Download presentation
Presentation is loading. Please wait.
Published byChristiana Norton Modified over 9 years ago
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.
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.