# 04 - CHAPTER 4 Applications of Differentiation Section 4.1...

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CHAPTER 4 Applications of Differentiation Section 4.1 Extrema on an Interval . . . . . . . . . . . . . . 248 Section 4.2 Rolle’s Theorem and the Mean Value Theorem . . 256 Section 4.3 Increasing and Decreasing Functions and the First Derivative Test . . . . . . . . . . . . . . 268 Section 4.4 Concavity and the Second Derivative Test . . . . 289 Section 4.5 Limits at Infinity . . . . . . . . . . . . . . . . . 306 Section 4.6 A Summary of Curve Sketching . . . . . . . . . 322 Section 4.7 Optimization Problems . . . . . . . . . . . . . . 343 Section 4.8 Differentials . . . . . . . . . . . . . . . . . . . . 363 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 369 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 387

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CHAPTER 4 Applications of Differentiation Section 4.1 Extrema on an Interval 248 1. none absolute maximum (and relative maximum) none none relative maximum relative minimum none G : F : E : D : C : B : A : 2. absolute minimum relative maximum none relative minimum relative maximum relative minimum none G : F : E : D : C : B : A : 4. f 9 s 2 d 5 0 f s 0 d 5 0 f s x d 52 p 2 sin x 2 f s x d 5 cos x 2 6. f 1 2 2 3 2 5 0 3 2 s x 1 1 d 2 1 y 2 s 3 x 1 2 d 3 2 s x 1 1 d 2 1 y 2 f x 1 2 s x 1 1 dg f s x d 3 x 3 1 2 s x 1 1 d 2 1 y 2 4 1 ± x 1 1 s 2 3 d f s x d 3 x ± x 1 1 3. f 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 5. 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 2 x 2 2 8. Using the limit definition of the derivative, does not exist, since the one-sided derivatives are not equal. f s 0 d lim x 0 1 f s x d 2 f s 0 d x 2 0 5 lim x 0 1 s 4 2 | x | d 2 4 x 2 0 1 lim x 0 2 f s x d 2 f s 0 d x 2 0 5 lim x 0 2 s 4 2 | x | d 2 4 x 5 1 10. Critical number: none x 5 0: x 5 0 7. 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 9. Critical number: absolute maximum x 5 2: x 5 2 11. Critical numbers: absolute maximum absolute minimum x 5 2: x 5 1, 3: x 5 1, 2, 3
Section 4.1 Extrema on an Interval 249 12. Critical numbers: none absolute maximum x 5 5: x 5 2: x 5 2, 5 13. Critical numbers: x 5 0, x 5 2 f 9 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 14. Critical numbers: x 5 0, x 5 ± ± 2 g s x d 5 4 x 3 2 8 x 5 4 x s x 2 2 2 d g s x d 5 x 2 s x 2 2 4 d 5 x 4 2 4 x 2 15. 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 17. 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 19. Critical number: x 5 0 5 2 x ln 2 3 ln s x 2 1 1 d 1 x 2 x 2 1 1 4 5 0 x 5 0 f s x d 5 2 x ln s x 2 1 1 d ln 2 1 x 2 2 x ln 2 s x 2 1 1 d f s x d 5 x 2 log 2 s x 2 1 1 d 5 x 2 ln s x 2 1 1 d ln 2 21. 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 16. Critical numbers: x 5 ± 1 f s x d 5 s x 2 1 1 ds 4 d 2 s 4 x ds 2 x d s x 2 1 1 d 2 5 4 s 1 2 x 2 d s x 2 1 1 d 2 f s x d 5 4 x x 2 1 1 18.

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## This note was uploaded on 11/13/2010 for the course MATH MAT 231 taught by Professor Thurber during the Spring '08 term at Thomas Edison State.

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04 - CHAPTER 4 Applications of Differentiation Section 4.1...

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