007A Sample Final 2

This is a sample, and is meant to represent the material usually covered in Math 7A 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  $(1,-2)$

 Problem 5 

The velocity  $V$  of the blood flow of a skier is modeled by

$V=375(R^{2}-r^{2})$

where  $R$  is the radius of the blood vessel,  $r$  is the distance of the blood flow from the center of the vessel and is a constant. Suppose the skier's blood vessel has radius  $R=0.08$  mm and that cold weather is causing the vessel to contract at a rate of  ${\frac {dR}{dt}}=-0.01$  mm per minute. How fast is the velocity of the blood changing?

 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.

 Problem 10 

Let

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

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

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

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

(d) Sketch the graph of  $y=f(x).$