Pre-Calc Homework Solutions 186

Pre-Calc Homework Solutions 186 - 186 dy dt Chapter 4...

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Unformatted text preview: 186 dy dt Chapter 4 Review (3x 2 dx dt dy dt dy dt d (uv) dt dv dt du dt 38. dy dx dx dt dx Since dt 4) 41. (a) 8 6x 2 cm/sec. u v 2 cm/sec, we have 6( 3)2 6(1)2 6(4)2 u(0.05v) 0.09uv 0.09y dy Since dt v(0.04u) (a) (b) (c) dy dt dy dt dy dt 8 8 8 46 cm/sec 2 cm/sec 88 cm/sec (b) 0.09y, the rate of growth of total production is 9% per year. dy dt d (uv) dt u dv dt v du dt 39. (a) The point being plotted would correspond to a point on the edge of the wheel as the wheel turns. (b) One possible answer is 16 t, where t is in seconds. (An arbitrary constant may be added to this expression, and we have assumed counterclockwise motion.) (c) In general, assuming counterclockwise motion: dx dt dy dt d 2 sin 2(sin )(16 ) dt d 2 cos 2(cos )(16 ) 32 dt 4 u(0.03v) v( 0.02u) 0.01uv 0.01y The total production is increasing at the rate of 1% per year. s Chapter 4 Review (pp. 242245) 1. y y x 2 x 1 32 sin cos x ( 1) ( 2 x)(1) At dx dt dy dt : 4 32 sin 32 cos 2 4 16 ( 16 ( 2) 2) 71.086 ft/sec 2 2 x x 2(2 x) 2 4 2 2 2 3x x 4 3 x 71.086 ft/sec At dx dt dy dt : 2 The first derivative has a zero at . 32 0 ft/sec 100.531 ft/sec Critical point value: Endpoint values: x x x 0 ft/sec 32 100.531 ft/sec 2 4 9 6 32 sin 32 cos : 32 sin 32 cos 2 4 3 y 2 y y at x 4 6 9 1.09 4 0 4 , and the global 3 At dx dt dy dt The global maximum value is minimum value is 4 at x 2. 40. (a) One possible answer: x 30 cos , y 40 2. Since y is a cubic function with a positive leading 30 sin coefficient, we have lim y x and lim y x . There are (b) Since the ferris wheel makes one revolution every 10 sec, we may let 0.2 t and we may write x 30 cos 0.2 t, y 40 30 sin 0.2 t. (This assumes that the ferris wheel revolves counterclockwise.) In general: dx dt dy dt no global extrema. 3. y (x 2)(e 1/x )( 2x 3) 2e 1/x 2 2 (e 1/x )(2x) 2 2e 1/x (x 2 1 x x x 1)(x 1) 30(sin 0.2 t)(0.2 ) 30(cos 0.2 t)(0.2 ) 5: 6 sin 6 cos 8: 6 sin 1.6 6 cos 1.6 0 ft/sec 6 ( 1) 6 sin 0.2 t Intervals 6 cos 0.2 t Sign of y Behavior of y Decreasing y 18.850 ft/sec 2 d [2e 1/x ( x 1 dx x 1 1 x 0 0 x 1 x 1 At t dx dt dy dt Increasing x)] ( x 2x 4 1 Decreasing Increasing (2e 1/x )(x (2e 1/x )(x 2 2 2 2 1) 1 2 x)(2e 1/x )( 2x 3) 2x 2 2 At t dx dt dy dt ) 17.927 ft/sec 2e 1/x 2 (x 4 x x4 2) 1.75] 5.825 ft/sec 2 2e 1/x [(x 2 0.5)2 x4 ...
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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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