Calculus Volume 1

814

Answer Key

347 . (sin2 x ) 4 x ⎡

⎣ 4· ln(sin2 x )+8 x ·cot2 x ⎤ ⎦

log 2 x

· 2ln x x ln2

349 . x

⎡ ⎣ −csc 2 x · ln x + cot x x

⎤ ⎦

351 . x cot x ·

⎡ ⎣ ⎢ −1

⎤ ⎦ ⎥

2/3

⎛ ⎝ x 2 +3

⎞ ⎠

x

353 . x −1/2

(3 x −4) 4 ·

2 x + 4

+ 12 3 x −4

⎛ ⎝ x 2 +3

⎞ ⎠

3

355 .

y = −1 ⎞ ⎠ 357 . a. x = e ~2.718 b. ( e , ∞), (0, e ) 5+5ln5 x + ⎛ ⎝ 5+ 1 5+5ln5

359 . a. P = 500,000(1.05) t individuals b. P ′( t ) = 24395 · (1.05) t individuals per year c. 39,737 individuals per year 361 . a. At the beginning of 1960 there were 5.3 thousand cases of the disease in New York City. At the beginning of 1963 there were approximately 723 cases of the disease in the United States. b. At the beginning of 1960 the number of cases of the disease was decreasing at rate of −4.611 thousand per year; at the beginning of 1963, the number of cases of the disease was decreasing at a rate of −0.2808 thousand per year. 363 . p = 35741(1.045) t 365 . Years since 1790 P ″

0

69.25

10

107.5

20

167.0

30

259.4

40

402.8

50

625.5

60

971.4

70

1508.5

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