Problem Set 56 PROBLEM SET 56 1. (a) 2x + y = 16 y = 16 - 2x V = x 2 y V = x 2 (16 - 2x) V = 16x 2 - 2x 3 Window: ::-::p"lin = [1, ::-::r·ol.3):: = 10 '.,.'p"lin = [1, '.,.'p"la::{ = 2~X1 v '" 151.7037 ft3 max Di . 16 ft 16 ft x 16 ft unensrons: 3 x 3 3 (b) V' = 32x - 6x 2 o = 2x(16 - 3x) x = 0 ~ , 3 V" = 32 - 12x v"C3 6 ) < 0 16 Thus x =- is a maximum" 3 V '" 151.7037 ft3 max Di . 16 ft x 16 ft x 16 ft unensions: 3 3 3 dy = 4x3 dx dt dt dy dt dy dt = 4(216)(2) it 4 = 1728 um S S 3. x(t) = t 3 + 2t 2 - 7t + 4 vet) = 3t 2 + 4t - 7 o = 3t 2 + 4t - 7 o = (3t + 7)(t - 1) 7 t = -"3' 1 120 v 2 7 At rest when 1 = - 3,1 7 Moving left when - 3 < 1 < 1 Moving right when 1 > 1 and when 1 < 7 3 4. h(t) = -16t 2 + 40t + 100 vet) = -32t + 40 o = -32t + 40 t = 1.25 s Maximum height at h(l.25) = 125 ft 5. ITr 2 sin x dx = -2 cos xe Tr/2 = 2 -Tr/2 6. r\1I2 dx = ~x3/2]4 J1 3 1 16 2 - -- 3 3 14 3 8. f~ sin 6 t cos t dt =~ sin 7 1 + C 4 28 f x dx 1 f 4x dx 9. 2x2 + 1 ="4 2x2 + 1 = !. In (2X2 + 1) + C 4 11. f 4 cos (3t) sin 2 (3t) dt = ~ f 3 cos (3t) sin 2 (3t) dt = ~ sirr' (3/) + C 12. f ~os (ax) dx ,}1 + sin (ax) ~f a cos (ax) [1 + sin (ax)]-1/2 dx ~[1 + sin (ax)]112 + C a Calculus, Second Edition
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PROBLEM SET - s3.amazonaws.com file13. y = sin (2x + 1) 2 2 + tanx x + 2 (x2 + 2) cos (2x + 1)(2) (x2 + 2)2 sin (2x + 1)(2x) 2 - (x2 + 2)2 + 2 sec x 2 cos (2x + 1) _ 2x sin (2x + 1)
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