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FDWKCalcSM_ch8 - 350 Section 8.1 51 y = sec 2 x so the area...

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350 Section 8.1 51. ′ = y x sec 2 , so the area is 2 1 2 2 0 4 π π tan sec , x x dx ( ) + ( ) which using NINT evaluates to 3 84 . . 52. x = 1 1 2 1 1 1 2 2 y x y y y and sothearea is ′= − + , π 2 1 2 dy , which using NINT evaluates to 5.02. 53. (a) The two curves intersect at x = 1.2237831. Store this value as A. Area = ( sin sec ) . . 2 1 366 0 + = x x dx A (b) Volume = π ( sin ) (sec ) . . 2 16 404 2 2 0 + ( ) = x x dx A (c) Volume = ( sin sec ) . . 2 1 629 2 0 + = x x dx A 54. (a) Average temp = ° 1 14 6 80 10 12 87 6 14 cos . π t dt F (b) F t t ( ) cos = 80 10 12 78 π for 5 2308694 18 766913 . . . t Store these two values as A and B . (c) Cost = 0 05 80 10 12 78 5 10 . cos . π t dt A B The cost was about $5.10. 55. (a) 15600 24 160 6004 2 9 17 ( ) t t dt + people. (b) 15 15600 24 160 11 15600 24 16 2 9 17 2 ( ) ( t t dt t t + + + 0 104 048 17 23 ) , dt dollars (c) = ≈ − H E L ( ) ( ) ( ) 17 17 17 380 people. H (17) is the number of people in the park at 5:00, and H ( ) 17 is the rate at which the number of people in the park is changing at 5:00. (d) When = = H t E t L t ( ) ( ) ( ) ; 0 that is, at t = 15.795 Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals Section 8.1 Sequences (pp. 435–443) Quick Review 8.1 1. f ( ) 5 5 5 3 5 8 = = + 2. f ( ) = + = − 2 2 2 3 2 3. + = 2 3 1 1 5 1 ( )( . ) 4. + = 7 5 1 3 5 ( )( ) 5. 1 5 2 12 4 1 . ( ) = 6. = − 2 1 5 4 5 3 1 ( . ) . 7. lim lim x x x x x x x x →∞ →∞ + + = = 5 2 3 16 5 3 0 3 2 4 2 3 4 8. lim sin( ) lim x x x x x x = = 0 0 3 3 3 9. lim sin lim x x x x x x →∞ →∞ = = 1 1 1 10. lim . x x x x x x →∞ + + = = 2 1 2 3 2 3 Does not exist, or Section 8.1 Exercises 1. 1 2 2 3 3 4 4 5 5 6 6 7 50 51 , , , , , ; 2. 2 5 2 8 3 11 4 14 5 17 6 149 50 , , , , , ; 3. 2 9 4 64 27 625 256 7776 3125 2 48832 , , , , . , 117649 46656 2 521626 51 50 2 691588 50 . , . 4. 2 2 0 4 10 18 2350 , , , , , , 5. 3 1 1 3 11 , , , ; 6. 2 1 0 1 5 , , , ; 7. 2 4 8 16 256 , , , ; 8. 10 11 12 1 13 31 19 487171 10 1 1 7 , , . , . ; . ( . ) 9. 1 1 2 3 21 , , , ; 10. 3 2 1 1 2 , , , ; 11. (a) 3 (b) a d + = − + = 7 2 7 3 19 ( ) (c) a a n n = + 1 3 (d) a n n n = − + = 2 1 3 3 5 ( )( ) 12. (a) 2 (b) a d + = + = 7 15 7 2 1 ( ) (c) a a n n = 1 2 (d) a n n n = + = − + 15 1 2 2 17 ( )( ) 13. (a) 1 2 (b) a d + = + = 7 1 7 1 2 9 2
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Section 8.1 351 13. Continued (c) a a n n = + 1 1 2 (d) a n n n = + = 1 1 1 2 1 2 ( ) ( ) + 14. (a) 0.1 (b) a d + = + = 7 3 7 0 1 3 7 ( . ) . (c) a a n n = + 1 0 1 . (d) a n n n = + = + 3 1 0 1 0 1 2 9 ( )( . ) . . 15. (a) 1 2 (b) 8 1 2 0 03125 8 = . (c) a a n n = 1 2 1 (d) a n n n = = 8 1 2 2 1 4 16. (a) 1 5 . (b) ( )( . ) . 1 1 5 25 6289 8 (c) a a n n = ( . ) 1 5 1 (d) a n n n = = ( )( . ) ( . ) 1 1 5 1 5 1 1 17. (a) 3 (b) ( ) , = − 3 19 683 9 (c) a a n n = − 3 1 (d) a n n n = − = − ( )( ) ( ) 3 3 3 1 18. (a) 1 (b) ( )( ) 5 1 5 8 = (c) a a n n =− 1 (d) a n n = 5 1 1 ( ) 19.
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