MATH 1100: QUANTITATIVE ANALYSIS
TEST #4 (VERSION A)
1. Find the general solution to
4x
dy
=
.
dx
y−3
Solution:
y =3±
p
4x2 + C
2. If consumption is $5 billion when disposable income is 0, and the marginal
propensity to consume is
dC
1
=1+ √
dy
y
find the national consumption function.
Solution:
√
C =5+y+2 y
3. Suppose that world population P (measured in billions of people) never
exceeds 100 and has rate of grow
dP
= K(100 − P ).
dt
Suppose that at time t = 0 there are 1 billion people, and the rate of growth
is
dP
= 90.
dt
Find P as a function of t.
Solution:
dP
= K dt.
100 − P
Now integrate. It is important to distinguish between the dP and the dt.
The integral
Z
dt
100 − P
is not the same as
Z
dP
.
100 − P
The final answer:
P = 100 − 99e−(10/11)t .
Date: November 2, 2001.
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MATH 1100: QUANTITATIVE ANALYSIS
TEST #4 (VERSION A)
4. Calculate the integral
Z
100
x2 + 1
x dx
Solution:
101
x2 + 1
+ C.
202
There is a danger here: don’t imagine that
Z
f (x)g(x) dx
will equal
Z
Z
f (x) dx
g(x) dx .
Instead, try u = x2 + 1, so du = 2x dx. Make sure that your final answer is
in terms of x, not u.
5. If the marginal revenue for a month is M R = x2 + 1, find the total revenue
from sales of 10 units.
Solution:
1030
R=
.
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