3.2: Rolle’s Theorem and the Mean Value Theorem

Slides:

Advertisements

Similar presentations

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

Advertisements

The Mean Value Theorem and Rolles Theorem Lesson 3.2 I wonder how mean this theorem really is?

Rolle’s Theorem and The Mean Value Theorem

The Mean Value Theorem Math 131 Spring Max-Min Theorem If f'(x) is defined on an open interval (a, b) and if f(x) has a relative extrema at a point.

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

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.

Miss Battaglia AP Calculus AB/BC.  Sketch a coordinate plane on a piece of paper. Label the points (1,3) and (5,3). Using a pencil, draw the graph of.

AP Calculus AB Chapter 3, Section 2 Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem and the Mean Value Theorem. Rolle’s Theorem  Let f be continuous on the closed interval [a, b] and differentiable on the open interval.

Rolle’s and The Mean Value Theorem BC Calculus. Mean Value and Rolle’s Theorems The Mean-Value Theorem ( and its special case ) Rolle’s Theorem are Existence.

Sec. 3.2: Rolle’s Theorem and MVT Rolle’s Theorem: Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If.

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

Simonds Portland Community College Given Information Gino drove exactly 60 miles between 3 PM and 4 PM. Question How fast was Gino driving right at 3:30.

Rolle’s theorem and Mean Value Theorem ( Section 3.2) Alex Karassev.

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.

Rolle’s and The Mean Value Theorem BC Calculus. Mean Value and Rolle’s Theorems The Mean-Value Theorem ( and its special case ) Rolle’s Theorem are Existence.

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.

Calculus Date: 12/10/13 Obj: SWBAT apply Rolle’s and the Mean Value Thm derivatives of absolute.

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.

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

Chapter Three: Section Two Rolle’s Theorem and the Mean Value Theorem.

1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such functions. Figure 1 (c) (b) (d) (a) Examples:

Warmup. Be seated before the bell rings DESK homework Warm-up (in your notes) Agenda : go over hw Hw quiz Notes lesson 3.2.

2. Rolle’s Theorem and Mean Value Theorem

 On any continuous function between points a and b there exists some point(s) c where the tangent line at this point has the same slope as the secant.

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

Section 3.2 Mean Value Theorem Math 1231: Single-Variable Calculus.

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.

3.8 Derivatives of Inverse Trig Functions

I wonder how mean this theorem really is?

4.2 - The Mean Value Theorem

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

Rolle’s Theorem & the Mean Value Theorem (3.2)

Table of Contents 25. Section 4.3 Mean Value Theorem.

Mean Value and Rolle’s Theorem

Rolle’s Theorem Section 3.2.

5-2 mean value theorem.

Table of Contents 21. Section 4.3 Mean Value Theorem.

Local Extrema & Mean Value Theorem

Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem and the Mean Value Theorem

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

Linear Plots Objective: Students will use tables and graphs to help determine the linear rule for a recursive function.

AP Calculus BC September 23, 2016.

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

2.1 The Derivative & the Tangent Line Problem

ROLLES THEOREM AND THE EXTREME VALUE THEOREM

5.2 Mean Value Theorem for Derivatives

3.2 Rolle’s Theorem and the Mean Value Theorem

5.2: Writing Linear Equations Given the Slope and a Point

Rolle’s Theorem and the Mean Value Theorem

ROLLES THEOREM AND THE EXTREME VALUE THEOREM

Today in Calculus Go over homework Trig Review Mean Value Theorem

Mindjog Find the domain of each function..

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

Section 4.2 Mean Value Theorem.

Do Now: Find all extrema of

Presentation transcript:

3.2: Rolle’s Theorem and the Mean Value Theorem Homework: p.174 9, 11, 15, 19, 23, 29, 31, 35, 39, 41, 53, 55 Standards EK2.4A1 – If a function f is continuous over the interval [a,b] and differentiable over the interval (a,b), the Mean Value Theorem guarantees a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval. Learning Objectives: Understand and use Rolle’s Theorem Understand and use the Mean Value Theorem

1 2 3 4 5 Con TE TEx2 SLE SLEx2

Concept 1: Rolle’s Theorem The Three Conditions of Rolle’s Theorem 𝑓(𝑥) must be continuous on [a,b] 𝑓(𝑥) must be differentiable on (a,b) 𝑓 𝑎 =𝑓(𝑏)

Example 1: Rolle’s Theorem with INTECEPTS Find the two x-intercepts of ℎ 𝑥 = 𝑥 2 −2𝑥−35 and show that ℎ ′ 𝑥 =0 at some point between the two intercepts.

Student Led Example 1: Rolle’s Theorem with INTERCEPTS Find the two x-intercepts of 𝑘 𝑥 = 𝑥 2 −3𝑥+2 and show that 𝑘 ′ 𝑥 =0 at some point between the two intercepts.

Example 2A: Rolle’s Theorem on a CLOSED INTERVAL Find a value 𝑐 such that ℎ ′ 𝑐 =0 if ℎ 1 𝑥 = 1 3 𝑥 3 − 𝑥 2 −3𝑥+4 on the interval [−3, 3]

Student Led Example 2A: Rolle’s Theorem on a CLOSED INTERVAL Find a value 𝑐 such that 𝑘 ′ 𝑐 =0 if 𝑘 𝑐 = 𝑥 2 +8𝑥+5 on the interval [2, 6]

Example 2B: Rolle’s Theorem on a CLOSED INTERVAL Can Rolle’s Theorem be applied to ℎ 2 𝑥 = 𝑥 2 +2𝑥+3 𝑥+2 on the interval −1, 3 ?

Student Led Example 2B: Rolle’s Theorem on a CLOSED INTERVAL Can Rolle’s Theorem be applied to 𝑘 2 𝑥 = 𝑥 2 +𝑥−6 𝑥−3 on the interval 0, 4 ?

Example 3: Rolle’s Theorem on a closed interval (TRIG) Find a value for 𝑐 such that 𝑓 ′ 𝑐 =0 If ℎ 𝑥 = sec 𝑥 On the interval −𝜋, 𝜋 Warning!

Technology Test You decide… Is this the graph of − 𝑥 2 +2𝑥? Warning!

Pictures are NOT proofs 1− 𝑥−1 2 − 1 1000 𝑥−1 1 7 +1

Concept 4: Mean Value Theorem (but really, it will be concept 5) The Three Conditions of the MVT 𝑓(𝑥) must be continuous on [a,b] 𝑓(𝑥) must be differentiable on (a,b) There exists a 𝑐 such that 𝑓 ′ 𝑐 = 𝑓 𝑏 −𝑓(𝑎) 𝑏−𝑎

Example 4: The Mean Value Theorem It’s basically the slope formula…moving on

Example 5: Instantaneous Rate of Change A plane begins its take-off at 2:00pm on a 2500 mile flight. After 5 ½ hours, the plane arrives at its destination. Explain why there are least two times during the flight the speed of the plane is 400 miles per hour.

Student Led Example 5: Instantaneous Rate of Change Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a semi-truck passes the first patrol car, its speed is clocked at 55 miles per hour. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 miles per hours. If the speed limit is 55 miles per hour, should the driver of the truck get a speeding ticket?

Can you do it? lim 𝑥→0 7𝑥− sin 𝑥 𝑥 2 + sin 3𝑥