ converges? Which diverges? Give reasons for your answers!

a)  $a_{n}={\bigg (}1+{\frac {1}{2n}}{\bigg )}^{n}$

b)  $a_{n}=\cos(n\pi ){\bigg (}{\frac {1+n}{n}}{\bigg )}^{n}$

 Problem 2 

Consider the series

$\sum _{n=2}^{\infty }{\frac {(-1)^{n}}{\sqrt {n}}}.$

a) Test if the series converges absolutely. Give reasons for your answer.

b) Test if the series converges conditionally. Give reasons for your answer.

 Problem 3 

Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test.

$\sum _{n=1}^{\infty }{\frac {n^{3}+7n}{\sqrt {1+n^{10}}}}$

 Problem 4 

Determine if the following series converges or diverges. Please give your reason(s).

a) $\sum _{n=1}^{+\infty }{\frac {n!}{(2n)!}}$

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

 Problem 5 

Consider the function

$f(x)=e^{-{\frac {1}{3}}x}$

a) Find a formula for the $n$th derivative $f^{(n)}(x)$ of $f$ and then find $f'(3).$

b) Find the Taylor series for $f(x)$ at $x_{0}=3,$ i.e. write $f(x)$ in the form

$f(x)=\sum _{n=0}^{\infty }a_{n}(x-3)^{n}.$

 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 $t_{0}={\frac {\pi }{4}}$.