# EVEN03 - CHAPTER 3 Applications of Differentiation Section...

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CHAPTER 3 Applications of Differentiation Section 3.1 Extrema on an Interval . . . . . . . . . . . . . . 378 Section 3.2 Rolle’s Theorem and the Mean Value Theorem . 381 Section 3.3 Increasing and Decreasing Functions and the First Derivative Test . . . . . . . . . . . . . . 387 Section 3.4 Concavity and the Second Derivative Test . . . . 394 Section 3.5 Limits at Infinity . . . . . . . . . . . . . . . . . 402 Section 3.6 A Summary of Curve Sketching . . . . . . . . . 410 Section 3.7 Optimization Problems . . . . . . . . . . . . . . 419 Section 3.8 Newton’s Method . . . . . . . . . . . . . . . . . 429 Section 3.9 Differentials . . . . . . . . . . . . . . . . . . . . 434 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 437 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 445

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CHAPTER 3 Applications of Differentiation Section 3.1 Extrema on an Interval Solutions to Even-Numbered Exercises 378 2. 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 4. 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 ± 1 ! x 1 1 s 2 3 d f s x d 3 x ! x 1 1 6. 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 8. Critical number: neither x 5 0: x 5 0. 10. Critical numbers: neither absolute maximum x 5 5: x 5 2: x 5 2, 5 12. 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 14. 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 16. On critical numbers: u 5 7 6 , 5 11 6 s 0, 2 d , 5 sec 2 s 2 sin 1 1 d 5 sec 3 2 1 sin cos 2 1 1 cos ± 5 sec s 2 tan 1 sec d f s d 5 2 sec tan 1 sec 2 f s d 5 2 sec 1 tan , 0 < < 2
Section 3.1 Extrema on an Interval 379 22. Left endpoint: Critical number: Minimum Right endpoint: Maximum Note: is not in the interval. x 52 2 s 4, 16 d s 2, 2 16 d s 0, 0 d f 9 s x d 5 3 x 2 2 12 5 3 s x 2 2 4 d f s x d 5 x 3 2 12 x , f 0, 4 g 24. Left endpoint: Minimum Critical number: Right endpoint: Maximum s 1, 1 d s 0, 0 d s 2 1, 2 1 d g s x d 5 1 3 x 2 y 3 g s x d 5 3 ! x , f 2 1, 1 g 26. From the graph, you see that is a critical number. Left endpoint: Minimum Right endpoint: Critical number: Maximum s 3, 3 d s 5, 1 d s 2 1, 2 1 d 4 15 4 t 5 3 y 5 3 2 | t 2 3 | , f 2 1, 5 g 28. Left endpoint: Maximum Right endpoint: Minimum 1 5, 5 3 2 s 3, 3 d h s t d 5 2 2 s t 2 2 d 2 h s t d 5 t t 2 2 , f 3, 5 g 30. Left endpoint: Right endpoint: Maximum Critical number: Minimum s 0, 1 d 1 p 3 , 2 2 1 2 6 , 2 ! 3 2 ± 1 2 6 , 1.1547 2 g s x d 5 sec x tan x g s x d 5 sec x , 3 2 6 , 3 ² 32. Left endpoint: Right endpoint: Maximum Critical number: Minimum s 0, 2 3 d s 3, 7.99 d s 2 1, 2 1.5403 d y 5 2 x 2 sin x y 5 x 2 2 2 2 cos x , f 2 1, 3 g 34. (a) Minimum: Maximum: (b) Maximum: (c) Minimum: (d) No extrema s 4, 1 d s 1, 4 d s 1, 4 d s 4, 1 d 36. (a) Minima: and Maximum: (b) Minimum: (c) Maximum: (d) Maximum: s 1, ! 3 d s 0, 2 d s 2 2, 0 d s 0, 2 d s 2, 0 d s 2 2, 0 d 18.

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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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EVEN03 - CHAPTER 3 Applications of Differentiation Section...

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