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Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. MCS 122 Chapter 1 Review.

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Presentation on theme: "Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. MCS 122 Chapter 1 Review."— Presentation transcript:

1 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. MCS 122 Chapter 1 Review of Limits Some of the material in these slides is from Calculus 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.

2 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Two Basic Problems of Calculus The solution to both of these problems requires the use of limits

3 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved (p. 70) Limits (An Informal View)‏

4 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 71)‏

5 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved (p. 72) One-Sided Limits (An Informal View)‏

6 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved (p. 73) The Relationship Between One-Sided and Two-Sided Limits

7 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 73)‏

8 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise (p. 77)‏

9 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise (p. 77)‏

10 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 74)‏

11 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved (p. 75) Infinite Limits (An Informal View)‏

12 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise (p. 77)‏

13 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise (p. 77)‏

14 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise (p. 78)‏

15 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 80)‏ Calculating Limits – Basic Cases

16 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem (p. 80)‏ Calculating Limits – Basic Cases

17 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Calculating Limits – Algebra Rules

18 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise (p. 87)‏

19 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem (p. 85)‏ Calculating Limits – Rational Functions

20 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved (p. 89) Limits at Infinity (An Informal View)‏ Calculating Limits – As x goes to +,- Infinity

21 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 89)‏

22 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 90)‏

23 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved (p. 91) Infinite Limits at Infinity (An Informal View)‏ Calculating Limits – As x goes to +,- Infinity

24 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise (p. 96)‏

25 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise (p. 96)‏

26 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise (p. 97)‏

27 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 100)‏ Calculating Limits – Rigorous Definition!!

28 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 101)‏ Calculating Limits – Rigorous Definition!!

29 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition (p. 101) Limit Definition Calculating Limits – Rigorous Definition!!

30 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Calculating Limits – Rigorous Definition!! We can calculate that Find a number δ> 0 such that |f(x) – 2| < 0.05 if 0 < |x-1| < δ. (Use the graphing capability of your calculator)

31 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition (p. 110)‏ Continuous Functions

32 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem (p. 113)‏ Continuous Functions

33 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem (p. 123)‏ Two useful Trig Limits

34 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 123)‏ Two useful Trig Limits


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