The Mean Value Theorem for Integrals – Average Value

Slides:

Advertisements

Similar presentations

Law of Sines Lesson Working with Non-right Triangles  We wish to solve triangles which are not right triangles B A C a c b h View Sine Law Spreadsheet.

Advertisements

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

Volume: The Shell Method Lesson 7.3. Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal.

1 Fundamental Theorem of Calculus Section The Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F.

The Area Between Two Curves Lesson When f(x) < 0 Consider taking the definite integral for the function shown below. The integral gives a negative.

Average Values & Other Antiderivative Applications Lesson 7.1.

The Fundamental Theorem of Calculus Lesson Definite Integral Recall that the definite integral was defined as But … finding the limit is not often.

4-4: The Fundamental Theorems Definition: If f is continuous on [ a,b ] and F is an antiderivative of f on [ a,b ], then: The Fundamental Theorem:

Derivatives of Products and Quotients Lesson 4.2.

The Fundamental Theorems of Calculus Lesson 5.4. First Fundamental Theorem of Calculus Given f is  continuous on interval [a, b]  F is any function.

Volumes of Revolution The Shell Method Lesson 7.3.

Volumes Lesson 6.2.

Arc Length and Surfaces of Revolution Lesson 7.4.

THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4. THE FUNDAMENTAL THEOREM OF CALCULUS Informally, the theorem states that differentiation and definite.

Improper Integrals Lesson 7.7. Improper Integrals Note the graph of y = x -2 We seek the area under the curve to the right of x = 1 Thus the integral.

Warm-Up- Books and calcs

I wonder how mean this theorem really is?

Mean Value Theorem 5.4.

The Area Between Two Curves

Linear Approximation and Differentials

Area and the Definite Integral

Real Zeros of Polynomial Functions

2.5 Zeros of Polynomial Functions

Lesson 11-7 Ratios of Areas (page 456)

Vectors and Angles Lesson 10.3b.

The Fundamental Theorems of Calculus

Derivatives of Products and Quotients

Intro to Limits 2.1.

Rates of Change Lesson 3.3.

Absolute Extrema Lesson 6.1.

Definition of the Derivative

Area and the Definite Integral

The Area Question and the Integral

A Preview of Calculus Lesson 2.1.

Intersection of Graphs of Polar Coordinates

Volumes – The Disk Method

Representation of Functions by Power Series

Derivatives of Parametric Equations

Solving Quadratic Functions

Linear Approximation and Differentials

Differentiation in Polar Coordinates

The Area Between Two Curves

Volumes of Revolution The Shell Method

Improper Integrals Lesson 8.8.

Applications Growth and Decay Math of Finance

Lesson 3: Definite Integrals and Antiderivatives

Taylor and MacLaurin Series

Average Values & Other Antiderivative Applications

Continuity Lesson 3.2.

The General Power Formula

Average Value of a Function

Arc Length and Surface Area

1. Reversing the limits changes the sign. 2.

Antiderivatives Lesson 7.1A.

The Fundamental Theorem of Calculus

Mean-Value Theorem for Integrals

Continuity and One-Sided Limits

Power Functions, Comparing to Exponential and Log Functions

The Fundamental Theorem of Calculus

The Area Between Two Curves

Derivatives of Inverse Functions

Area as the Limit of a Sum

Limits Lesson 3.1.

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

The Indefinite Integral

The Fundamental Theorems of Calculus

UNIT IV Derivatives of Products and Quotients

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

The Natural Log Function: Integration

Concavity and Rates of Change

Presentation transcript:

The Mean Value Theorem for Integrals – Average Value Just as mean as before The Mean Value Theorem for Integrals – Average Value Lesson 5.7

Average Value of a Function Consider recording a temperature each hour and taking an average If we take it more often and take a limit …

Average Value of a Function Now apply the concept to a continuous function f(x) is the height of the "box" which is equal to the area under the curve. a b

Average Value of a Function Find the average value of these functions

Finding Where This Happens This height will be f(c) for some x = c (at least 1) And we can solve for that value of c f(x) a b c

Try It Out Given f(x) = x2 + 4x +1 on [0, 2] Find a value for c such that

Application Suppose a study comes up with a model that says t years from now CO2 in the area of a city will be What is the average level in the first 3 years? At what point in time does that average level actually occur

Assignment Lesson 5.7 Page 332 Exercises 1 – 35 odd