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ISM_T11_C03_B - Section 3.3 The Derivative as a Rate of...

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Section 3.3 The Derivative as a Rate of Change 139 27. (a) y 6 1 6 1 1 œ œ Ê œ ˆ Š t t t t 1 6 144 dt 12 dy # # (b) The largest value of is 0 m/h when t 12 and the fluid level is falling the slowest at that time. The dy dt œ smallest value of is 1 m/h, when t 0, and the fluid level is falling the fastest at that time. dy dt œ (c) In this situation, 0 the graph of y is dy dt Ÿ Ê always decreasing. As increases in value, dy dt the slope of the graph of y increases from 1 to 0 over the interval 0 t 12. Ÿ Ÿ 28. (a) V r 4 r 4 (2) 16 ft /ft œ Ê œ Ê œ œ 4 dV dV 3 dr dr 1 1 1 1 $ # # $ ¸ r=2 (b) When r 2, 16 so that when r changes by 1 unit, we expect V to change by approximately 16 . œ œ dV dr 1 1 Therefore when r changes by 0.2 units V changes by approximately (16 )(0.2) 3.2 10.05 ft . Note 1 1 œ ¸ $ that V(2.2) V(2) 11.09 ft . ¸ $ 29. 200 km/hr 55 m/sec m/sec, and D t V t. Thus V t t 25 sec. When œ œ œ Ê œ œ Ê œ Ê œ 5 500 10 20 500 20 500 9 9 9 9 9 9 9 # t 25, D (25) m œ œ œ 10 6250 9 9 # 30. s v t 16t v v 32t; v 0 t ; 1900 v t 16t so that t 1900 œ Ê œ œ Ê œ œ œ Ê œ ! ! ! # # # v v 32 32 3 64 v v v (64)(1900) 80 19 ft/sec and, finally, 238 mph. Ê œ œ ¸ ! È È 80 19 ft sec 1 min 1 hr 5280 ft 60 sec 60 min 1 mi È 31. (a) v 0 when t 6.25 sec œ œ (b) v 0 when 0 t 6.25 body moves up; v 0 when 6.25 t 12.5 body moves down Ÿ Ê Ÿ Ê (c) body changes direction at t 6.25 sec œ (d) body speeds up on (6.25 12.5] and slows down on [0 6.25) ß ß (e) The body is moving fastest at the endpoints t 0 and t 12.5 when it is traveling 200 ft/sec. It's œ œ moving slowest at t 6.25 when the speed is 0. œ (f) When t 6.25 the body is s 625 m from the origin and farthest away. œ œ
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140 Chapter 3 Differentiation 32. (a) v 0 when t sec œ œ 3 # (b) v 0 when 0 t 1.5 body moves down; v 0 when 1.5 t 5 body moves up Ÿ Ê Ÿ Ê (c) body changes direction at t sec œ 3 # (d) body speeds up on and slows down on ˆ 3 3 # # ß & (e) body is moving fastest at t 5 when the speed v(5) 7 units/sec; it is moving slowest at œ œ œ k k t when the speed is 0 œ 3 # (f) When t 5 the body is s 12 units from the origin and farthest away. œ œ 33. (a) v 0 when t sec œ œ 6 15 3 È (b) v 0 when t body moves left; v 0 when 0 t or t 4 Ê Ÿ Ÿ 6 15 6 15 6 15 6 15 3 3 3 3 È È È È body moves right Ê (c) body changes direction at t sec œ 6 15 3 È (d) body speeds up on and slows down on 0 . Š Š Š 6 15 6 15 6 15 6 15 3 3 3 3 È È È È ß # ß % ß (e) The body is moving fastest at t 0 and t 4 when it is moving 7 units/sec and slowest at t sec œ œ œ 6 15 3 È (f) When t the body is at position s 6.303 units and farthest from the origin. œ ¸ 6 15 3 È
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Section 3.4 Derivatives of Trigonometric Functions 141 34. (a) v 0 when t œ œ 6 15 3 È (b) v 0 when 0 t or t 4 body is moving left; v 0 when Ÿ Ÿ Ê 6 15 6 15 3 3 È È t body is moving right 6 15 6 15 3 3 È È Ê (c) body changes direction at t sec œ 6 15 3 È (d) body speeds up on and slows down on Š Š Š 6 15 6 15 6 15 6 15 3 3 3 3 È È È È ß # ß % (e) The body is moving fastest at 7 units/sec when t 0 and t 4; it is moving slowest and stationary at œ œ t œ 6 15 3 È (f) When t
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