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Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Derivatives of Algebraic Functions Prepared.

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Presentation on theme: "Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Derivatives of Algebraic Functions Prepared."— Presentation transcript:

1 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Derivatives of Algebraic Functions Prepared by: Midori Kobayashi Humber College C27

2 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.1 Limits

3 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.1-EXAMPLE 3-Page 765 7

4 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.1-EXAMPLE 3-Page 765-Continued

5 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.1-EXAMPLE 3-Page 765-Continued – 3

6 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.1-EXAMPLE 5-Page 766 The highest power of x

7 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.1-EXAMPLE 5-Page 766-Continued

8 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.1-EXAMPLE 6-Page 767 Limit may exist!

9 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.1-EXAMPLE 6-Page 767-Continued

10 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2 The Derivative

11 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 13-Page 775 f(x)=x 2 Expand using (A+B) 2 = A 2 +2AB + B 2

12 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 13-Page 775-Continued Factored by Δx Δx cancelled

13 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 14-Page 775 Substitute x+Δx into y = 3x 2

14 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 14-Page 775-Continued Note: y = 3x 2

15 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 14-Page 775-Continued Divide by Δx

16 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 14-Page 775-Continued Let Δx approachs zero.

17 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 14-Page 775-Continued

18 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 15-Page 777 Substitute x+Δx into x

19 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 15-Page 777-Continued

20 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 15-Page 777-Continued

21 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.2-EXAMPLE 15-Page 777-Continued

22 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.3 Rules for Derivatives 27.3 Rules for Derivatives

23 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.3-EXAMPLE 21-Page 782 2π 2 is a constant

24 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.3-EXAMPLE 22-Page 783

25 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.3-EXAMPLE 22-Page 783-Continued

26 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.3-EXAMPLE 22-Page 783-Continued

27 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.3-EXAMPLE 27-Page 785

28 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.4 27.4 Derivative of a Function Raised to a Power

29 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.4-EXAMPLE 30-Page 789 Don’t forget!

30 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.4-EXAMPLE 32-Page 789 Don’t forget!

31 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.4-EXAMPLE 33-Page 790 Don’t forget!

32 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.4-EXAMPLE 33-Page 790-Continued

33 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.5 27.5 Derivatives of Products and Quotients

34 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.5-EXAMPLE 34-Page 792 u v So u’ = 2x v’ = 1

35 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.5-EXAMPLE 35-Page 792 u v So u’ = 1 v’ = ½(x-3) - ½

36 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.5-EXAMPLE 35-Page 792 u v So u’= 6x 2 v’= 4

37 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.5-EXAMPLE 40-Page 795 u v So u’= 2(t 3 -3)(3t 2 ) v’= ½(t+1) –½

38 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.5-EXAMPLE 40-Page 795-Continued

39 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.6 27.6 Derivatives of Implicit Relations

40 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.6-EXAMPLE 43-Page 797 1

41 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.6-EXAMPLE 45-Page 798 11

42 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.6-EXAMPLE 48-Page 799 11 Product Rule 1y’y’ y’y’ y’y’

43 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.6-EXAMPLE 48-Page 799-Continued

44 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.6-EXAMPLE 49-Page 800 Don’t forget to place ( )

45 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.7 27.7 Higher-Order Derivatives

46 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.7-EXAMPLE 51-Page 802 First derivative Second derivative Third derivative Fourth derivative Fifth derivative

47 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.7-EXAMPLE 52-Page 802 u v So u’ = 1 v’ = ½(x-3) - ½

48 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 27.7-EXAMPLE 52-Page 802-Continued u v So u’ = 1 v’ = -½(x-3) -3/2

49 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Copyright Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Copyright © 2008 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (The Canadian Copyright Licensing Agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.

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