# 007A 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  ${\displaystyle \lim _{x\rightarrow 3}{\bigg (}{\frac {f(x)}{2x}}+1{\bigg )}=2,}$  find  ${\displaystyle \lim _{x\rightarrow 3}f(x).}$

(b) Evaluate  ${\displaystyle \lim _{x\rightarrow 2}{\frac {2-x}{x^{2}-4}}.}$

(c) Find  ${\displaystyle \lim _{x\rightarrow 0}{\frac {\tan(4x)}{\sin(6x)}}.}$

## Problem 2

Suppose the size of a population at time  ${\displaystyle t}$  is given by

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

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

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

## Problem 3

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

(a)  ${\displaystyle f(x)={\frac {(3x-5)(-x^{-2}+4x)}{x^{\frac {4}{5}}}}}$

(b)  ${\displaystyle g(x)={\sqrt {x}}+{\frac {1}{\sqrt {x}}}+{\sqrt {\pi }}}$  for  ${\displaystyle x>0.}$

(c)  ${\displaystyle h(x)={\bigg (}{\frac {3x^{2}}{x+1}}{\bigg )}^{4}}$

## Problem 4

Consider the circle  ${\displaystyle x^{2}+y^{2}=25.}$

(a)  Find  ${\displaystyle {\frac {dy}{dx}}.}$

(b)  Find the equation of the tangent line at the point  ${\displaystyle (4,-3).}$

## Problem 5

At time  ${\displaystyle t,}$  the position of a body moving along the  ${\displaystyle s-}$axis is given by  ${\displaystyle s=t^{3}-6t^{2}+9t}$ (in meters and seconds).

(a)  Find the times when the velocity of the body is equal to  ${\displaystyle 0.}$

(b)  Find the body's acceleration each time the velocity is  ${\displaystyle 0.}$

(c)  Find the total distance traveled by the body from time  ${\displaystyle t=0}$  second to  ${\displaystyle t=2}$  seconds.