4.2 The Mean Value Theorem.

Slides:

Advertisements

Similar presentations

We Calculus!!! 3.2 Rolle s Theorem and the Mean Value Theorem.

Advertisements

Rolle’s Theorem and The Mean Value Theorem

Mean Value Theorem for Derivatives4.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

1 Local Extrema & Mean Value Theorem Local Extrema Rolle’s theorem: What goes up must come down Mean value theorem: Average velocity must be attained Some.

Rolle’s Theorem and the Mean Value Theorem3.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by.

If f (x) is a differentiable function over [ a, b ], then at some point between a and b : Mean Value Theorem for Derivatives.

The Mean Value Theorem Lesson 4.2 I wonder how mean this theorem really is?

Section 3.2 – Rolle’s Theorem and the Mean Value Theorem

APPLICATIONS OF DIFFERENTIATION The Mean Value Theorem APPLICATIONS OF DIFFERENTIATION In this section, we will learn about: The significance of.

CHAPTER Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h  0 ( f(a + h) – f(a))

4.2 - The Mean Value Theorem

Chapter 4: Applications of Derivatives Section 4.2: Mean Value Theorem

Mean Value Theorem for Derivatives.

CHAPTER 3 SECTION 3.2 ROLLE’S THEOREM AND THE MEAN VALUE THEOREM

 Exploration:  Sketch a rectangular coordinate plane on a piece of paper.  Label the points (1, 3) and (5, 3).  Draw the graph of a differentiable.

Index FAQ Rolle and Mean Value Theorem. Index FAQ APPLICATIONS OF DIFFERENTIATION We will see that many of the results of this chapter depend on one central.

Today in Calculus Go over homework Derivatives by limit definition Power rule and constant rules for derivatives Homework.

Calculus 1 Rolle’s Theroem And the Mean Value Theorem for Derivatives Mrs. Kessler 3.2.

A car accelerates from a stop to 45 m/sec in 4 sec. Explain why the car must have been accelerating at exactly m/sec at some moment. 2 Do Now.

Lesson 4-2 Mean Value Theorem and Rolle’s Theorem.

Applications of Differentiation Section 4.2 The Mean Value Theorem

APPLICATIONS OF DIFFERENTIATION 4. We will see that many of the results of this chapter depend on one central fact—the Mean Value Theorem.

3.2 Rolle’s Theorem and the Mean Value Theorem. After this lesson, you should be able to: Understand and use Rolle’s Theorem Understand and use the Mean.

4.2 – The Mean Value Theorem

Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem 3.6.

APPLICATIONS OF DIFFERENTIATION 4. We will see that many of the results of this chapter depend on one central fact—the Mean Value Theorem.

If f (x) is continuous over [ a, b ] and differentiable in (a,b), then at some point, c, between a and b : Mean Value Theorem for Derivatives.

Calculus and Analytical Geometry Lecture # 15 MTH 104.

Theorems Lisa Brady Mrs. Pellissier Calculus AP 28 November 2008.

If f(x) is a continuous function on a closed interval x ∈ [a,b], then f(x) will have both an Absolute Maximum value and an Absolute Minimum value in the.

4.2A Rolle’s Theorem* Special case of Mean Value Theorem Example of existence theorem (guarantees the existence of some x = c but does not give value of.

4.2 The Mean Value Theorem.

3.2 Rolle’s Theorem and the

4.2 The Mean Value Theorem In this section, we will learn about:

Mean Value Theorem 5.4.

4.2 The Mean Value Theorem State Standard

Hypothesis: Conclusion:

Table of Contents 25. Section 4.3 Mean Value Theorem.

Rolle’s Theorem Section 3.2.

Lesson 3.2 Rolle’s Theorem Mean Value Theorem 12/7/16

Mean Value Theorem.

4.4 The Fundamental Theorem of Calculus

5-2 mean value theorem.

Table of Contents 21. Section 4.3 Mean Value Theorem.

Mean Value Theorem for Derivatives

Local Extrema & Mean Value Theorem

Rolle’s Theorem and the Mean Value Theorem

Mean Value & Rolle’s Theorems

CHAPTER 3 SECTION 3.2 ROLLE’S THEOREM AND THE MEAN VALUE THEOREM

Rolle’s Theorem.

3.2 Rolle’s Theorem and the

Applications of Derivatives

Average Rate vs. Instantaneous Rate

Important Values for Continuous functions

1. Be able to apply The Mean Value Theorem to various functions.

Section 3.2 Differentiability.

4.6 The Mean Value Theorem.

Rolle's Theorem Objectives:

The Intermediate Value Theorem

§4.2 Mean value theorem(MVT)

Warmup 1. What is the interval [a, b] where Rolle’s Theorem is applicable? 2. What is/are the c-values? [-3, 3]

Rolle’s Theorem and the Mean Value Theorem

APPLICATIONS OF DIFFERENTIATION

The Fundamental Theorems of Calculus

Mean Value Theorem for Derivatives

3.2. Definition of Derivative.

Copyright © Cengage Learning. All rights reserved.

Section 4.2 Mean Value Theorem.

Do Now: Find all extrema of

Presentation transcript:

4.2 The Mean Value Theorem

Rolle’s Theorem Let f be a function that satisfies the following three conditions: f is continuous on the closed interval [a,b] . f is differentiable on the open interval (a,b) . f(a) = f(b) . Then there is a number c in (a,b) such that f ′(c)=0. Examples on the board.

If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b: Mean Value Theorem for Derivatives The Mean Value Theorem says that at some point in the interval, the actual slope equals the average slope. Note: The Mean Value Theorem only applies over a closed interval.

An illustration of the Mean Value Theorem. Tangent parallel to chord. Slope of tangent: Slope of chord:

Corollaries of the Mean Value Theorem Corollary 1: If f ′(x)=0 for all x in an interval (a,b), then f is constant on (a,b) . Corollary 2: If f ′ (x) = g′ (x) for all x in an interval (a,b), then f - g is constant on (a,b) , that is f(x)= g(x) + c where c is a constant. (see the next slide for an illustration of Corollary 2)

These two functions have the same slope at any value of x. Functions with the same derivative differ by a constant.