SECTION 4-4 A Second Fundamental Theorem of Calculus.

Slides:

Advertisements

Similar presentations

Follow the link to the slide. Then click on the figure to play the animation. A Figure Figure

Advertisements

Applying the well known formula:

Rizzi – Calc BC.  Integrals represent an accumulated rate of change over an interval  The gorilla started at 150 meters The accumulated rate of change.

Example, Page 321 Draw a graph of the signed area represented by the integral and compute it using geometry. Rogawski Calculus Copyright © 2008 W. H. Freeman.

Section 4.3 Indefinite Integrals and Net Change Theorem Math 1231: Single-Variable Calculus.

3.3 –Differentiation Rules REVIEW: Use the Limit Definition to find the derivative of the given function.

Chapter 6 The Integral Sections 6.1, 6.2, and 6.3

The First Fundamental Theorem of Calculus. First Fundamental Theorem Take the Antiderivative. Evaluate the Antiderivative at the Upper Bound. Evaluate.

CALCULUS II Chapter 5. Definite Integral Example.

Let c (t) = (t 2 + 1, t 3 − 4t). Find: (a) An equation of the tangent line at t = 3 (b) The points where the tangent is horizontal. ` ` `

The Fundamental Theorem of Calculus Inverse Operations.

1 5.4 – Indefinite Integrals and The Net Change Theorem.

Fundamental Theorems of Calculus 6.4. The First (second?) Fundamental Theorem of Calculus If f is continuous on, then the function has a derivative at.

Warm Up. 6.4 Fundamental Theorem of Calculus If you were being sent to a desert island and could take only one equation with you, might well be your.

4-3 DEFINITE INTEGRALS MS. BATTAGLIA – AP CALCULUS.

Section 5.3 – The Definite Integral

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

5.c – The Fundamental Theorem of Calculus and Definite Integrals.

Integration 4 Copyright © Cengage Learning. All rights reserved.

Section 5.4a FUNDAMENTAL THEOREM OF CALCULUS. Deriving the Theorem Let Apply the definition of the derivative: Rule for Integrals!

Section 5.6 Integration: “The Fundamental Theorem of Calculus”

4.4c 2nd Fundamental Theorem of Calculus. Second Fundamental Theorem: 1. Derivative of an integral.

7.4: The Fundamental Theorem of Calculus Objectives: To use the FTC to evaluate definite integrals To calculate total area under a curve using FTC and.

Antiderivatives An antiderivative of f(x) is any function F(x) such that F’(x) = f(x)

Chapter 5 Integration Third big topic of calculus.

CHAPTER 4 SECTION 4.4 THE FUNDAMENTAL THEOREM OF CALCULUS.

5.4 Fundamental Theorem of Calculus. It is difficult to overestimate the power of the equation: It says that every continuous function f is the derivative.

Section 4.4 The Fundamental Theorem of Calculus Part II – The Second Fundamental Theorem.

Integration Copyright © Cengage Learning. All rights reserved.

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.

6/3/2016 Perkins AP Calculus AB Day 10 Section 4.4.

4.4 The Fundamental Theorem of Calculus

Integration 4 Copyright © Cengage Learning. All rights reserved.

4-4 THE FUNDAMENTAL THEOREM OF CALCULUS MS. BATTAGLIA – AP CALCULUS.

Miss Battaglia AB/BC Calculus.  Connects differentiation and integration.  Integration & differentiation are inverse operations. If a function is continuous.

4009 Fundamental Theorem of Calculus (Part 2) BC CALCULUS.

MAT 1235 Calculus II 4.4 Part II Indefinite Integrals and the Net Change Theorem

5.4: Fundamental Theorem of Calculus Objectives: Students will be able to… Apply both parts of the FTC Use the definite integral to find area Apply the.

SECTION 5.4 The Fundamental Theorem of Calculus. Basically, (definite) integration and differentiation are inverse operations.

Mathematics. Session Definite Integrals –1 Session Objectives  Fundamental Theorem of Integral Calculus  Evaluation of Definite Integrals by Substitution.

Particle A particle is an object having a non zero mass and the shape of a point (zero size and no internal structure). In various situations we approximate.

MAT 212 Brief Calculus Section 5.4 The Definite Integral.

Section 6.1 Antiderivatives Graphically and Numerically.

ANTIDERIVATIVES AND INDEFINITE INTEGRATION AB Calculus.

Section 3.9 Antiderivatives

2.1 Position, Velocity, and Speed 2.1 Displacement  x  x f - x i 2.2 Average velocity 2.3 Average speed  

5.3 Definite Integrals and Antiderivatives. What you’ll learn about Properties of Definite Integrals Average Value of a Function Mean Value Theorem for.

7.1 Integral As Net Change. Net Change Theorem If f’(x) is a rate of change, then the net change of f(x) over the interval [a,b] is This tells us the.

4033-Properties of the Definite Integral (5.3) AB Calculus.

Integration 4 Copyright © Cengage Learning. All rights reserved.

Calculus Section 5.3 Differentiate exponential functions If f(x) = e x then f’(x) = e x f(x) = x 3 e x y= √(e x – x) Examples. Find the derivative. y =

Ch. 8 – Applications of Definite Integrals 8.1 – Integral as Net Change.

5.3 Definite Integrals and Riemann Sums. I. Rules for Definite Integrals.

Net Change as the Integral of a Rate Section 5.5 Mr. Peltier.

Calculus 6.1 Antiderivatives and Indefinite Integration.

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

Integration Copyright © Cengage Learning. All rights reserved.

Average Value Theorem.

Definite Integration Say good-bye to C

4.4 The Fundamental Theorem of Calculus

The Fundamental Theorems of Calculus

Unit 6 – Fundamentals of Calculus Section 6

Accumulation and Particle Motion

The Fundamental Theorem of Calculus (FTC)

Packet #24 The Fundamental Theorem of Calculus

Total Distance Traveled

Sec 5.4: INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM

Calculus I (MAT 145) Dr. Day Wednesday April 17, 2019

Accumulation and Particle Motion

Presentation transcript:

SECTION 4-4 A Second Fundamental Theorem of Calculus

1) Find by working from inside out: - First integrate - Then differentiate constant Recall: Derivatives and Integrals undo each other. They are inverses

Second Fundamental Theorem of Calculus If f(x) is continuous on [a,b], then the derivative of is

What if both limits have variables?

2) Find 3) Find

4) Find 5) Find

What if the upper limit is a function other than x? 6) Find ifthen

7) Find 8) Find

9) Find ifthen What if both limits have variables?

10) Find 11) Find

Net Change Theorem The definite integral of the rate of change of a quantity F’(x) gives the total, or net change, in that quantity on the interval [a, b].

12) Find the displacement (net change) of a particle moving along a line so that its velocity is feet per second. a) What is the displacement on the time interval [1, 5]

12) cont b) What is the total distance traveled by the particle [1, 5]

Assignment: Page 295 # 75-92, 97, 98 and 99