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a151q5wi - angle of the elevation of the Sun is 30 ◦ and...

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MATH 151 Answers to Assignment 5 Autumn 2009 1. The side of a cube is reported as 20 ± 0 . 05 mm. Use di ff erentials to estimate the percentange error in the volume of the cube. V = x 3 ; dV = 3 x 2 dx . Here x = 20 and dx = 0 . 05. So dV V = 3 · 20 2 · 0 . 05 20 3 . Answer : 0 . 75% 2. Find d 2 dx 2 ln( e x + e - x ). Simplify your answer as much as possible. d dx ln( e x + e - x ) = e x + e - x ( - 1) e x + e - x d 2 dx 2 ln( e x + e - x ) = d dx e x - e - x e x + e - x = ( e x - e - x ( - 1))( e x + e - x ) - ( e x - e - x )( e x + e - x ( - 1)) ( e x + e - x ) 2 = ( e 2 x + 1 + 1 + e - 2 x ) - ( e 2 x - 1 - 1 + e - 2 x ) ( e x + e - x ) 2 Answer : 4 ( e x + e - x ) 2 3. Use logarithmic di ff erentiation to find d dx (1 - sin x ) 3 x . Let y = (1 - sin x ) 3 x . Then ln y = 3 x ln(1 - sin x ). Di ff erentiating both sides gives y y = 3 ln(1 - sin x ) + 3 x - cos x 1 - sin x Answer : (1 - sin x ) 3 x 3 ln(1 + sin x ) - 3 x cos x 1+sin x 4. Find d dt ln 3 (1 + sin 2 t ). Answer : 3 ln 2 (1 + sin 2 t ) 2 sin t cos t 1 + sin 2 t (over) Typeset using A M S -T E X.
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MATH 151 Answers to Assignment 5, p . 2 Autumn 2009 5.
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Unformatted text preview: angle of the elevation of the Sun is 30 ◦ and is increasing at 0 . 1 ◦ per minute. Let the length of the shadow of the tower be x feet and the angle of elevation of the Sun θ (in radians). dθ dt = π 180 . 1 = π 1800 per minute. We need to Fnd dx dt ± ± ± ± θ = π/ 6 . 200 x = tan θ ; x = 200 tan θ = 200 cot θ . dx dt =-200 csc 2 θ dθ dt . dx dt ± ± ± ± θ = π/ 6 =-200 ft csc 2 π 6 × π 1800 1 min =-² 1 2 ³ 2 π 9 12 in min A C B θ x 200feet Answer : π 3 inch per minute end...
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