# ODD03 - CHAPTER 3 Applications of Differentiation Section...

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CHAPTER 3 Applications of Differentiation Section 3.1 Extrema on an Interval . . . . . . . . . . . . . . 103 Section 3.2 Rolle’s Theorem and the Mean Value Theorem . 107 Section 3.3 Increasing and Decreasing Functions and the First Derivative Test . . . . . . . . . . . . . . 113 Section 3.4 Concavity and the Second Derivative Test . . . . 121 Section 3.5 Limits at Infinity . . . . . . . . . . . . . . . . . 129 Section 3.6 A Summary of Curve Sketching . . . . . . . . . 136 Section 3.7 Optimization Problems . . . . . . . . . . . . . . 145 Section 3.8 Newton’s Method . . . . . . . . . . . . . . . . . 155 Section 3.9 Differentials . . . . . . . . . . . . . . . . . . . . 160 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 163 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 172

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103 CHAPTER 3 Applications of Differentiation Section 3.1 Extrema on an Interval Solutions to Odd-Numbered Exercises 1. f 9 s 0 d 5 0 f s x d 5 s x 2 1 4 ds 2 x d 2 s x 2 ds 2 x d s x 2 1 4 d 2 5 8 x s x 2 1 4 d 2 f s x d 5 x 2 x 2 1 4 3. f s 3 d 5 1 2 27 3 3 5 1 2 1 5 0 f s x d 5 1 2 27 x 2 3 5 1 2 27 x 3 f s x d 5 x 1 27 2 x 2 5 x 1 27 x x 2 2 5. is undefined. f s 2 2 d f s x d 5 2 3 s x 1 2 d 2 1 y 3 f s x d 5 s x 1 2 d 2 y 3 7. Critical numbers: absolute maximum x 5 2: x 5 2 9. Critical numbers: absolute maximum absolute minimum x 5 2: x 5 1, 3: x 5 1, 2, 3 11. Critical numbers: x 5 0, x 5 2 f s x d 5 3 x 2 2 6 x 5 3 x s x 2 2 d f s x d 5 x 2 s x 2 3 d 5 x 3 2 3 x 2 13. Critical number is t 5 8 3 . 5 8 2 3 t 2 ! 4 2 t 5 1 2 s 4 2 t d 2 1 y 2 f 2 t 1 2 s 4 2 t dg g s t d 5 t 3 1 2 s 4 2 t d 2 1 y 2 s 2 1 d 4 1 s 4 2 t d 1 y 2 g s t d 5 t ! 4 2 t , t < 3 15. On critical numbers: x 5 p 3 , x 5 , x 5 5 3 s 0, 2 d , h s x d 5 2 sin x cos x 2 sin x 5 sin x s 2 cos x 2 1 d h s x d 5 sin 2 x 1 cos x , 0 < x < 2 17. Left endpoint: Maximum Right endpoint: Minimum s 2, 2 d s 2 1, 8 d f s x d 52 2 No critical numbers f s x d 5 2 s 3 2 x d , f 2 1, 2 g 19. Left endpoint: Minimum Critical number: Maximum Right endpoint: Minimum s 3, 0 d s 3 2 , 9 4 d s 0, 0 d f s x d 2 x 1 3 f s x d x 2 1 3 x , f 0, 3 g
21. Left endpoint: Minimum Right endpoint: Maximum Critical number: Critical number: ± 1, 2 1 2 2 s 0, 0 d s 2, 2 d ± 2 1, 2 5 2 2 f 9 s x d 5 3 x 2 2 3 x 5 3 x s x 2 1 d f s x d 5 x 3 2 3 2 x 2 , f 2 1, 2 g 23. Left endpoint: Maximum Critical number: Minimum Right endpoint: s 1, 1 d s 0, 0 d s 2 1, 5 d f s x d 5 2 x 2 1 y 3 2 2 5 2 s 1 2 3 ! x d 3 ! x f s x d 5 3 x 2 y 3 2 2 x , f 2 1, 1 g 25. Left endpoint: Maximum Critical number: Minimum Right endpoint: Maximum ± 1, 1 4 2 s 0, 0 d ± 2 1, 1 4 2 g s t d 5 6 t s t 2 1 3 d 2 g s t d 5 t 2 t 2 1 3 , f 2 1, 1 g 27. Left endpoint: Maximum Right endpoint: Minimum s 1, 2 1 d ± 0, 2 1 2 2 h s s d 5 2 1 s s 2 2 d 2 h s s d 5 1 s 2 2 , f 0, 1 g 29. Left endpoint: Maximum Right endpoint: Minimum ± 1 6 , ! 3 2 2 s 0, 1 d f s x d 52 p sin x f s x d 5 cos x , 3 0, 1 6 4 31. On the interval this equation has no solutions. Thus, there are no critical numbers. Left endpoint: Maximum Right endpoint: Minimum s 2, 3 d s 1, ! 2 1 3 d < s 1, 4.4142 d f 1, 2 g , 8 sec 2 x 8 5 4 x 2 y 5 2 4 x 2 1 8 sec 2 x 8 5 0 y 5 4 x 1 tan x 8 , f 1, 2 g 33. (a) Minimum: Maximum: (b) Minimum: (c) Maximum: (d) No extrema s 2, 1 d s 0, 2 3 d s 2, 1 d s 0, 2 3 d 35.

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## This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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ODD03 - CHAPTER 3 Applications of Differentiation Section...

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