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Practice Problem MATH 1502- Test 1

Practice Problem MATH 1502- Test 1 - Practice Problem for...

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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 4% per year. Let P ( t ) be the population of this country at time t . (a) Write down the differential 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: (a) 2 ty 0 + t 2 y = t 2 , y (0) = 4 (b) dy dx = x 2 y (1+ x 3 ) , y (0) = - 3 3. Find the following limits. Show your work/reasoning! (a) lim t 1 t 3 - 1 t 2 - 1 (b) lim t →∞ t ( e 2 /t - 1 ) (c) lim x 0 e x x (d) lim x →∞ ( cos 1 x ) x 3. Find the general solution to the following differential equations: (a) y 00 - 2 y 0 - 3 y = 0 (b) y 00 + 8 y 0 + 16 y = 0 (c) y 0 + 2 t y = cos t t 2 4. Determine if the integral converges and, if so, calculate the integral. Show your work/reasoning! (a) R 1 0 1 (1 - t ) 2 dt (b) R 1 e - x x dx (c) R 9 0 1 ( x - 1) 2 / 3 dx 5. Determine if the following series converges or diverges. If the series contains positive and negative
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