ODD01 - CHAPTER 1 Limits and Their Properties Section 1.1...

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CHAPTER 1 Limits and Their Properties Section 1.1 A Preview of Calculus . . . . . . . . . . . . . . . . . . . . 27 Section 1.2 Finding Limits Graphically and Numerically . . . . . . . . 27 Section 1.3 Evaluating Limits Analytically . . . . . . . . . . . . . . . 31 Section 1.4 Continuity and One-Sided Limits . . . . . . . . . . . . . . 37 Section 1.5 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . 42 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
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27 CHAPTER 1 Limits and Their Properties Section 1.1 A Preview of Calculus Solutions to Odd-Numbered Exercises 1. Precalculus: s 20 ft y sec ds 15 seconds d 5 300 feet 3. Calculus required: slope of tangent line at is rate of change, and equals about 0.16. x 5 2 5. Precalculus: sq. units Area 5 1 2 bh 5 1 2 s 5 ds 3 d 5 15 2 7. Precalculus: cubic units Volume 5 s 2 ds 4 ds 3 d 5 24 9. (a) (b) The graphs of are approximations to the tangent line to at (c) The slope is approximately 2. For a better approximation make the list numbers smaller: H 0.2, 0.1, 0.01, 0.001 J x 5 1. y 1 y 2 8 2 4 6 (1, 3) 11. (a) (b) (c) Increase the number of line segments. < 2.693 1 1.302 1 1.083 1 1.031 < 6.11 D 2 5 ! 1 1 s 5 2 d 2 1 ! 1 1 s 5 2 2 5 3 d 2 1 ! 1 1 s 5 3 2 5 4 d 2 1 ! 1 1 s 5 4 2 1 d 2 D 1 5 ! s 5 2 1 d 2 1 s 1 2 5 d 2 5 ! 16 1 16 < 5.66 Section 1.2 Finding Limits Graphically and Numerically 1. s Actual limit is 1 3 . d lim x 2 x 2 2 x 2 2 x 2 2 < 0.3333 x 1.9 1.99 1.999 2.001 2.01 2.1 0.3448 0.3344 0.3334 0.3332 0.3322 0.3226 f s x d 3. Actual limit is d 1 y s 2 ! 3 d . s lim x 0 ! x 1 3 2 ! 3 x < 0.2887 x 0.1 0.01 0.001 0.001 0.01 0.1 0.2911 0.2889 0.2887 0.2887 0.2884 0.2863 f s x d 2 2 2
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5. Actual limit is d 2 1 16 . s lim x 3 f 1 y s x 1 1 2 s 1 y 4 d x 2 3 < 2 0.0625 x 2.9 2.99 2.999 3.001 3.01 3.1 2 0.0610 2 0.0623 2 0.0625 2 0.0625 2 0.0627 2 0.0641 f s x d 7. (Actual limit is 1.) (Make sure you use radian mode.) lim x 0 sin x x < 1.0000 x 0.1 0.01 0.001 0.001 0.01 0.1 0.9983 0.99998 1.0000 1.0000 0.99998 0.9983 f s x d 2 2 2 9. lim x 3 s 4 2 x d 5 1 11. lim x 2 f s x d 5 lim x 2 s 4 2 x d 5 2 15. tan x does not exist since the function increases and decreases without bound as x approaches p y 2. lim x y 2 17. does not exist since the function oscillates between and 1 as x approaches 0. 2 1 lim x 0 cos s 1 y x d (b) (c) does not exist. The values of C jump from 1.75 to 2.25 at t 5 3. lim t 3 C s t d lim t 3.5 C s t d 5 2.25 t 2 2.5 2.9 3 3.1 3.5 4 C 1.25 1.75 1.75 1.75 2.25 2.25 2.25 19. (a) 0 0 5 3 C s t d 5 0.75 2 0.50 v 2 s t 2 1 d b t 3 3.3 3.4 3.5 3.6 3.7 4 C 1.75 2.25 2.25 2.25 2.25 2.25 2.25 21. You need to find such that implies That is, 1 9 > x 2 1 > 2 1 11 . 10 9 2 1 > x 2 1 > 10 11 2 1 10 9 > x > 10 11 9 10 < 1 x < 11 10 1 2 0.1 < 1 x < 1 1 0.1 2 0.1 < 1 x 2 1 < 0.1 ² f s x d 2 1 ² 5 ² 1 x 2 1 ² < 0.1. 0 < ² x 2 1 ² < d So take Then implies Using the first series of equivalent inequalities, you obtain ² f s x d 2 1 ² 5 ² 1 x 2 1 ² < e < 0.1. 2 1 11 < x 2 1 < 1 9 . 2 1 11 < x 2 1 < 1 11 0 < ² x 2 1 ² < 5 1 11 . 13. does not exist. For values of x to the left of 5, equals whereas for values of x to the right of 5, equals 1. ² x 2 5 ² y s x 2 5 d 2 1, ² x 2 5 ² y s x 2 5 d lim x 5 ² x 2 5 ² x 2 5 28 Chapter 1 Limits and Their Properties
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23. Hence, if you have ± f s x d 2 L ± < 0.01 ± s 3 x 1 2 d 2 8 ± < 0.01 ± 3 x 2 6 ± < 0.01 3 ± x 2 2 ± < 0.01 0 < ± x 2 2 ± < d 5 0.01 3 , 0 < ± x 2 2 ± < 0.01 3 < 0.0033 5 3 ± x 2 2 ± < 0.01 ± 3 x 2 6 ± < 0.01 ± s 3 x 1 2 d 2 8 ± < 0.01 lim x 2 s 3 x 1 2 d 5 8 5 L 25.
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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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ODD01 - CHAPTER 1 Limits and Their Properties Section 1.1...

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