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Math 1100 Practice Exam 3 1. Be able to define/explain/identify in a picture/draw the following (a) Definite integral (b) Indefinite integral (c) Average value of a function (d) Fundamental Theorem of Calculus (e) Differential equation (f) antiderivative (g) differential (h) separable differential equation (i) improper integral 2. Evaluate the following indefinite integrals. (a) Z 2x(x2 + 3)3 dx (b) Z 4 dx x5 23 November, 2011 (c) Z xe−x dx Z x+1 dx x3 (d) (e) Z x3 − x 2 + 1 dx x−1 (f) Z 3x4 dx (2x5 − 5)4 (g) Z x2 ex dx (h) Z x2 ln xdx (i) 3x2 − 2 dx x3 − 2x Z (j) Z 2 xe2x dx (k) Z x2 x dx +1 (l) Z √ 6 4 xdx (m) Z x3 + 6x + 1dx (n) Z 3e3x dx 3. Evaluate the following definite integrals. (a) 1 Z 8xdx 0 (b) Z 1 e 4 dx x (c) Z 0 2 3 8x2 e−x dx (d) 2 Z e4x dx 0 (e) √ x x − 4dx 6 Z 4 (f) Z 4 9 √ 3 xdx (g) Z 1 3x3 dx 4x4 + 9 0 (h) 2 Z 1 5 dx x3 4. Evaluate the following improper integrals, or show that they diverge. (a) Z 1 ∞ 1 dx x6 (b) Z ∞ 1 1 √ dx x (c) Z −2 −∞ (d) Z x √ dx x2 − 1 ∞ −∞ 4 x3 e−x dx 5. Find the general solution of each differential equation. (a) dy = 3x3 + x2 dx (b) dy x2 = dx y (c) dx = x3 ydy (d) ydx = xdy (e) dx = (x2 y 2 + x2 )dy 6. Find the particular solution to each differential equation. (a) dy x+1 = when x = 1, y = 3 dx xy (b) 2y 2 dx = 3x2 dy when x = 2, y = −1 7. If the marginal cost for a product is M C = 4x + 40 and the total cost of producing 25 units is $3600, what will be the cost of producing 30 units? 8. Find the area between y = 4 and y = 4x − x2 from x = 0 to x = 2. 9. Suppose R3 1 f (x)dx = 6 and R4 3 f (x)dx = −2. Find R4 1 f (x)dx. 10. For what value of c is the function f (x) = c x3 if x ≥ 100 0 otherwise a probability density function? 11. Find the area of the region bounded by each pair of curves. (a) y = 2x and y = x3 − 2x (b) y = x − x2 and y = x2 − 4x 12. Evaluate without taking antiderivatives: (a) Z −6 √ x3 + 1dx = −6 (b) Z 3 9 2x Z x e dx + 2 13. The graph below represents the function f (x). Evaluate the following. (Hint: count squares) R2 (a) −2 f (x)dx (b) R −3 (c) R4 0 1 f (x)dx f (x)dx 3 2 x9 e2x dx = 14. When the interest on an investment is compounded continuously, the investment grows at a rate proportional to the amount of money in the account. If $20,000 is invested at time t = 0 and the amount in the account after 22 years is $280,264 find the function that gives the value of the investment as a function of t. 15. Use three rectangles to approximate the area under the curve f (x) = x3 from x = 0 to x = 3. Use right-hand endpoints to determine the height of each rectangle. 16. Find dy if: (a) y = x3 (b) y = ex (c) y = √ x 17. Find the average value of the given function on the given interval. (a) f (x) = 2x − x2 on [0, 2] (b) y = √ 3 x on [−8, −1] 18. Find the area of the region under the graph of f (x) and to the right of x = 1 when f (x) = x√1 x R1 19. Use four rectangles to approximate −1 x2 + x + 1dx. Evaluate at left-hand endpoints to determine the heights of the rectangles. 20. Suppose that the growth of a population of bacteria satisfies dy = ky where y is the dt number of organisms and t the number of hours. If there are initially 10,000 organisms and the number triples after two hours, how long will it be before the population reaches 100 times its original size?