V =! Things to remember: E(p) = - pf'(p)

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1 dx (B) From (2), d!(4x + 5) (5x + 100) Setting x 150 in (1), we get 45, ,000 or !15 ± 55 2 "15 ± and 2 Since 0, 20. Now, for x 150, 20 and dx d -[4(150) + 5(20)]("6) 5(150) + 100(20) , 20-6, we have Thus, the rice is increasing at the rate of $1.53 er month. 31. Volume V πr 2 h, where h thickness of the circular oil slick. Since h 0.1 1, we have: 10 V! 10 R2 Differentiating with resect to t: # d % " dv $ Given: dr dv! 5 & 10 R2 ( '! dr 2R 10! 5 R dr 0.32 when R 500. Therefore, (500)(0.32) 100π(0.32) cubic feet er minute. EXERCISE 4-7 Things to remember: 1. RELATIVE AND PERCENTAGE RATES OF CHANGE The RELATIVE RATE OF CHANGE of a function f(x) is f'(x) f(x). The PERCENTAGE RATE OF CHANGE is 100 f'(x) f(x). 2. ELASTICITY OF DEMAND If rice and demand are related by x f(), then the ELASTICITY OF DEMAND is given by E() - f'() f() 198 CHAPTER 4 ADDITIONAL DERIVATIVE TOPICS

2 3. INTERPRETATION OF ELASTICITY OF DEMAND E() Demand Interretation 0 < E() < 1 Inelastic Demand is not sensitive to changes in rice. A change in rice roduces a smaller change in demand. E() > 1 Elastic Demand is sensitive to changes in rice. A change in rice roduces a larger change in demand. E() 1 Unit A change in rice roduces the same change in demand. 4. REVENUE AND ELASTICITY OF DEMAND If R() f() is the revenue function, then R'() and [1 - E()] always have the same sign. Demand is inelastic [E() < 1, R'() > 0]: A rice increase will increase revenue. A rice decrease will decrease revenue. Demand is elastic [E() > 1, R'() < 0]: A rice increase will decrease revenue. A rice decrease will increase revenue. 1. f(x) 25 f (x) 0 Relative rate of change of f: f "(x) f(x) f(x) 30x f (x) 30 Relative rate of change of f: f "(x) f(x) 30 30x 1 x 5. f(x) 10x f'(x) 10 Relative rate of change of f: f'(x) f(x) 10 10x x f(x) 100x - 0.5x 2 f'(x) x Relative rate of change of f: f'(x) f(x) 100! x 100x! 0.5x 2 9. f(x) 4 + 2e -2x f'(x) -4e -2x Relative rate of change of f: f'(x) f(x)!4e!2x 4 + 2e!2x - 2e!2x 2 + e!2x " e 2x e 2x e 2x EXERCISE

3 11. f(x) 25x + 3x ln x f'(x) ln x ln x Relative rate of change of f: f'(x) f(x) 13. x f() 12, f'() -20 Elasticity of demand: E()!f'() f() ln x 25x + 3x ln x ,000! (A) At 10: E(10) 12,000! , Demand is inelastic (B) At 20: E(20) 12,000! ; unit elasticity ,000 (C) At 30: E(30) 12,000! 9,000 18,000 3,000 6 Demand is elastic. 15. x f() f'() Elasticity of demand: E()!f'() f() ! 2! (A) At 30: E(30) 950! 60! Demand is inelastic (B) At 50: E(50) 950! 100! ; unit elasticity (C) At 70: E(70) 950! 140! Demand is elastic x 30 (A) x 30! , 0 30 (B) f() f'() -200 Elasticity of demand: E()!f'() f() ! ! 10 (C) At 10: E(10) 30! If the rice increases by 10%, the demand will decrease by aroximately 0.5(10%) 5%. 200 CHAPTER 4 ADDITIONAL DERIVATIVE TOPICS

4 25 (D) At 25: E(25) 30! 25 5 If the rice increases by 10%, the demand will decrease by aroximately 5(10%) 50%. 15 (E) At 15: E(15) 30! 15 1 If the rice increases by 10%, the demand will decrease by aroximately 10% x + 60 (A) x 60! , 0 60 (B) R() ( ) (C) f() f'() -50 Elasticity of demand: E()!f'() f() ! 50 (D) Elastic: E() Inelastic: E() 60! > 1 > 60 - > 30, 30 < < 60 60! < 1 < 60 - < 30, 0 < < 30 60! (E) R'() f() [1 - E()] (equation (9)) R'() > 0 if E() < 1; R'() < 0 if E() > 1 Therefore, revenue is increasing for 0 < < 30 and decreasing for 30 < < 60. (F) If $10 and the rice is decreased, revenue will also decrease. (G) If $40 and the rice is decreased, revenue will increase. 21. x f() 10( - 30) 2, 0 30 f'() 20( - 30)![20(! 30)] Elasticity of demand: E() 10(! 30) 2 Elastic: E() - Inelastic: E() -!2! 30 2! 30 > 1-2 < - 30 ( - 30 < 0 reverses inequality) -3 < -30 > 10; 10 < < 30 2! 30 < 1-2 > - 30 ( - 30 < 0 reverses inequality) -3 > -30 < 10; 0 < < 10 EXERCISE

5 23. x f() 144 " 2, 0 72 f'() 1 2 (144-2)-1/2 (-2) Elasticity of demand: E() Elastic: E() Inelastic: E() 144! 2 "1 144 " 2 144! 2 > 1 > > 144 > 48, 48 < < ! 2 < 1 < < 144 < 48, 0 < < x f() 2,500 " f'() 1 2 (2, ) -1/2 (-4)!2 (2, 500! 2 2 ) Elasticity of demand: E() 2,500! ,250! 2 2 Elastic: E() 1,250! 2 > 1 Inelastic: E() 2 > 1, > 1,250 2 > 625 > 25, 25 < < ,250! 2 < 1 2 < 1, < 1,250 2 < 625 < 25, 0 < < CHAPTER 4 ADDITIONAL DERIVATIVE TOPICS

6 27. x f() 20(10 - ) 0 10 R() f() 20(10 - ) R'() Critical value: R'() ; 5 Sign chart for R'(): Test Numbers R'() R() Increasing Decreasing Demand: Inelastic Elastic x R'() 0 200(+) 10!200(!) 29. x f() 40( - 15) R() f() 40( - 15) 2 R'() 40( - 15) (2)( - 15) 40( - 15)[ ] 40( - 15)(3-15) 120( - 15)( - 5) Critical values [in (0, 15)]: 5 Sign chart for R'(): Test Numbers R'() R() Increasing Decreasing Demand: Inelastic Elastic x R'() 0 (+) 10 (!) 31. x f() R() f() R'() / Critical values: R'() Sign chart for R'(): R'() R() Increasing Decreasing Demand: Inelastic Elastic x 2; 4 Test Numbers R'() 0 30(+) 5 (!) EXERCISE

7 33. g(x) x g'(x) -0.1 E(x) - g(x) xg'(x) - 50! 0.1x!0.1x E(200) g(x) 50-2 x 500 x - 1 g'(x) - 1 x E(x) - g(x) xg'(x) - 50 " 2 x # x % & " ( E(400) $ 1 x ' 50 x x f() A -k, A, k ositive constants f'() -Ak -k-1 E()!f'() f() Ak!k A!k k 39. The comany's daily cost is increasing by 1.25(20) $25 er day. 41. x ,000 x f() 2, f'() -400 Elasticity of demand: E() E(2) 2 3 < ,000! 400 5! The demand is inelastic; a rice increase will increase revenue. 43. x + 1, x f() 800-1,000 f'() -1,000 Elasticity of demand: E() 1, ! 1, ! 5 E(0.30) 1.5 4! < 1 The demand is inelastic; a rice decrease will decrease revenue. 45. From Problem 41, R() f() 2, R'() 2, Critical values: R'() 2, R"() -800 Since 2.50 is the only critical value and R"(2.50) -800 < 0, the maximum revenue occurs when the rice $ CHAPTER 4 ADDITIONAL DERIVATIVE TOPICS

8 y 47. f(t) 0.34t , 0 t 50 f'(t) Percentage rate of change: 100 f'(t) f(t) t r(t) ln t r'(t) t Relative rate of change of f(t): f'(t)!3.6 f(t) t 11.3! 3.6 ln t! t! 3.6 t ln t C(t) Relative rate of change in 2002: C(12)! (12)! 3.6(12)ln(12) The relative rate of change for robberies annually er 1,000 oulation age 12 and over is aroximately t CHAPTER 4 REVIEW 1. A(t) 2000e 0.09t A(5) 2000e 0.09(5) 2000e or $ A(10) 2000e 0.09(10) 2000e or $ A(20) 2000e 0.09(20) 2000e , or $12, (4-1) 2. d dx (2 ln x + 3ex ) 2 d dx ln x + 3 d dx ex 2 x + 3ex (4-2) 3. d dx e2x-3 e 2x-3 d (2x - 3) (by the chain rule) dx 2e 2x-3 (4-4) 4. y ln(2x + 7) 1 y' 2x + 7 (2) 2 2x + 7 (by the chain rule) (4-4) 5. y ln u, where u 3 + e x. (A) y ln[3 + e x ] (B) dy dx dy du du dx 1 u (ex ) e x (ex ) e x 3 + e x (4-4) CHAPTER 4 REVIEW 205