MATH-UA 132: MFE II
Spring 2024
Written Homework 9
Due: Wednesday, April 24, 11:59 pm via Gradescope
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MATH-UA.0132 - Written Homework 9
Exercise I: A sneaky integral
We wish to compute, for n = 0, 1, 2, . . . , the integral
In =
Z 1
0
(1 x2)n dx.
1. (3 pts) Compute I0 and I1.
2. (2 pts) Why can we not substitute u = 1 x2 in order to compute In?
3. (3 pts) Compute Z
x(1 x2)n1 dx.
2
I 1 iidx I fci dx
f.d flidx.gl idx
x
x ⼀后吼
1 0
1 1 1in
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du 2ㄨdx
的 ⼼装
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x cannot be eliminated 2variables
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MATH-UA.0132 - Written Homework 9
4. (3 pts) Use the previous question and IBP to express the integralZ 1
0
x2(1 x2)n1 dx
in terms of In. Hint: x2(1 x2)n1 = x⇥ x(1 x2)n1.
5. (3 pts) Deduce a formula for In in terms of In1. Hint:
(1 x2)n = (1 x2)(1 x2)n1 = (1 x2)n1 x2(1 x2)n1.
3
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以 1
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噐 In In 1
In In 1器
MATH-UA.0132 - Written Homework 9
6. (3 pts) Deduce the value of In. You should write it as a big product, like 2⇥ 4⇥ · · ·⇥ (2n).
4
In In 1 笳
I 1
I ⼆⾔ 1x⾔
左⼆ 告 I 1 ⾔ 告
Ii 器 在⼆ 千 Isil x前告⽚
In ⼆哈哈 x_x 器
MATH-UA.0132 - Written Homework 9
Exercise II: Present value
Recall from MFE I that if an amount A is invested and compounded continuously at a rate r > 0
per year, then the amount in the bank after t years is
Aert.
In other words, an amount A obtained now is worth more than an amount A obtained later. Con-
versely, it means that an amount B obtained in t years is worth less today. More precisely, it is
worth
Bert
today. This is called the present (discounted) value of the amount B.
1. (3 pts) Assume that instead of investing an amount A at time 0, you invest a continuous stream
f(t) during the time interval [0, T ]. This means that you invest f(t) at each time t, and more
precisely that you invest an amount f(t) dt during the small time interval [t, t + dt]. Explain
why the present value of this stream is given by the formula
P =
Z T
0
f(t)ert dt.
2. (5 pts) Assume that you invest a constant stream f(t) = 100 for 3 years at a rate r = 2%.
Compute its present value, and give a numerical approximation rounded to the nearest integer.
Is it better to receive this constant flow or the full amount right away?
5
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MATH-UA.0132 - Written Homework 9
3. (3 pts) More generally, if you receive a constant stream f(t) = c for T years at a rate r, what
is its present value?
4. (3 pts) A perpetuity is a constant stream of payments f(t) = c that continues forever. Find
the present value of a perpetuity. You should see that even though a perpetuity allows you to
get the same amount of money forever, it is still not worth an infinite amount of money!
5. (6 pts) Find the present value of an oscillating stream of payments f(t) = 1 + cos t that
continues forever. In other words, computeZ 1
0
(1 + cos t) ert dt.
No need to formally take a limit here: you can pretend FTC 2 applies directly, and use the
fact that
lim
t!1
cos(t)ert = lim
t!1
sin(t)ert = 0
(by squeeze theorem).
6
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