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

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

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

 Problem 2 

Consider the following function  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:}

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

(a) Find  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 \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  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 t}   is the time in seconds.

Determine the position and velocity of the object when  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 t=\frac{\pi}{8}.}

Contributions to this page were made by Kayla Murray