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PracticeProblems1Fall2012.pdf
PracticeProblems1Fall2012.pdf

PracticeProblems1Fall2012.pdf

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Georgia Institute of Technology

Mathematics

MATH 1502

Blekherman

Fall

Description

Practice Problem for Midterm #1, Math 1502
1. The population of a country grows at a rate proportional to the current population. Numerical
data show that the proportionality factor is 6% per year. Let P(t) be the population of this country
at time t.
(a) Write down the di▯erential equation for P(t).
(b) Find a formula for P(t).
(c) How many years does it take for the population of this country to double? Find the exact
formula.
2. Solve the following initial value problems:
0 2 2
(a) 2ty + t y = t ; y(0) = 4
dy x2
(b) dx = y(1+x ); y(0) = ▯3
3. Find the following limits. Show your work/reasoning!
t ▯1
(a) limt!1 t ▯1
▯ 2=t ▯
(b) limt!1 t e ▯ 1
ex
(c) limx!0 x
▯ ▯
1 x
(d) limx!1 cos x
3. Find the general solution to the following di▯erential equations:
00 0
(a) y ▯ 2y ▯ 3y = 0
(b) y + 8y + 16y = 0
(c) y ▯ 2y + 2y = 0
0 2 cos t
(d) y + ty = t2
4. Determine if the integral converges and, if so, calculate the integral whenever possible. Show
your work/reasoning!
R 1
(a) 1 2dt
0 (1▯t)
p
R 1 ep x
(b) 1 x dx
R 9 1
(c) 0 (x▯1)2=3dx
R 2
(d) 1 x3+1 dx
10 x ▯2
5. Determine if the following series converges or diverges. If the series contains positive and negative
terms determine whether it converges absolutely or conditionally. Show your work/reasoning!
P 1 k +3
(a) k=0 k
3
1 P 1 kpn k
(b) k=1(▯1) k
P 1 1
(c) n=1 (ln n)
P 1 n!
(d) n=1 2
▯ ▯
P 1 k k
(e) k=1 2k+3
P 1 1
(f) n=1 n(n+2)(n+4)
P 1 1
6. a) Is the seriek=1 k +1an overestimate or an underestimate for the following integral:
Z
1 1
dx?
1 x + 1
Explain!
P 1 1
b) How does the series k=2k +1 compare to the same integral? Explain!
c) If

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