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 

Consider the power series

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

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

(b) Find the interval of convergence of the above power series.

(c) Find the closed formula for the function  $f(x)$  to which the power series converges.

(d) Does the series

$\sum _{n=0}^{\infty }{\frac {1}{(n+1)3^{n+1}}}$

converge?

 Problem 7 

A curve is given in polar coordinates by

$r=1+\cos ^{2}(2\theta ).$

(a) Show that the point with Cartesian coordinates  $(x,y)={\bigg (}{\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}{\bigg )}$  belongs to the curve.

(b) Sketch the curve.

(c) In Cartesian coordinates, find the equation of the tangent line at  ${\bigg (}{\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}{\bigg )}.$

 Problem 8 

A curve is given in polar coordinates by  $r=4+3\sin \theta .$

(a) Sketch the curve.

(b) Find the area enclosed by the curve.

 Problem 9 

A wheel of radius 1 rolls along a straight line, say the  $x$-axis. A point  $P$  is located halfway between the center of the wheel and the rim; assume  $P$  starts at the point  ${\bigg (}0,{\frac {1}{2}}{\bigg )}.$  As the wheel rolls,  $P$  traces a curve. Find parametric equations for the curve.

 Problem 10 

A curve is described parametrically by

$x=t^{2}$

$y=t^{3}-t$

(a) Sketch the curve for  $-2\leq t\leq 2.$

(b) Find the equation of the tangent line to the curve at the origin.