# 007A Sample Midterm 2

This is a sample, and is meant to represent the material usually covered in Math 7A 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  ${\displaystyle \lim _{x\rightarrow 2}{\frac {{\sqrt {x^{2}+12}}-4}{x-2}}}$

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

(c) Evaluate  ${\displaystyle \lim _{x\rightarrow 0}x^{2}\cos {\bigg (}{\frac {1}{x}}{\bigg )}}$

## Problem 2

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

## Problem 3

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

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

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

(c)   ${\displaystyle f(x)={\sqrt {3x^{2}+5x-7}}}$  where  ${\displaystyle x>0}$

## Problem 4

Assume  ${\displaystyle N(t)}$  denotes the size of a population at time  ${\displaystyle t}$  and that  ${\displaystyle N(t)}$  satisfies the equation:

${\displaystyle {\frac {dN}{dt}}=3N{\bigg (}1-{\frac {N}{20}}{\bigg )}.}$

Let  ${\displaystyle f(N)=3N{\bigg (}1-{\frac {N}{20}}{\bigg )},~N\geq 0.}$  Graph  ${\displaystyle f(N)}$  as a function of  ${\displaystyle N}$  and identify all equilibria. That is, all points where  ${\displaystyle {\frac {dN}{dt}}=0.}$

## Problem 5

A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing when 50 (meters) of the string has been let out?