02 - CHAPTER 2 Limits and Their Properties Section 2.1...

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CHAPTER 2 Limits and Their Properties Section 2.1 A Preview of Calculus . . . . . . . . . . . . . . . . . . . . 71 Section 2.2 Finding Limits Graphically and Numerically . . . . . . . . 72 Section 2.3 Evaluating Limits Analytically . . . . . . . . . . . . . . . 83 Section 2.4 Continuity and One-Sided Limits . . . . . . . . . . . . . . 94 Section 2.5 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . 105 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
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CHAPTER 2 Limits and Their Properties Section 2.1 A Preview of Calculus 71 1. Precalculus: s 20 ft y sec ds 15 seconds d 5 300 feet 2. Calculus: velocity is not constant Distance < 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 4. Precalculus: rate of change slope 0.08 5 5 5. Precalculus: sq. units Area 5 1 2 bh 5 1 2 s 5 ds 3 d 5 15 2 6. Calculus required: 5 5 sq. units < 2 s 2.5 d Area 5 bh 7. (a) (b) (c) At the slope is 2. You can improve your approximation of the slope at by considering values very close to 1. x - x 5 1 P s 1, 3 d m 5 3 2 0.5 5 2.5 x 5 0.5: m 5 3 2 1.5 5 1.5 x 5 1.5: m 5 3 2 2 5 1 x 5 2: x Þ 1 5 s x 2 1 ds 3 2 x d x 2 1 5 3 2 x , slope 5 m 5 s 4 x 2 x 2 d 2 3 x 2 1 123 3 4 P y x f s x d 5 4 x 2 x 2 9. (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 8. (a) (b) (c) At the slope is You can improve your approximation of the slope at by considering values very close to 4. x - x 5 4 1 ! 4 1 2 5 1 4 5 0.25. P s 4, 2 d m 5 1 ! 5 1 2 < 0.2361 x 5 5: m 5 1 ! 3 1 2 < 0.2679 x 5 3: m 5 1 ! 1 1 2 5 1 3 x 5 1: x Þ 4 5 ! x 2 2 s ! x 1 2 ds ! x 2 2 d 5 1 ! x 1 2 , slope 5 m 5 ! x 2 2 x 2 4 x y 12345 2 P (4, 2) f s x d 5 ! x
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72 Chapter 2 Limits and Their Properties 10. (a) For the figure on the left, each rectangle has width For the figure on the right, each rectangle has width (b) You could obtain a more accurate approximation by using more rectangles. You will learn later that the exact area is 2. 5 ! 3 1 2 6 p < 1.9541 5 6 3 1 2 1 ! 3 2 1 1 1 ! 3 2 1 1 2 4 Area < 6 3 sin 6 1 sin 3 1 sin 2 1 sin 2 3 1 5 6 1 sin 4 6 . 5 ! 2 1 1 4 < 1.8961 5 4 3 ! 2 2 1 1 1 ! 2 2 4 Area < 4 3 sin 4 1 sin 2 1 sin 3 4 1 sin 4 4 . 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 2.2 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 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 x 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 3. Actual limit is d 2 1 16 . s lim x 3 ± 1 y s x 1 1 dg 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
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Section 2.2 Finding Limits Graphically and Numerically 73 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 5. (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 6.
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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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02 - CHAPTER 2 Limits and Their Properties Section 2.1...

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