Odd04

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Unformatted text preview: CHAPTER Integration Section 4.1 Section 4.2 Section 4.3 Section 4.4 Section 4.5 Section 4.6 4 Antiderivatives and Indefinite Integration . . . . . . . . . 177 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Riemann Sums and Definite Integrals . . . . . . . . . . . 188 The Fundamental Theorem of Calculus . . . . . . . . . . 192 Integration by Substitution . . . . . . . . . . . . . . . . . 197 Numerical Integration . . . . . . . . . . . . . . . . . . . 204 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 CHAPTER Integration Section 4.1 4 Antiderivatives and Indefinite Integration Solutions to Odd-Numbered Exercises 1. d3 dx x3 dy dt y C d 3x dx 3 C 9x 4 9 x4 3. d 13 x dx 3 4x C x2 4 x 2x 2 5. 3t2 t3 C C 3t2 7. dy dx y x3 2 d3 t Check: dt 25 x 5 2 C 2 Check: d 25 x dx 5 C x3 2 Given 9. 3 Rewrite x1 3 dx Integrate x4 3 43 x 12 Simplify 34 x 4 3 x dx C C 11. 1 xx dx x 32 dx 12 1x2 22 C 2 x 1 4x2 C 13. 1 dx 2x3 x2 2 3x 1 x 2 3 dx C C 15. x 3 dx d x2 dx 2 3x C C x 3 17. 2x Check: 3x 2 dx d2 x dx x2 x3 x3 C C 2x 3x 2 Check: 19. x3 Check: 2 dx d 14 x dx 4 14 x 4 2x 2x C C x3 2 21. x3 2 2x d 25 x dx 5 1 dx 2 25 x 5 x2 x 2 x2 C x x3 2 C 2x 1 Check: 23. 3 x2 dx x2 3 dx d 35 x dx 5 3 x5 3 53 C x2 3 C 3 35 x 5 x2 3 C 25. 1 dx x3 Check: d dx x 3 dx x 2 2 C C 1 x3 1 2x2 C Check: 1 2x2 177 178 Chapter 4 Integration 27. x2 x x 1 dx 2 x3 23 x 3 2 x1 2 x 2 12 dx x3 2 25 x 5 x1 2 23 x 3 x 2 2x1 x2 2 C x x 1 21 x 15 2 3x2 5x 15 C Check: d 25 x dx 5 2 2x1 C 2 12 29. x 1 3x 2 dx x3 3x2 12 x 2 C x 2 dx 2x 3x2 x C x 1 3x 2 2 31. y2 y dy Check: d 27 y dy 7 y5 2 dy 2 27 y 7 y5 2 C y2 y C 2 Check: d3 x dx 12 x 2 2x 33. dx Check: 1 dx d x dx x C C 1 35. 2 sin x Check: d dx 3 cos x dx 2 cos x 2 cos x 3 sin x C 3 sin x 2 sin x C 3 cos x 37. 1 Check: csc t cot t dt d t dt csc t t C csc t 1 C csc t cot t 39. sec2 Check: sin d tan d d cos tan C cos sec2 C sin 41. tan2 y 1 dy C sec2 y dy sec2 y tan y tan2 y C 1 43. f x y cos x d Check: tan y dy 3 2 C 2C 3 x 2 2 0 2 3 C 45. f x fx 2 2x y 5 4 47. f x C f )x) 2x 2 1 x x3 3 x2 x3 3 x y 4 49. C dy dx y 1 2x 2x 1 x2 2 1, 1, 1 1 dx 1 x 1 x2 C⇒C x C 1 fx f )x) f )x) x3 3 x 3 y f′ 3 f )x) 2x 3 2 x 3 2 1 2 3 3 2 1 3 2 x Answers will vary. f Answers will vary. S ection 4.1 Antiderivatives and Indefinite Integration 179 51. dy dx y 4 y cos x, 0, 4 cos x dx sin 0 sin x sin x C 4 53. (a) Answers will vary. y 5 C⇒C 4 −3 x 5 −3 (b) dy dx y 2 2 y 1 x 2 x2 4 42 4 C x2 4 8t3 8t3 4 2 5t 32 1, 4, 2 x 4 C −4 6 8 −2 C x 2 55. f x fx f0 fx 4x, f 0 4x dx 6 2x 2 20 6 2 6 2x 2 C 6 57. h t ht h1 ht 61. f x f4 f0 5, h 1 5 dt 5 11 2t4 4 5t C 11 C⇒C C⇒C 2t4 x 2 0 x 2 2 59. f x f2 f2 fx f2 fx fx f2 fx 2 5 10 2 dx 4 2x 2x 6 x 2 2x C1 1 fx f4 x 4 C2 fx fx f0 fx 0 32 dx C1 3 12 2x 12 C1 3 2 x C1 C1 1 5 ⇒ C1 2 ⇒ C1 1 dx x2 2 x 2x 0 4x1 2 C2 x 10 ⇒ C2 4 3 dx 4x1 2 3x C2 C2 3x 0 ⇒ C2 4x 0 3x 63. (a) h t h0 ht (b) h 6 0 1.5t 0 5 dt C 5t 2 0.75t 2 12 ⇒ C 12 5t 12 C 0.75t2 0.75 6 56 12 69 cm 180 Chapter 4 Integration 65. f 0 (a) f 4 4. Graph of f is given. 1.0 (f) f is a minimum at x (g) 6 4 2 x −2 3. (b) No. The slopes of the tangent lines are greater than 2 on 0, 2 . Therefore, f must increase more than 4 units on 0, 4 . (c) No, f 5 < f 4 because f is decreasing on 4, 5 . (d) f is an maximum at x 3.5 because f 3.5 the first derivative test. 0 and y 2 4 6 8 (e) f is concave upward when f is increasing on ,1 and 5, . f is conca...
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This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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