Presentation is loading. Please wait.

Presentation is loading. Please wait.

Chapter 4: The Mean-Value Theorem & Application Topics

Published byRosalind Murphy Modified over 6 years ago

Similar presentations

Presentation on theme: "Chapter 4: The Mean-Value Theorem & Application Topics"— Presentation transcript:

1 Chapter 4: The Mean-Value Theorem & Application Topics
Theorem 4.1.2, p. 155 Rolle’s theorem, p. 156, figure 4.1.2 The mean-value theorem, p. 154, figure 4.1.1 Increasing and Decreasing functions Definition 4.2.1, p. 161, figure 4.2.1 Theorem 4.2.2, p. 161 f is constant on I if f’(x)=0 for all x in I, p. 162 Theorem 4.2.3, p. 162 (4.2.4), p. xxx Theorem 4.2.4, p. 165, figure Local Extreme Values Definition 4.3.1, p. 167&168, figure 4.3.1 Theorem 4.3.2, p. 168 Critical number (4.3.3), p. 168 The first derivative test, p. 170, figures The second derivative test, p. 171 Endpoint and Absolute Extreme Values Endpoint extreme values (4.4.1), p. 174–175, figures Absolute extreme values (4.4.2), p. 175 Behavior of f as x→ ∞, p. 117 Summary for finding extreme values, p. 179 Max-Min Problems Strategy for solving max-min problems, p. 183 Concavity and Points of Inflection Definition 4.6.1, p. 191, figure 4.6.1 Definition 4.6.2, p. 191 Theorem 4.6.3, p. 192 Theorem 4.6.4, p. 192 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps Vertical asymptote, p. 195–196, figures Horizontal asymptote, p. 197, figures Rational functions and horizontal asymptotes, p. xxx Vertical tangent, p. 199, figure Vertical cusp, p. 199, figure Curve Sketching Procedure, p. 201 Velocity and Acceleration; Speed Velocity, p. 209 b. Acceleration and Speed, p. 210 c. Free fall near the surface of the Earth, p. 213 Related Rates of Change Per Unit Time Strategies for solving related rates problems, p. 219 Differentials Definition, p. 224 Newton-Raphson Approximations a. The Newton-Raphson method, p. 230 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

2 The Mean-Value Theorem
Theorem 4.1.2, p. 155 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

3 The Mean-Value Theorem
Rolle’s theorem, p. 156, figure 4.1.2 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

4 The Mean-Value Theorem
The mean-value theorem, p. 154, figure 4.1.1 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

5 Increasing and Decreasing Functions
Definition 4.2.1, p. 161, figure 4.2.1 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

6 Increasing and Decreasing Functions
Theorem 4.2.2, p. 161 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

7 Increasing and Decreasing Functions
f is constant on I iff f’(x)=0 for all x in I, p. 162 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

8 Increasing and Decreasing Functions
Theorem 4.2.3, p. 162 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

9 Increasing and Decreasing Functions
Theorem 4.2.4, p. 165, figure Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

10 Local Extreme Values Definition 4.3.1, p. 167&168, figure 4.3.1
Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

11 Local Extreme Values Theorem 4.3.2, p. 168
Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

12 Local Extreme Values Critical number (4.3.3), p. 168
Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

13 Local Extreme Values The first derivative test, p. 170, figures Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

14 Local Extreme Values The second derivative test, p. 171
Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

15 Endpoint and Absolute Extreme Values
Endpoint extreme values (4.4.1), p. 174–175, figures Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

16 Endpoint and Absolute Extreme Values
Absolute extreme values (4.4.2), p. 175 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

17 Endpoint and Absolute Extreme Values
Behavior of f as x=+/-, p Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

18 Endpoint and Absolute Extreme Values
Summary for finding all extreme values, p. 179 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

19 Max-Min Problems Strategy for solving max-min problems, p. 183
Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

20 Concavity and Points of Inflection
Definition 4.6.1, p. 191, figure 4.6.1 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

21 Concavity and Points of Inflection
Definition 4.6.2, p. 191 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

22 Concavity and Points of Inflection
Definition 4.6.3, p. 192 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

23 Concavity and Points of Inflection
Definition 4.6.4, p. 192 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

24 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps
Vertical asymptote, p. 195–196, figures Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

25 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps
Horizontal asymptote, p. 197, figures Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

26 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps
Vertical tangent, p. 199, figure Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

27 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps
Vertical cusp, p. 199, figure Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

28 Curve Sketching Procedure, p. 201
Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

29 Velocity Velocity and acceleration; speed p. 209
Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

30 Acceleration and Speed
Velocity and acceleration; speed p. 210 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

31 Free Fall Near the Surface of the Earth
Free fall, p. 213 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

32 Related Rates of Change per Unit Time
Strategy for solving rate of change problems, p.219 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

33 Differentials Differentials (4.11.2) p. 224, figure 4.11.1
Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

34 Newton-Raphson Approximations
The Newton-Raphson method, p. 230, figure Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Download ppt "Chapter 4: The Mean-Value Theorem & Application Topics"

Similar presentations

4.3 Derivatives and the shapes of graphs 4.4 Curve Sketching

4.3 Derivatives and the shapes of graphs 4.4 Curve Sketching
Maximum and Minimum Values

Maximum and Minimum Values
4.1 Extreme Values for a function Absolute Extreme Values (a)There is an absolute maximum value at x = c iff f(c)  f(x) for all x in the entire domain.

4.1 Extreme Values for a function Absolute Extreme Values (a)There is an absolute maximum value at x = c iff f(c)  f(x) for all x in the entire domain.
© 2010 Pearson Education, Inc. All rights reserved.

© 2010 Pearson Education, Inc. All rights reserved.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 1.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 1.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 1.
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Indeterminate Form.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Indeterminate Form.
1 Example 1 Sketch the graph of the function f(x) = x 3 -3x 2 -45x+47. Solution I. Intercepts The x-intercepts are the solutions to 0= f(x) = x 3 -3x 2.

1 Example 1 Sketch the graph of the function f(x) = x 3 -3x 2 -45x+47. Solution I. Intercepts The x-intercepts are the solutions to 0= f(x) = x 3 -3x 2.
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Double Integrals a. (16.2.1),

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Double Integrals a. (16.2.1),
Applications of Differentiation Curve Sketching. Why do we need this? The analysis of graphs involves looking at “interesting” points and intervals and.

Applications of Differentiation Curve Sketching. Why do we need this? The analysis of graphs involves looking at “interesting” points and intervals and.
The Shape of the Graph 3.3. Definition: Increasing Functions, Decreasing Functions Let f be a function defined on an interval I. Then, 1.f increases on.

The Shape of the Graph 3.3. Definition: Increasing Functions, Decreasing Functions Let f be a function defined on an interval I. Then, 1.f increases on.
Review Derivatives When you see the words… This is what you know…  f has a local (relative) minimum at x = a  f(a) is less than or equal to every other.

Review Derivatives When you see the words… This is what you know…  f has a local (relative) minimum at x = a  f(a) is less than or equal to every other.
Applications of Derivatives

Applications of Derivatives
AP/Honors Calculus Chapter 4 Applications of Derivatives Chapter 4 Applications of Derivatives.

AP/Honors Calculus Chapter 4 Applications of Derivatives Chapter 4 Applications of Derivatives.
CHAPTER 2 2.4 Continuity Derivatives and the Shapes of Curves.

CHAPTER Continuity Derivatives and the Shapes of Curves.
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Elementary Examples a.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Elementary Examples a.
A Model Solution and More. Sketch the graph of y = Y- intercepts: For y-intercepts, set x = 0 X- intercepts: For X-intercepts, set y = 0 The x-intercept.

A Model Solution and More. Sketch the graph of y = Y- intercepts: For y-intercepts, set x = 0 X- intercepts: For X-intercepts, set y = 0 The x-intercept.
Graphing. 1. Domain 2. Intercepts 3. Asymptotes 4. Symmetry 5. First Derivative 6. Second Derivative 7. Graph.

Graphing. 1. Domain 2. Intercepts 3. Asymptotes 4. Symmetry 5. First Derivative 6. Second Derivative 7. Graph.
Definition of the Natural Exponential Function

Definition of the Natural Exponential Function
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Chapter 12: Vectors Cartesian.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Chapter 12: Vectors Cartesian.

Similar presentations

Ads by Google