Exam #4
Math 115
Winter 2012
Name: ________________________________________ This is a closed book exam. You may
use a calculator and the formulas handed out with the exam.. Show all work and explain any
reasoning which is not clear from the computations. (This is particularly important if I am to be
able to give you part credit if you make a mistake.) Don’t crowd your work, i.e. make it as easy
as possible for me to follow your work. Turn in this exam along with your answers. However,
don't write your answers on the exam itself; leave them on the pages with your work. Also turn
in the formulas; put them on the formula pile.
1. (15 points) A piece of wire 10 inches long is bent into two pieces. One piece is bent into a
square and the other piece is bent into a circle. How should the wire be cut so that the total
area enclosed by both the square and the circle is a minimum.
2. (15 points) Consider the population P of the United States. Let R denote the instantaneous
growth rate of the U.S. population. Suppose R is given by
R = 1,000,000 + 3,000 t + 10,000 sin t
where t is the time in years measured from Jan 1, 2000 and R is measured in people per year.
(So t = ½ corresponds to July 1, 2000 and t = 2 corresponds to Jan 1, 2002.) Suppose the
population on Jan 1, 2001 is 170,000,000. Find a formula for P in terms of t. Recall that the
instantaneous growth rate R of the population is the rate of change of P with respect to t.
3.
a.
(7 points) Write the following sum using sigma notation.
4
9 16 25  81 100
+ +
+ +
+ +
20 18 16 14
6
4
12
b. (8 points) Write the sum
 303 +- 2kk in expanded form (like in part a) showing the
k=3
values of the first three terms and last two terms. It is not necessary to find the value of
sum. Just indicate what the values of the first three and last two terms are.
5
2
4. (25 points) Find 
 (2 - x ) dx using the definition of the integral as a limit of approximating
1
sums. The formula sheet should be helpful. NOTE: You can not do this by finding an
antiderivative.
4
5. (15 points) Find 
 [

1
+ x x ] dx
x
1
x
6. (15 points) Let f(x) = x 

2
2
x
df
1 + t dt. Find dx.. Your answer will contain 

2
2
1 + t2 dt.