Table of Contents 25. Section 4.3 Mean Value Theorem.

Slides:

Advertisements

Similar presentations

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.

Advertisements

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

Aim: Rolle’s Theorem Course: Calculus Do Now: Aim: What made Rolle over to his theorem? Find the absolute maximum and minimum values of y = x 3 – x on.

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

4.2 The Mean Value Theorem.

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

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.

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.

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.

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.

In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs,

Section 4.2 Mean Value Theorem What you’ll learn Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences Why?

Rolle’s Theorem/Mean-Value Theorem Objective: Use and interpret the Mean-Value Theorem.

AP Calculus Unit 4 Day 5 Finish Concavity Mean Value Theorem Curve Sketching.

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.

When you see… Find the zeros You think…. To find the zeros...

AP CALCULUS AB FINAL REVIEW APPLICATIONS OF THE DERIVATIVE.

Ch. 5 – Applications of Derivatives 5.2 – Mean Value Theorem.

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.

Warm-upWarm-up 1.Find all values of c on the interval that satisfy the mean value theorem. 2. Find where increasing and decreasing.

4.2 The Mean Value Theorem.

4.2 - The Mean Value Theorem

3.2 Rolle’s Theorem and the

Rolle’s theorem and Mean Value Theorem (Section 4.2)

Rolle’s Theorem/Mean-Value Theorem

3.2 Rolle’s Theorem and the Mean Value Theorem

4.2 The Mean Value Theorem State Standard

3.2 Rolle’s Theorem and the Mean Value Theorem

Lesson 63 Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem Section 3.2.

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

Calculus Index Cards Front And Back.

5-2 mean value theorem.

Table of Contents 21. Section 4.3 Mean Value Theorem.

Relative Extrema and More Analysis of Functions

4.3 Using Derivatives for Curve Sketching.

Lesson 4-QR Quiz 1 Review.

Local Extrema & Mean Value Theorem

Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem and the Mean Value Theorem

4.2 Mean Value Theorem.

Kuan Liu, Ryan Park, Nathan Saedi, Sabrina Sauri & Ellie Tsang

Sec 2 Cont: Mean Value Theorem (MVT)

Mean Value & Rolle’s Theorems

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

3.2 Rolle’s Theorem and the

Applications of Derivatives

II. differentiable at x = 0 III. absolute minimum at x = 0

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

5.2 Section 5.1 – Increasing and Decreasing Functions

Section 4.2 Mean value theorem.

Rolle's Theorem Objectives:

5.2 Mean Value Theorem for Derivatives

3.2 Rolle’s Theorem and the Mean Value Theorem

4.3 Connecting f’ and f’’ with the graph of f

The Intermediate Value Theorem

Lesson 2: Mean Value Theorem

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

Increasing and Decreasing Functions and the First Derivative Test

Derivatives and Graphing

Increasing and Decreasing Functions and the First Derivative Test

Applications of Derivatives

Mindjog Find the domain of each function..

Applications of Derivatives

Rayat Shikshan Sanstha’s S.M.Joshi College, Hadapsar -28

Section 4.2 Mean Value Theorem.

Do Now: Find all extrema of

Unit 4: Applications of Derivatives

Presentation transcript:

Table of Contents 25. Section 4.3 Mean Value Theorem

Mean Value Theorem Essential Question – What 2 concepts that we have learned about does the mean value theorem connect?

Mean Value Theorem Connects average rate of change over an interval with the instantaneous rate of change at a point within that interval Somewhere between points a and b on a differentiable curve, there is at least one tangent line parallel to chord AB.

Mean Value Theorem If f(x) is continuous at every point on [a,b] and differentiable on (a,b), then there is at least one point c in (a,b) where Rolle’s Theorem is the special case of the MVT where f(a) = f(b)

Example Show that f(x) = x2 satisfies the Mean Value Theorem on [0,2]. Continuous on [0,2] and differentiable on [0,2]

Example Why does the mean value theorem not work on [-1,1] for the following functions? Not differentiable at x=0 Not continuous at x=1

Example Find a tangent that is parallel to secant AB

Uses of Mean Value Theorem Speed Mean Value Theorem says that the instantaneous rate of change (velocity) at some point must equal the average rate of change over the whole interval Example A car accelerating from zero takes 8 sec to go 352 feet. The average velocity is 352/8 = 44 ft/sec or 30 mph. At some point during the acceleration, the car’s speed must be exactly 30 mph.

Uses of Mean Value Theorem Increasing and Decreasing When the derivative is positive, function is increasing When the derivative is negative, function is decreasing If function only increases or decreases on an interval, then it is called monotonic Example Where is y = x2 increasing and decreasing?

Example Where is f(x)=x3-4x increasing and decreasing? Increasing

Testing Critical points At points where f has a minimum value, f’<0 to the left and f’>0 to the right (falling on left and rising to right) Similarly, where f has a maximum value, f’>0 to the left and f’<0 to the right (rising on left and falling on right)

1st derivative test for local extrema At a critical point c If f’ changes sign from positive to negative at c, then f has a local max If f’ changes sign from negative to positive at c, then f has a local min If f’ doesn’t change sign, then no local extrema At left endpoint if f’<0 on interval to the right, f has a local max At left endpoint if f’>0 on interval to the right, f has a local min At right endpoint if f’<0 on interval to the left, f has a local min At right endpoint if f’>0 on interval to the right, f has a local max

Example -2 2 Plug in values in each interval to f’ Local max Local min

Example -3 1 Plug in values in each interval to f’ Local max Local min

Assignment Pg. 236: #1-9 odd, 13-29 odd, 41, 45, 57