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Georgia Institute of Technology
MATH 1502

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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