8 Problems on the Differential Equations - Review Sheet | MATH 247, Study notes of Mathematics

Download 8 Problems on the Differential Equations - Review Sheet | MATH 247 and more Study notes Mathematics in PDF only on Docsity! Math 247 Review for Final 1. Let F (x) = ∫ 10 x2 sin(t2 + 1) dt. Compute F ′(x). 2. Compute the average value of f(x) = ln(x) over the interval [1, e2]. 3. Compute the area between the curves y = x2 + 1 and y = x+ 3. 4. Evaluate the integrals: (you may want to use the table on pages 333–334) a) ∫ cos ( ln(3x) ) dx b) ∫ 1√ x2 + 7 dx c) ∫ 6xe−x 2 dx d) ∫ 1 0 x2 ( 1− x3 )7 dx e) ∫ π/2 0 cos(x) 1 + sin2(x) dx f) ∫ x2 sin(x) dx g) ∫ xe−5x dx 5. Compute the value of the improper integrals: a) ∫ ∞ 0 e−2x dx b) ∫ 2 0 1 (x− 1)2/3 dx 6. Use comparison to show the convergence/divergence of the following improper integrals: a) ∫ ∞ 1 1√ x+ x4 dx b) ∫ 1 0 1√ x+ x4 dx c) ∫ ∞ 2 1√ x− 1 dx 7. Solve the following differential equations: a) dy dx = 2xy, y(0) = 7 b) dy dx = y3 8. Suppose that dy dx = y(2− y)(y − 5). Find the equilibria and determine their stability.