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  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\sqrt{3x-5}.}

(b) Find the equation of the tangent line to  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\sqrt{3x-5}}   at  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2,1).}

 Problem 5 

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

(a)   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\sqrt{x}(x^2+2)}

(b)   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)=\frac{x+3}{x^{\frac{3}{2}}+2}} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>0}

(c)   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x)=\frac{e^{-5x^3}}{\sqrt{x^2+1}}}

Contributions to this page were made by Kayla Murray