Calculus Instructions

INSTRUCTIONS: Whenever we evaluate a definite integral, assume the instructions say: “Evaluate the integral, if it exists”. When evaluating an indefinite integral, you may assume that the function is continuous on its domain. 11. What was the Mean Value Theorem for Integrals? R∞ 12. Evaluate: 0 te−st dt, where s is a positive constant. 13. Write the following limit as nite integral: on the given    n  X πi 2 + πi π 2+ sin · . lim n→∞ n n n i=1 1. Set up an integral for the volume of the solid obtained by rotating the region defined by y = √ x − 1, y = 0 and x = 5 about the y−axis. √ 2. Write the area under y = 3 x, 0 ≤ x ≤ 8 as the limit of a Riemann sum (use right endpoints). 14. If f (x) = 1 + x + x2 + x3 + . . . = 3. Find the volume of the solid obtained by rotating the region bounded by: y = x1 , y = 0, x = 1, x = 3 about y = −1. R5 4. The integral π 2 y dy represented the volume of a solid. Describe the solid. P∞ a defiinterval: n=0 xn , (a) Find the interval on which the sum converges. (b) Viewing this as a geometric series, what is the formula for f ? (c) Find f 0 (x) by differentiating the power series. If you also differentiate your answer to part (b), then you’ll get a formula for the new power series. R (d) Find f (x) dx by integrating the power series. What is the formula for the resulting sum (using the antiderivative of part (b))? 5. Write the appropriate partial fraction expansion for the following expression (do not solve for the constants): 1+x (x − 1)2 (x2 + 1) 6. Let R be the region in the first quadrant bounded by y = x3 and y = 2x − x2 . Calculate: (a) The area of R, (b) Volume obtained by rotating R about the x−axis (c) Volume obtained by rotating R about the y−axis 15. Find the area between the curves y 2 = x and x − 2y = 3. 16. RWrite the following R 7 difference as a single integral: 10 f (x) dx − f (x) dx 2 2 7. Determine a region whose area is equal to the following r limit (do not evaluate the limit): n X 3 3i lim 1+ n→∞ n n i=1 17. If R4 3 18. If 8. What is the derivative of sin−1 (x)? tan−1 (x)? What are their antiderivatives? 9. What is the derivative of e−2x ? The antiderivative of e−3x ? Same questions for sin(3x). R1 R4 f (x)dx = 2, 0 f (x)dx R3 f (x) dx = 1, find 1 f (x) dx. 0 R1 0 19. Compute dF dx , if F (x) = 20. Compute dg dy , if g(y) = 21. Find 10. Suppose you are integrating P (x)/Q(x), where P and Q are polynomials. Explain the process by which we integrate this expression. (Consider the degrees of P and Q). f (x) dx = 31 , what is dy dx , if y = R 5x cos(x) R1 R2 x R √y 3 0 = −6, and 5 − 6f (x) dx? cos(t2 ) dt cos(t) t dt. cos(t2 ) dt
n 22. If an = n+1 , what is the limit of the sen quence? Would the associated series converge? 1 23. Evaluate the limit by recognizing the sum as the Riemann sum to an associated integral: r r r ! 1 2 n 1 lim + + ... + n→∞ n n n n 36. Compute: Z sin(x) d (a) t3 dt dx 3x Z 5 d (b) x3 dx dx 1 Z 5 d 3 (c) x dx dx 1 24. Write the following integral as theZlimit of a Rie5 (1+2x3 ) dx mann sum (use right endpoints): 0 25. Given that R9√ 4 x dx = 38 3 , what is R4√ 9 37. Define the integral of f : t dt? (a) if f is continuous on [a, b] (as a Riemann Sum) x 26. Let f (x) = e on the interval [0, 2]. (a) Find the average value of f . (b) Find c such that favg = c. (b) On the interval [a, b] if f has a vertical asymptote at x = a, but is continuous on (a, b]. 27. The velocity function is v(t) = 3t − 5, 0 ≤ t ≤ 3 (a) Find the displacement, (b) Find the distance traveled. (c) On the interval [a, ∞), if f is continuous there. 28. Exercise 7, pg. 427 (There are some graphs to consider). 38. What is the difference between a sequence, a series, and a power series? 29. Suppose h(1) = −2, h0 (1) = 2, h00 (1) = 3, h(2) = 6, h0 (2) = 5, and h00 (2) = 13, and h00 R2 is continuous. Evaluate 1 h00 (u) du. 39. What does it mean for a infinite series to ”converge”? Be specific in your answer using sn as the sum for k = 1 to n. 30. Find the area between the curves y = |x| and y = x2 − 2. R9 31. If f is continuous and 0 f (x) dx = 4, find R3 xf (x2 ) dx. 0 40. What does it mean (graphically) for a sequence to ”converge”? n! , does the sequence converge? If 41. If an = (n+2)! so, to what does it converge? 32. For each function, find the Taylor series for f (x) centered at the given value of a: 42. If we think of an as f (n), then what is the relaRT PT tionship between n=1 an and 1 f (x) dx? You may assume f is decreasing and positive. Hint: There are two possibilities, where you use either right endpoints or left endpoints in a Riemann sum. (a) f (x) = 1 + x + x2 at a = 2 (b) f (x) = ex at a = 3. (c) f (x) = 1 x at a = 1. 33. Problem 41, p. 439 (Pictures) 43. The function f (x) is given as straight lines going through the points (0, 1), (2, 3), and (5, 0). R5 Compute 0 f (x) dx using geometry. 34. Find a so that half the area under the curve y = 1 x2 lies in the interval [1, a] and half of the area lies in the interval [a, 4]. Practice with Series: Converge (absolute or conditional) or Diverge? 35. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by y = x, y = 4x − x2 , about x = 7. 1. 2 ∞ X n−1 2+n n n=1 n ∞  X 3n 2. 1 + 8n n=1 3. 4. ∞ X 10n n! n=1 ∞ X 2. ∞ X xn n2 n=1 3. ∞ X nn xn n=1 n2 e−n 3 4. ∞ X n2 xn 10n n=1 n=1 5. ∞ X (−1)n−1 √ n−1 n=1 5. ∞ X (3x − 2)n n3n n=1 6. ∞ X k+5 6. ∞ X ∞ X 7. ∞ X (−1)n x2n−1 (2n − 1)! n=1 8. ∞ X (4x + 1)n n2 n=1 5k k=1 n=1 7. (−1)n n (n + 1)(n + 2) n=1 8. ∞ X n2 − 1 3 n + 2n2 + 5 n=1 9. ∞ X cos(n/2) n2 + 4n n=1 √ 10. ∞ X n2 + 1 5n n=1 11. ∞ X 9. Evaluate: Z sec2 (x) (a) dx 1 − tan(x) Z ∞ 1 (b) dx (x + 2)(x + 3) 0 Z (c) ln(x) dx Z dx (d) x ln(x) Z 1 (e) dy y 2 − 4y − 12 Z (f) (1 − t)(2 + t2 ) dt Z √ √ (g) u( u + 3 u) du Z ∞ 1 (h) dx (2x + 1)3 1 Z 3 1 √ dx (i) x 0 Z 4 1/x e (j) dx x2 1 √ k (−1) k=1 k k+5 12. ∞ X (2n)n n2n n=1 13. ∞ X 3n n2 n! n=1 14. ∞ X k −1.7 k=1 Practice with Power Series: Find the interval of convergence. 1. n3 (x − 5)n ∞ X (−1)n xn n+1 n=1 3 (k) Z e−x sin(2x) dx x2 dx (4 − x2 )3/2 Z 1 (m) dx 1 + ex Z tan−1 (x) (n) dx 1 + x2 Z a p (o) x x2 − a2 dx (l) Z (p) Z −a 2 0 dx dx (2x − 3)2 Z sin2 x cos5 x dx Z √ 2 9x − 4 (r) dx x Z 1 √ (s) dx 2 x − 4x Z (t) x4 ln(x) dx Z 2 2x + 3 (u) + dx 3x + 1 x2 + 9 Z (v) x2 cos(3x) dx (q) 4