MATH 1100: QUANTITATIVE ANALYSIS
TEST #5 (VERSION A)
1. Suppose that a business has continuous income stream with an annual rate
of flow at time t given by f (t) = 10000. The interest rate is r = 10%,
compounded continuously. Find the capital value of the income stream.
Solution:
Z ∞
CV =
10000e−0.1t dt = $100000.
0
2. Suppose that
(
ce−kx
f (x) =
0
if x > 0
otherwise
where k > 0. What must the value of c be (in terms of k) so that f (x) is a
probability density function?
Solution: c = k
3. Suppose that the supply function for a product is p = x4 + 1, and the
equilibrium price is $2. Find the producer’s surplus.
Solution: At p = $2 = x4 + 1, we have x = 1.
Z
P S = x1 p1 − p(x) dx
Z 1
= (1)(2) −
x4 + 1 dx
0
= 2 − (1/5 + 1)
= 4/5
= .80
4. Suppose that
(
f (x) =
4
9π
√
9 − x2
0
if 0 < x < 3
otherwise.
Find the mean value of a random variable with f (x) as probability density
function.
Solution:
Z 3 p
4
µ=
x 9 − x2 dx
9π 0
4
=
9
9π
4
= .
π
Date: November 16, 2001.
1
2
MATH 1100: QUANTITATIVE ANALYSIS
TEST #5 (VERSION A)
To do the integral, try u = 9 − x2 , so du = −2x dx.
5. Using n rectangles of width 1/n whose left shoulder touches the graph of
f (x) = 2x, approximate the area under the graph of f (x) between x = 0
and x = 1. You might need to know that
n
X
1=n
k=1
n
X
k = n(n + 1)/2.
k=1
Solution:
The area of all of the rectangles is
1
1− .
n