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 following function $f:$

$f(x)=\left\{{\begin{array}{lr}x^{2}&{\text{if }}x<1\\{\sqrt {x}}&{\text{if }}x\geq 1\end{array}}\right.$

a) Find $\lim _{x\rightarrow 1^{-}}f(x).$

b) Find $\lim _{x\rightarrow 1^{+}}f(x).$

c) Find $\lim _{x\rightarrow 1}f(x).$

d) Is $f$ continuous at $x=1?$ Briefly explain.

 Problem 3 

Let $y={\sqrt {3x-5}}.$

a) Use the definition of the derivative to compute ${\frac {dy}{dx}}$ for $y={\sqrt {3x-5}}.$

b) Find the equation of the tangent line to $y={\sqrt {3x-5}}$ at $(2,1).$

 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 $t={\frac {\pi }{8}}.$