MATH 1100: QUANTITATIVE ANALYSIS
FALL 2001 FINAL EXAM
Name:
This test has 9 pages and 9 problems.
Solve any 8 of the 9 problems. No calculators are allowed. You are permitted only
a pencil or pen. Work out everything as far as you can before making decimal
approximations.
1. Completely calculate the reduced form, and then find all solutions to
x + 2y − z = 1
3x + y = 7
2x − y + z = 2.
Date: December 14, 2001.
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MATH 1100: QUANTITATIVE ANALYSIS
FALL 2001 FINAL EXAM
2. Suppose that a business has continuous income stream with an annual
rate of flow at time t given by f (t) = 100. The interest rate is r = 10%,
compounded continuously. Find the capital value of the income stream.
MATH 1100: QUANTITATIVE ANALYSIS
FALL 2001 FINAL EXAM
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3. Find the quantities x and y of two products which you should buy to
maximize utility U = x2 y 2 subject to all three of the budget constraints
x≥0
y≥0
x + 2y ≤ 20.
Hint: these constraints form a triangle. You have to check for critical points
of U inside the triangle, using the first derivatives, check for critical points
on the sides of the triangle using Lagrange multipliers (or perhaps just
common sense), and also check the corners of the triangle. Make sure your
answer is a point (x, y) satisfying all of the budget constraints.
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MATH 1100: QUANTITATIVE ANALYSIS
FALL 2001 FINAL EXAM
4. Suppose that the joint cost function of two products is C(x, y) =
Find the marginal cost with respect to each of x and y.
ln(x+1)
y+1 .
MATH 1100: QUANTITATIVE ANALYSIS
FALL 2001 FINAL EXAM
5. Show that
(
3e−3t
f (t) =
0
is a probability density function.
if t ≥ 0
otherwise
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MATH 1100: QUANTITATIVE ANALYSIS
FALL 2001 FINAL EXAM
6. Using n rectangles of width 1/n whose left shoulder touches the graph of
f (x) = 2x + 1, approximate the area under the graph of f (x) between
x = 0 and x = 1. DO NOT use any other method to solve this problem:
use rectangles. You might need to know that
n
X
1=n
k=1
n
X
k=1
k = n(n + 1)/2.
MATH 1100: QUANTITATIVE ANALYSIS
7. Suppose that
x + exy = y 2 + ln(y).
Find dy/dx when x = 0 and y = 1.
FALL 2001 FINAL EXAM
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MATH 1100: QUANTITATIVE ANALYSIS
FALL 2001 FINAL EXAM
8. Suppose that $2 is invested at a 10% nominal interest rate, compounded
continuously.
(a) Write an equation describing the value of the investment at t years
since the principal was invested.
(b) Approximate the value of the investment after 30 years, using the
approximation e3 ∼ 20.
(c) How many years does it take for the investment to multiply in value
twenty times over?
(d) How many years does it take for the investment to multiply in value
ninety times over? Use the approximation ln(90) ∼ 4.5.
MATH 1100: QUANTITATIVE ANALYSIS
FALL 2001 FINAL EXAM
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9. Suppose that national consumption is $5 billion when disposable income
is zero. Measuring national consumption in billions, suppose that it varies
according to
dC
4y
.
=
dy
C −3
Find the national consumption as a function of disposable income.