chapter_5-1

chapter_5-1 - Chapter 5 Integration 5.1...

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184 Chapter 5 Integration 5.1 Antidifferentiation: The Indefinite Integral 2. 1 0 1 x dx x dx C x C == + = + ∫∫ 4. 32 12 2 3 t tdt t dt C + 6. 33 3 xx x ed x x e C + 8. 0.7 0.3 0.7 10 0.7 7 x xd x C x C =+ = + 10. () 23 2 11 2 dx x x dx C C x x −− ⎛⎞ −= ⎜⎟ ⎝⎠ =−+ =− + 12. 13 4/3 5 3 36 39 6 45 d x xC −+ + + 14. 2 121 2 123 2 1 2 ln 6 2 dy y y y yd y y y d y yy C y −−− + + + ∫∫∫ 16. 3 2 1 2 2 1 2 2 2 2 5 x x d x xx x + + + 18. 6 2l n 2 26 l n( l n 2 ) u u u u euu C ++ + + 20. 2 52 2 24 5 x d x x x xd x x d x x C +− + 22. 53 3 2 d y C y += + + 24. 2 2 4 21 44 1 4 d x x d x x x x + + =++ + + 26. 0.02 0.13 0.15 0.02 0.15 0.02 0.15 0.02 4 4 4 0.15 0.02 20 200 3 tt ee d t d t C C + + + 28. 2 2 2 1 1 2 n 2 x dx dx x x x C + = + + + 30. 2 3 2 ln( ) 3 x x x x d x C = +
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Chapter 5. Integration 185 32. x dy e dx = () xx yx e dx e C −− == + (0) 1 3 4 () 4 x yC C e =− + = = =− 34. 1 dy x dx x + = 1/2 3/2 1 2 2 3 x d x x d x C C + = =+ =++ + 16 13 (4) 4 5 33 21 3 2 C x x + = 36. 2 (3 2 ) 3 = =−+ fx f xd x xdx xx C f (0) = 0 0 + C = 1 C = 1 2 () 3 1 =−− x x 38. 2 32 (3 6 2) x xxd x xxx C = −+ (0) 0 0 0 6 6 3 2 6 fC C fx x x x =+−+ = = 40. 2 2 2 x d x x xC = + 2 11 (1) 2 2 22 1 () 2 C x x =+ + = 42. 3 4 3ln 4 x dx x C = ⎛⎞ ⎜⎟ ⎝⎠ + 0 4 0 4 l n 4 4 C x x =−+ = = + 44. 2 23 (4 1.2 ) 20 . 4 = + Rq R qdq qq d q C (20) 800 3,200 30,000 32,400 ( ) 2 0.4 32,400 (40) 10,000 RC C q q R + = = + = Thus $10,000 can be expected from producing 40 units. 46. (a) 2/5 7/5 ( 10) ( 10) 50 7 St t d t t C tC + + (0) 10,000 50 () 10 ,000 7 SC t (b) In two years sales will be approximately 50 (24) 10,000 24 7 9,388.82 dollars. S = (c) The store remains profitable provided 50 ( ) 10,000 8,000 7 280 55.972 t t t so profitability will last just under 56 months.
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186 Chapter 5. Integration 48. 23 12 () 0 .2 ht t t =+ () 53 32 0.2 0.12 0.67 h t d t tt d t C = =++ (0) 2, ( ) 0.12 0.67 2 and (27) 125.16 ft. hCh t t t h == = + + = 50. 0.2 .5 = t Mt e 0.2 0.2 0.2 0 0.5 0.2 5 2 = t t t e d t eC 0.2(2) 0.2(1) 0.4 0.2 (2) (1) 55 22 5 2 0.6761 ⎛⎞ + ⎜⎟ ⎝⎠ =− MM ee The mass increases by 0.6761 grams during the second hour. 52. (a) Let () Pt denote the population of the endangered species. Since the species is growing at 0.03 ( ) 0.51 t e = per year, () is the antiderivative of 0.03 0.51 . t e 0.03 0.03 ( ) 0.51 17 . e d t e C −− + Since 0.03 (0) 500, 500 17 or 517. t Pe C C −+ = 0.3 (10) 517 17 504.41 = so the species will be 504 strong 10 years from now. (b) Writing exercise; answers will vary. 54. ( ) 240 4 R xx 2 (240 4 ) 240 2 Rx R xdx xd x xxC = + 2 Since (0) 0, ( ) 240 2 . (5) 1,150 5 R x x Rp where p is the price per unit. Thus the price per unit is $230. 56. (a) 2/3 1/3 ( ) 200 200 600 QK K dK K C KC = We have (8) 1,200 5,500 QC = so 4,300 C = and ( ) 600 4,300 K . (b) (27) 600(3) 4,300 6,100 Q = units. (c) 600 4,300 7,000 600 2,700 4.5 91.125 K K K K += = = = so $91,125 are needed to produce 7,000 units. 58. Since profit = revenue cost, marginal profit = marginal revenue marginal cost.
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This note was uploaded on 05/20/2011 for the course MAC 2233 taught by Professor Royer during the Spring '08 term at FIU.

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chapter_5-1 - Chapter 5 Integration 5.1...

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