Calculus: Early Transcendentals, by Anton, 7th Edition,ch02

Calculus - Early Transcendentals

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44 CHAPTER 2 Limits and Continuity EXERCISE SET 2.1 1. (a) 1 (b) 3 (c) does not exist (d) 1 (e) 1 (f) 3 2. (a) 2 (b) 0 (c) does not exist (d) 2 (e) 0 (f) 2 3. (a) 1 (b) 1 (c) 1 (d) 1 (e) −∞ (f) + 4. (a) 3 (b) 3 (c) 3 (d) 3 (e) + (f) + 5. (a) 0 (b) 0 (c) 0 (d) 3 (e) + (f) + 6. (a) 2 (b) 2 (c) 2 (d) 3 (e) −∞ (f) + 7. (a) −∞ (b) + (c) does not exist (d) undef (e) 2 (f) 0 8. (a) + (b) + (c) + (d) undef (e) 0 (f) 1 9. (a) −∞ (b) −∞ (c) −∞ (d) 1 (e) 1 (f) 2 10. (a) 1 (b) −∞ (c) does not exist (d) 2 (e) + (f) + 11. (a) 0 (b) 0 (c) 0 (d) 0 (e) does not exist (f) does not exist 12. (a) 3 (b) 3 (c) 3 (d) 3 (e) does not exist (f) 0 13. for all x 0 = 4 14. for all x 0 = 6 , 3 19. (a) 2 1 . 5 1 . 1 1 . 01 1 . 001 0 0 . 5 0 . 9 0 . 99 0 . 999 0 . 1429 0 . 2105 0 . 3021 0 . 3300 0 . 3330 1 . 0000 0 . 5714 0 . 3690 0 . 3367 0 . 3337 1 0 0 2 The limit is 1 / 3.
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Exercise Set 2.1 45 (b) 2 1 . 5 1 . 1 1 . 01 1 . 001 1 . 0001 0 . 4286 1 . 0526 6 . 344 66 . 33 666 . 3 6666 . 3 50 0 1 2 The limit is + . (c) 0 0 . 5 0 . 9 0 . 99 0 . 999 0 . 9999 1 1 . 7143 7 . 0111 67 . 001 667 . 0 6667 . 0 0 -50 0 1 The limit is −∞ . 20. (a) 0 . 25 0 . 1 0 . 001 0 . 0001 0 . 0001 0 . 001 0 . 1 0 . 25 0 . 5359 0 . 5132 0 . 5001 0 . 5000 0 . 5000 0 . 4999 0 . 4881 0 . 4721 0.6 0 -0.25 0.25 The limit is 1/2. (b) 0 . 25 0 . 1 0 . 001 0 . 0001 8 . 4721 20 . 488 2000 . 5 20001 100 0 0 0.25 The limit is + .
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46 Chapter 2 (c) 0 . 25 0 . 1 0 . 001 0 . 0001 7 . 4641 19 . 487 1999 . 5 20000 0 -100 -0.25 0 The limit is −∞ . 21. (a) 0 . 25 0 . 1 0 . 001 0 . 0001 0 . 0001 0 . 001 0 . 1 0 . 25 2 . 7266 2 . 9552 3 . 0000 3 . 0000 3 . 0000 3 . 0000 2 . 9552 2 . 7266 3 2 -0.25 0.25 The limit is 3. (b) 0 0 . 5 0 . 9 0 . 99 0 . 999 1 . 5 1 . 1 1 . 01 1 . 001 1 1 . 7552 6 . 2161 54 . 87 541 . 1 0 . 1415 4 . 536 53 . 19 539 . 5 60 -60 -1.5 0 The limit does not exist. 22. (a) 0 0 . 5 0 . 9 0 . 99 0 . 999 1 . 5 1 . 1 1 . 01 1 . 001 1 . 5574 1 . 0926 1 . 0033 1 . 0000 1 . 0000 1 . 0926 1 . 0033 1 . 0000 1 . 0000 1.5 1 -1.5 0 The limit is 1.
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Exercise Set 2.1 47 (b) 0 . 25 0 . 1 0 . 001 0 . 0001 0 . 0001 0 . 001 0 . 1 0 . 25 1 . 9794 2 . 4132 2 . 5000 2 . 5000 2 . 5000 2 . 5000 2 . 4132 1 . 9794 2.5 2 -0.25 0.25 The limit is 5/2. 23. The height of the ball at time t = 0 . 25 + ∆ t is s (0 . 25 + ∆ t ) = 16(0 . 25 + ∆ t ) 2 + 29(0 . 25 + ∆ t ) + 6, so the distance traveled over the interval from t = 0 . 25 t to t = 0 . 25 + ∆ t is s (0 . 25 + ∆ t ) s (0 . 25 t ) = 64(0 . 25)∆ t + 58∆ t . Thus the average velocity over the same interval is given by v ave = [ s (0 . 25 + ∆ t ) s (0 . 25 t )] / 2∆ t = ( 64(0 . 25)∆ t + 58∆ t ) / 2∆ t = 21 ft/s, and this will also be the instantaneous velocity, since it happens to be independent of ∆ t . 24. The height of the ball at time t = 0 . 75 + ∆ t is s (0 . 75 + ∆ t ) = 16(0 . 75 + ∆ t ) 2 + 29(0 . 75 + ∆ t ) + 6, so the distance traveled over the interval from t = 0 . 75 t to t = 0 . 75 + ∆ t is s (0 . 75 + ∆ t ) s (0 . 75 t ) = 64(0 . 75)∆ t + 58∆ t . Thus the average velocity over the same interval is given by v ave = [ s (0 . 75 + ∆ t ) s (0 . 75 t )] / 2∆ t = ( 64(0 . 75)∆ t + 58∆ t ) / 2∆ t = 5 ft/s, and this will also be the instantaneous velocity, since it happens to be independent of ∆ t . 25. (a) 100 , 000 , 000 100 , 000 1000 100 10 10 100 1000 2 . 0000 2 . 0001 2 . 0050 2 . 0521 2 . 8333 1 . 6429 1 . 9519 1 . 9950 100 , 000 100 , 000 , 000 2 . 0000 2 . 0000 40 -40 -14 6 asymptote y = 2 as x → ±∞ (b) 100 , 000 , 000 100 , 000 1000 100 10 10 100 1000 20 . 0855 20 . 0864 20 . 1763 21 . 0294 35 . 4013 13 . 7858 19 . 2186 19 . 9955 100 , 000 100 , 000 , 000 20 . 0846 20 . 0855 70 0 -160 160 asymptote y = 20 . 086.
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48 Chapter 2 (c) 100 , 000 , 000 100 , 000 1000 100 10 10 100 1000 100 , 000 100 , 000 , 000 100 , 000 , 001 100 , 000 1001 101 . 0 11 . 2 9 . 2 99 . 0 999 . 0 99 , 999 99 , 999 , 999 50 –50 -20 20 no horizontal asymptote 26. (a) 100 , 000 , 000 100 , 000 1000 100 10 10 100 1000 100 , 000 100 , 000 , 000 0 . 2000 0 . 2000 0 . 2000 0 . 2000 0 . 1976 0 . 1976 0 . 2000 0 . 2000 0 . 2000 0 . 2000 0.2 -1.2 -10 10 asymptote y = 1 / 5 as x → ±∞ (b) 100 , 000 , 000 100 , 000 1000 100 10 10 100 0 . 0000 0 . 0000 0 . 0000 0 . 0000 0 . 0016 1668 . 0 2 . 09 × 10 18 1000 100 , 000 100 , 000 , 000 1 . 77 × 10 301 ? ?
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