007B Sample Final 1

This is a sample, and is meant to represent the material usually covered in Math 7B 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 

Divide the interval  $[-1,1]$  into four subintervals of equal length  ${\frac {1}{2}}$  and compute the left-endpoint Riemann sum of  $y=1-x^{2}.$

 Problem 2 

Evaluate the following integrals.

(a)  $\int _{0}^{\frac {\sqrt {3}}{4}}{\frac {1}{1+16x^{2}}}~dx$

(b)  $\int {\frac {\sqrt {x+1}}{x}}~dx$

(c)  $\int _{1}^{e}{\frac {\cos(\ln(x))}{x}}~dx$

 Problem 3 

The rate  $r$  at which people get sick during an epidemic of the flu can be approximated by

$r=1600te^{-0.2t}$

where  $r$  is measured in people/day and  $t$  is measured in days since the start of the epidemic.

(a) Sketch a graph of  $r$  as a function of  $t.$

(b) When are people getting sick fastest?

(c) How many people get sick altogether?

 Problem 4 

Find the volume of the solid obtained by rotating about the  $x$-axis the region bounded by  $y={\sqrt {1-x^{2}}}$  and  $y=0.$

 Problem 5 

Find the following integrals.

(a)  $\int x\cos(x)~dx$

(b)  $\int \sin ^{3}(x)\cos ^{2}(x)~dx$

 Problem 6 

Does the following integral converge or diverge? Prove your answer!

$\int _{1}^{\infty }{\frac {\sin ^{2}(x)}{x^{3}}}~dx$

 Problem 7 

Solve the following differential equations:

(a)  ${\frac {dy}{dx}}=3y,$  where  $y_{0}=2$  for  $x_{0}=0$

(b)  ${\frac {dy}{dx}}=(y-1)(y-2)$  where  $y_{0}=0$  for  $x_{0}=0$