Math 222-004 – Spring 2014
Practice Final
1. Circle TRUE if the statement is always true and circle FALSE otherwise. If the statement is false,
explain why or give an example.
P
P
P
(a) If
an does not converge and
bn does not converge, then
an b n
does not converge
True
False
True
False
True
False
True
False
(b) The position vectors for the points (1, 4, −1) and (2, 1, 6) are perpendicular.
(c) The integral
Z
7
−7
cos
πx 14
(1 + x2 )3/2 dx is positive.
(d) If a sequence sn is convergent, then the terms sn tend to 0 as n tends
to ∞
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2. The marginal propensity to consume, c, is the percentage of every dollar that consumers will spend
(rather than save). Suppose that the U.S. government estimates that spending will decrease by 900
billion dollars over the next year. In order to compensate for this, the government passes a 780
billion dollar stimulus package. So the government will spend 780 billion dollars. Each dollar spent
is received by someone, who will spend c of that dollar. So in addition to 780 billion dollars, c780
billion dollars are also spent. Of the c780 billion dollars spent, each dollar is received, and c of each
of those dollars is spent, so an additional c2 780billion dollars is spent, and so on.
(a) If c = 0.5, what is the total contribution to spending from the stimulus package?
(b) Given your answer to the previous question, will the stimulus package be sufficient to overcome
the 900 billion dollar decrease in spending?
(c) Suppose that the government measured c and chose the 780 billion dollar stimulus plan to
exactly compensate for the 900 billion dollar drop in spending. What value of c did they
obtain?
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3. Newton’s law of universal gravitation states that the force due to gravity on an object r meters
from the center of the earth changes at a rate proportional to the force times the inverse of the
distance.
(a) Write a differential equation for the force F on an object r meters from the center of the earth.
(b) At the surface of the earth (6, 378, 100 meters from the center), the force due to gravity on an
object of 10kg mass is 980 Newtons. Use this information to solve an initial value problem for
the force on the object. You will have one unknown constant remaining.
(c) What happens to the force due to gravity on this object as r goes to infinity?
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4. Find the radius and interval of convergence for the following series.
∞
X
2n xn
(a)
n!n2
n=0
(b)
∞
X
(−1)n xn
n ln n
n=2
(c)
∞
X
xn
n2 5 n
n=1
4
5. When dealing with the positions of objects in a distant solar system, place the origin at the star at
the center.
(a) If two planets, P1 and P2 , are at points (10, 1, 5) and (−2, 4, 1). Write the position vectors p~1
and p~2 for the two planets.
(b) Solar systems organize themselves into planes. Write an expression for plane containing both
planets and the star.
(c) Comets travel in enormously large elliptic orbits in the universe. On the scale of solar systems,
the trajectory of a comet looks like a line. If the trajectory of a comet near the solar system
described in this problem is given by




4
1
~c(t) =  2  + t  1  ,
1
−2
does the comet ever lie in the same plane as the planets and star?
5
6. Given a function f (t), we can find the average value of the function over the interval [0, x] using
the following expression:
Z
1 x
A(x) =
f (t)dt.
x 0
(a) Find a differential equation satisfied by A(x).
(b) What kind of differential equation did you find in (a)?
(c) Given that A(0) = 1, solve your differential equation. How does your answer compare to the
given A(x)?
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7. A badger, in Peru for a study-abroad semester, discovers a pond hidden high in the Andes mountains. The pond has perfectly circular
p horizontal cross-sections, and its radii r at different depths
y are given by the function r(y) = 36 − y 2 . At the surface, the pond has a radius of 6 meters.
(a) Sketch the pond, clearly labeling your variables. Which direction should you slice, and what
shape do you see?
(b) Set up an integral that gives the total volume of the pond. Evaluate the integral to find the
total volume of the pond.
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