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)

\lim _{x\rightarrow 4}{\frac {{\sqrt {x+5}}-3}{x-4}}

b)

\lim _{x\rightarrow 0}{\frac {\sin ^{2}x}{3x}}

c)

\lim _{x\rightarrow -\infty }{\frac {\sqrt {x^{2}+2}}{2x-1}}

Problem 2

Let

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 $x$ is $f$ continuous?

Problem 3

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

a)

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

b)

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

c)

y=\sin ^{-1}x

Problem 4

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

3x^{2}+xy+y^{2}=5

at the point

(1,-2)

A lighthouse is located on a small island 3km away from the nearest point $P$ 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 $P?$

Problem 6

Find the absolute maximum and absolute minimum values of the function

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

on the interval $[0,2].$

Problem 7

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

Problem 8

Compute

a)

\lim _{x\rightarrow \infty }{\frac {x^{-1}+x}{1+{\sqrt {1+x}}}}

b)

\lim _{x\rightarrow 0}{\frac {\sin x}{\cos x-1}}

c)

\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.