# 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

Sketch the graph of  $f.$ At each point of discontinuity, state whether  $f$ is left or right continuous.

$f(x)={\begin{array}{cc}{\Bigg \{}&{\begin{array}{cc}x^{3}+1&x\leq 0\\-x+1&0 ## Problem 3

Let  $y=3{\sqrt {2x+5}},x\geq 0.$ (a) Use the definition of the derivative to compute   ${\frac {dy}{dx}}.$ (b) Find the equation of the tangent line to  $y=3{\sqrt {2x+5}}$ at  $(2,9).$ ## Problem 4

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

(a)  $f(x)={\frac {(3x-5)(-x^{-2}+4x)}{x^{\frac {4}{5}}}}$ (b)  $g(x)={\sqrt {x}}+{\frac {1}{\sqrt {x}}}+{\sqrt {\pi }}$ for  $x>0.$ ## Problem 5

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

(a)  $f(x)=\sin {\bigg (}{\frac {x^{-3}}{e^{-x}}}{\bigg )}$ (b)  $g(x)={\sqrt {\frac {x^{2}+2}{x^{2}+4}}}$ (c)  $h(x)=(x+\cos ^{2}x)^{8}$ 