EVEN01 - 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 . . . . . . . . . . . . . . . . . . . 305 Section 1.2 Finding Limits Graphically and Numerically . . . . . . . 305 Section 1.3 Evaluating Limits Analytically . . . . . . . . . . . . . . . 309 Section 1.4 Continuity and One-Sided Limits . . . . . . . . . . . . . 315 Section 1.5 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . 320 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
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305 CHAPTER 1 Limits and Their Properties Section 1.1 A Preview of Calculus Solutions to Even-Numbered Exercises 2. Calculus: velocity is not constant Distance < s 20 ft y sec ds 15 seconds d 5 300 feet 4. Precalculus: rate of change slope 0.08 5 5 6. Precalculus: 5 2 p Area 5 s ! 2 d 2 8. Precalculus: Volume 5 s 3 d 2 6 5 54 10. (a) (b) You could improve the approximation by using more rectangles. Area < 1 2 1 5 1 5 1.5 1 5 2 1 5 2.5 1 5 3 1 5 3.5 1 5 4 1 5 4.5 2 < 9.145 Area < 5 1 5 2 1 5 3 1 5 4 < 10.417 Section 1.2 Finding Limits Graphically and Numerically 2. s Actual limit is 1 4 . d lim x 2 x 2 2 x 2 2 4 < 0.25 x 1.9 1.99 1.999 2.001 2.01 2.1 0.2564 0.2506 0.2501 0.2499 0.2494 0.2439 f s x d 4. 2 1 4 . d s Actual limit is lim x 2 3 ! 1 2 x 2 2 x 1 3 < 2 0.25 x 3.1 3.01 3.001 2.999 2.99 2.9 2 0.2516 2 0.2502 2 0.2500 2 0.2500 2 0.2498 2 0.2485 f s x d 2 2 2 2 2 2 6. Actual limit is d 1 25 . s lim x 4 f x y s x 1 1 2 s 4 y 5 d x 2 4 < 0.04 x 3.9 3.99 3.999 4.001 4.01 4.1 0.0408 0.0401 0.0400 0.0400 0.0399 0.0392 f s x d 8. (Actual limit is 0.) (Make sure you use radian mode.) lim x 0 cos x 2 1 x < 0.0000 x 0.1 0.01 0.001 0.001 0.01 0.1 0.0500 0.0050 0.0005 0.0005 0.0050 0.0500 2 2 2 f s x d 2 2 2
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10. lim x 1 s x 2 1 2 d 5 3 12. lim x 1 f s x d 5 lim x 1 s x 2 1 2 d 5 3 14. does not exist since the function increases and decreases without bound as x approaches 3. lim x 3 1 x 2 3 16. lim x 0 sec x 5 1 18. lim x 1 sin s p x d 5 0 20. (a) (b) (c) does not exist. The values of C jump from to at t 5 3. 0.71 0.59 lim t 3.5 C s t d lim t 3.5 C s t d 5 0.71 0 0 5 1 C s t d 5 0.35 2 0.12 v 2 s t 2 1 d b t 34 0.71 0.71 0.71 0.71 0.71 0.71 0.59 C s t d 3.7 3.6 3.5 3.4 3.3 t 3 3 4 0.71 0.71 0.71 0.59 0.59 0.59 0.47 C s t d 3.5 3.1 2.9 2.5 22. You need to find such that implies That is, So take Then implies Using the first series of equivalent inequalities, you obtain ± f s x d 2 3 ± 5 ± x 2 2 4 ± < e 5 0.2. ! 3.8 2 2 < x 2 2 < ! 4.2 2 2. 2 s ! 4.2 2 2 d < x 2 2 < ! 4.2 2 2 0 < ± x 2 2 ± < d 5 ! 4.2 2 2 < 0.0494. 2 0.2 < 4 2 0.2 < 3.8 < ! 3.8 < ! 3.8 2 2 < x 2 2 4 x 2 x 2 x x 2 2 < 0.2 < 4 1 0.2 < 4.2 < ! 4.2 < ! 4.2 2 2 ± f s x d 2 3 ± 5 ± x 2 2 1 2 3 ± 5 ± x 2 2 4 ± < 0.2. 0 < ± x 2 2 ± < 24. Hence, if you have ± f s x d 2 L ± < 0.01 ± 1 4 2 x 2 2 2 2 ± < 0.01 ± 2 2 x 2 ± < 0.01 ± 2 1 2 s x 2 4 d ± < 0.01 0 < ± x 2 4 ± < 5 0.02, 0 < ± x 2 4 ± < 0.02 5 ± 2 1 2 s x 2 4 d ± < 0.01 ± 2 2 x 2 ± < 0.01 ± 1 4 2 x 2 2 2 2 ± < 0.01 lim x 4 1 4 2 x 2 2 5 2 306 Chapter 1 Limits and Their Properties
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26. If we assume 4 < x < 6, then Hence, if you have ± f s x d 2 L ± < 0.01 ± s x 2 1 4 d 2 29 ± < 0.01 ± x 2 2 25 ± < 0.01 ± x 2 5 ±± x 1 5 ± < 0.01 ± x 2 5 ± < 0.01 11 < 1 ± x 1 5 ± s 0.01 d 0 < ± x 2 5 ± < d 5 0.01 11 , 5 0.01 y 11 < 0.0009. ± x 2 5 ± < 0.01 ± x 1 5 ± ± s x 1 5 ds x 2 5 d ± < 0.01 ± x 2 2 25 ± < 0.01 ± s x 2 1 4 d 2 29 ± < 0.01 lim x 5 s x 2 1 4 d 5 29 28. Given Hence, let Hence, if you have ± f s x d 2 L ± < e ± s 2 x 1 5 d 2 s 2 1 d ± < ± 2 x 1 6 ± < ± x 1 3 ± < 2 0 < ± x 1 3 ± < 5 2 , 5 y 2.
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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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EVEN01 - CHAPTER 1 Limits and Their Properties Section 1.1...

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