009C Sample Midterm 1

This is a sample, and is meant to represent the material usually covered in Math 9C for the midterm. An actual test may or may not be similar. Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

Does the following sequence converge or diverge? If the sequence converges, also find the limit of the sequence.

Be sure to jusify your answers!

$a_{n}={\frac {\ln n}{n}}$

 Problem 2 

Consider the infinite series $\sum _{n=2}^{\infty }2{\bigg (}{\frac {1}{2^{n}}}-{\frac {1}{2^{n+1}}}{\bigg )}.$

a) Find an expression for the $n$th partial sum $s_{n}$ of the series.

b) Compute $\lim _{n\rightarrow \infty }s_{n}.$

 Problem 3 

Determine whether the following series converges absolutely, conditionally or whether it diverges.

Be sure to justify your answers!

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

 Problem 4 

Find the derivatives of the following functions. Do not simplify.

a) $f(x)={\sqrt {x}}(x^{2}+2)$

b) $g(x)={\frac {x+3}{x^{\frac {3}{2}}+2}}$ where $x>0$

c) $h(x)={\frac {e^{-5x^{3}}}{\sqrt {x^{2}+1}}}$

 Problem 5 

The displacement from equilibrium of an object in harmonic motion on the end of a spring is:

$y={\frac {1}{3}}\cos(12t)-{\frac {1}{4}}\sin(12t)$

where $y$ is measured in feet and $t$ is the time in seconds. Determine the position and velocity of the object when $t={\frac {\pi }{8}}.$