Pre-Calc Homework Solutions 446

Pre-Calc Homework Solutions 446 - 446 Cumulative Review 65....

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Unformatted text preview: 446 Cumulative Review 65. Use integration by parts. cos 2x dx 1 sin 2x 2 1 x e sin 2x dx 2 60. Use integration by parts. u du e x e x dv x u du x2 2x dx dv v x 2 cos x sin x dx cos x 2x cos x dx e dx v cos 2x dx 1 x e sin 2x 2 x 2 sin x dx Now let Now let u du Then e x e x dv x e dx v 1 sin 2x dx 2 1 cos 2x 4 u du x dx dv v 2 cos x dx 2 sin x 2x sin x 2x sin x 2 sin x dx 2 cos x C C x 2 sin x dx cos 2x dx 1 x e cos 2x 4 e 5 x x 2 cos x x cos x 2 1 x e sin 2x 2 1 e x cos 2x dx 4 dy dx (2 x 2) cos x 2x sin x so e 61. x x2 x The graph of the slope field of the differential equation cos 2x dx (2 sin 2x cos 2x) A x 1 x B 6 C x 2 sin x and the antiderivative y (2 x 2) cos x 2 5x 6 (x x 2 1)(x 6) 2x sin x is shown below. x 2 A (x B 6) B(x 1, B 6A 1) (A B) x (B 1 ,B 7 6A) 8 so 7 [ 5, 5] by [ 10, 10] Solving A x x2 2 yields A 2 8 1 . Then 5x 6 7(x 6) 7(x 1) x 2 8 1 dx dx x 2 5x 6 7(x 6) 7(x 1) 8 1 1 (x 6)8 ln x 6 ln x 1 C ln 7 7 7 x 1 5 [3 2 66. Use integration by parts. C 359; u du x dx xe x e x dx dv v xe x e x dx ex ex C e x(x 1) C 62. Area Volume 63. y C 64. y y y y (t 2(8.3) 359 1 2(9.9) 8975 ft 3 ... 2(8.3) 3] 25 1) x e x dx 1 2 Confirm by differentiation: 7 and y 2 1 2t e C; y (0) 2 1 2t 7 1 e . 2 2 t 1 1 C 2, so d x [e (x dx 1) C] ex (x 1)e x xe x Ce k(5). 67. (a) y 1 cos 2 sin 2 1 cos 2 sin 2 1 sin 2 cos 4 1 sin 2 cos 4 Ce kt, with 6,000 10,000 6,000 ln 3 3 5 Ce k(2) and 10,000 5 3 C1, and y 1 . 2 1 2 1 2 2 0 Then e k(5 2) , so e 3k and therefore C2, and y 4 2 0 k 0.170. 6,000 e 2k Furthermore, C 4268. The approximate number 4268e 0.170t. of bacteria is given by y (b) About 4268 ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.

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