009A Sample Midterm 1

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) Find  $\lim _{x\rightarrow 2}g(x),$  provided that  $\lim _{x\rightarrow 2}{\bigg [}{\frac {4-g(x)}{x}}{\bigg ]}=5.$

(b) Find  $\lim _{x\rightarrow 0}{\frac {\sin(4x)}{5x}}$

(c) Evaluate  $\lim _{x\rightarrow -3^{+}}{\frac {x}{x^{2}-9}}$

 Problem 2 

Suppose the size of a population at time  $t$  is given by

$N(t)={\frac {1000t}{5+t}},~t\geq 0.$

(a) Determine the size of the population as  $t\rightarrow \infty .$  We call this the limiting population size.

(b) Show that at time  $t=5,$  the size of the population is half its limiting size.

 Problem 3 

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 4 

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 5 

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