Chapter 4: The Mean-Value Theorem & Application Topics

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Presentation transcript:

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 4.2.11 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 4.3.7-10 The second derivative test, p. 171 Endpoint and Absolute Extreme Values Endpoint extreme values (4.4.1), p. 174–175, figures 4.4.1-4 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 4.7.1-4 Horizontal asymptote, p. 197, figures 4.7.7-8 Rational functions and horizontal asymptotes, p. xxx Vertical tangent, p. 199, figure 4.7.11 Vertical cusp, p. 199, figure 4.7.12 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.

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.

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.

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.

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.

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.

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.

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.

Increasing and Decreasing Functions Theorem 4.2.4, p. 165, figure 4.2.11 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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.

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.

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.

Local Extreme Values The first derivative test, p. 170, figures 4.3.7-10 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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.

Endpoint and Absolute Extreme Values Endpoint extreme values (4.4.1), p. 174–175, figures 4.4.1-4 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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.

Endpoint and Absolute Extreme Values Behavior of f as x=+/-, p. 177 - 178 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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.

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.

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.

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.

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.

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.

Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps Vertical asymptote, p. 195–196, figures 4.7.1-4 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps Horizontal asymptote, p. 197, figures 4.7.7-8 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps Vertical tangent, p. 199, figure 4.7.11 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps Vertical cusp, p. 199, figure 4.7.12 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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

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

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.

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.

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.

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.

Newton-Raphson Approximations The Newton-Raphson method, p. 230, figure 4.12.1 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.