# 009A Sample Final 2

This is a sample, and is meant to represent the material usually covered in Math 9A for the final. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

## Problem 1

Compute

(a)  ${\displaystyle \lim _{x\rightarrow 4}{\frac {{\sqrt {x+5}}-3}{x-4}}}$

(b)  ${\displaystyle \lim _{x\rightarrow 0}{\frac {\sin ^{2}x}{3x}}}$

(c)  ${\displaystyle \lim _{x\rightarrow -\infty }{\frac {\sqrt {x^{2}+2}}{2x-1}}}$

## Problem 2

Let

${\displaystyle f(x)=\left\{{\begin{array}{lr}{\frac {x^{2}-2x-3}{x-3}}&{\text{if }}x\neq 3\\5&{\text{if }}x=3\end{array}}\right.}$

For what values of  ${\displaystyle x}$  is  ${\displaystyle f}$  continuous?

## Problem 3

Compute   ${\displaystyle {\frac {dy}{dx}}.}$

(a)  ${\displaystyle y={\bigg (}{\frac {x^{2}+3}{x^{2}-1}}{\bigg )}^{3}}$

(b)  ${\displaystyle y=x\cos({\sqrt {x+1}})}$

(c)  ${\displaystyle y=\sin ^{-1}x}$

## Problem 4

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

${\displaystyle 3x^{2}+xy+y^{2}=5}$  at the point  ${\displaystyle (1,-2)}$

## Problem 5

A lighthouse is located on a small island 3 km away from the nearest point  ${\displaystyle P}$  on a straight shoreline and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline on a point 1km away from  ${\displaystyle P?}$

## Problem 6

Find the absolute maximum and absolute minimum values of the function

${\displaystyle f(x)={\frac {1-x}{1+x}}}$

on the interval  ${\displaystyle [0,2].}$

## Problem 7

Show that the equation  ${\displaystyle x^{3}+2x-2=0}$  has exactly one real root.

## Problem 8

Compute

(a)  ${\displaystyle \lim _{x\rightarrow \infty }{\frac {x^{-1}+x}{1+{\sqrt {1+x}}}}}$

(b)  ${\displaystyle \lim _{x\rightarrow 0}{\frac {\sin x}{\cos x-1}}}$

(c)  ${\displaystyle \lim _{x\rightarrow 1}{\frac {x^{3}-1}{x^{10}-1}}}$

## Problem 9

A plane begins its takeoff at 2:00pm on a 2500-mile flight. After 5.5 hours, the plane arrives at its destination. Give a precise mathematical reason using the mean value theorem to explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.

## Problem 10

Let

${\displaystyle f(x)={\frac {4x}{x^{2}+1}}}$

(a) Find all local maximum and local minimum values of  ${\displaystyle f,}$  find all intervals where  ${\displaystyle f}$  is increasing and all intervals where  ${\displaystyle f}$  is decreasing.

(b) Find all inflection points of the function  ${\displaystyle f,}$  find all intervals where the function  ${\displaystyle f}$  is concave upward and all intervals where  ${\displaystyle f}$  is concave downward.

(c) Find all horizontal asymptotes of the graph  ${\displaystyle y=f(x).}$

(d) Sketch the graph of  ${\displaystyle y=f(x).}$