009A Sample Midterm 3

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

Find the following limits:

a) If $\lim _{x\rightarrow 3}{\bigg (}{\frac {f(x)}{2x}}+1{\bigg )}=2,$ find $\lim _{x\rightarrow 3}f(x).$

b) Find $\lim _{x\rightarrow 0}{\frac {\tan(4x)}{\sin(6x)}}.$

c) Evaluate $\lim _{x\rightarrow \infty }{\frac {-2x^{3}-2x+3}{3x^{3}+3x^{2}-3}}.$

 Problem 2 

The position function $s(t)=-4.9t^{2}+200$ gives the height (in meters) of an object that has fallen from a height of 200 meters. The velocity at time $t=a$ seconds is given by:

$\lim _{t\rightarrow a}{\frac {s(t)-s(a)}{t-a}}$

a) Find the velocity of the object when $t=3$

b) At what velocity will the object impact the ground?

 Problem 3 

Use the definition of the derivative to compute ${\frac {dy}{dx}}$ for $y=3{\sqrt {-2x+5}}.$

 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 $t={\frac {\pi }{8}}.$