009A Sample Midterm 2

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 

Evaluate the following limits.

(a) Find  $\lim _{x\rightarrow 2}{\frac {{\sqrt {x^{2}+12}}-4}{x-2}}$

(b) Find  $\lim _{x\rightarrow 0}{\frac {\sin(3x)}{\sin(7x)}}$

(c) Evaluate  $\lim _{x\rightarrow ({\frac {\pi }{2}})^{-}}\tan(x)$

 Problem 2 

The function  $f(x)=3x^{7}-8x+2$  is a polynomial and therefore continuous everywhere.

(a) State the Intermediate Value Theorem.

(b) Use the Intermediate Value Theorem to show that  $f(x)$  has a zero in the interval  $[0,1].$

 Problem 3 

Use the definition of the derivative to find   ${\frac {dy}{dx}}$   for the function  $y={\frac {1+x}{3x}}.$

 Problem 4 

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

(a)   $f(x)=x^{3}(x^{\frac {4}{3}}-1)$

(b)   $g(x)={\frac {x^{3}+x^{-3}}{1+6x}}$ where $x>0$

 Problem 5 

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

(a)   $f(x)=\tan ^{3}(7x^{2}+5)$

(b)   $g(x)=\sin(\cos(e^{x}))$

(c)   $h(x)={\frac {(5x^{2}+7x)^{3}}{\ln(x^{2}+1)}}$