The Fundamental Theorem of Calculus (FTC)

Slides:

Advertisements

Similar presentations

Average Value and the 2 nd Fundamental Theorem. What is the area under the curve between 0 and 2? f(x)

Advertisements

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

5.4 The Fundamental Theorem. The Fundamental Theorem of Calculus, Part 1 If f is continuous on, then the function has a derivative at every point in,

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

Warmup 1) 2). 5.4: 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)

4.4 The Fundamental Theorem of Calculus

SECTION 4-4 A Second Fundamental Theorem of Calculus.

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.

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.

Warm-Up: (let h be measured in feet) h(t) = -5t2 + 20t + 15

Math – Antiderivatives 1. Sometimes we know the derivative of a function, and want to find the original function. (ex: finding displacement from.

Chapter 5 – The Definite Integral. 5.1 Estimating with Finite Sums Example Finding Distance Traveled when Velocity Varies.

Section 6.1 Antiderivatives Graphically and Numerically.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6.

5.a – Antiderivatives and The Indefinite Integral.

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

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

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

Copyright © Cengage Learning. All rights reserved. 4.4 Indefinite Integrals and the Net Change Theorem.

MTH 252 Integral Calculus Chapter 6 – Integration Section 6.6 – The Fundamental Theorem of Calculus Copyright © 2005 by Ron Wallace, all rights reserved.

Integration Copyright © Cengage Learning. All rights reserved.

The Fundamental Theorem of Calculus is appropriately named because it establishes connection between the two branches of calculus: differential calculus.

5.4 The Fundamental Theorem of Calculus. I. The Fundamental Theorem of Calculus Part I. A.) If f is a continuous function on [a, b], then the function.

Indefinite Integrals or Antiderivatives

4.4 The Fundamental Theorem of Calculus

6 Integration Antiderivatives and the Rules of Integration

MTH1170 The Fundamental Theorem of Calculus

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Definite Integration Say good-bye to C

4.4 The Fundamental Theorem of Calculus

Fundamental Concepts of Integral Calculus

6-4 Day 1 Fundamental Theorem of Calculus

Estimating with Finite Sums

Ch. 6 – The Definite Integral

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. {image}

The Fundamental Theorems of Calculus

4.9 – Antiderivatives.

The Fundamental Theorem of Calculus Part 1 & 2

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. {image}

Unit 6 – Fundamentals of Calculus Section 6

Section 4.3 – Area and Definite Integrals

THE FUNDAMENTAL THEOREM OF CALCULUS

Lesson 5-R Review of Chapter 5.

Section 5.3 Definite Integrals and Antiderivatives

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

Sec 5.3: THE FUNDAMENTAL THEOREM OF CALCULUS

The Fundamental Theorem of Calculus

Warm Up 1. Find 2 6 2

Definite Integrals and Antiderivatives

Packet #24 The Fundamental Theorem of Calculus

Lesson 5-4b Net Change Theorem.

Chapter 7 Integration.

Definite Integrals & Antiderivatives

Distance vs. Displacement & Properties of Integrals

Warm Up 1. Find 2 6 2

Indefinite Integration 4.1, 4.4, 4.5

Sec 5.4: INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM

Fundamental Theorem of Calculus

More Definite Integrals

The Fundamental Theorem of Calculus (4.4)

Fundamental Theorem of Calculus

Presentation transcript:

The Fundamental Theorem of Calculus (FTC) Section 4.4 – Part 2 The Fundamental Theorem of Calculus (FTC)

FTC Part 2 Definite Integral as a number To understand FTC part 2, we need to think of the integral as a function. Definite Integral as a number Definite Integral as a Function F is a function of x Constant f is a function of x f is a function of t Constant Constant

Ex: Evaluate the following function at

In the previous example, the derivative of F is the original integrand with only the variable changed. This brings us to FTC Part 2.

The second Fundamental Theorem of Calculus If f is continuous on an open interval containing a, then, for every x in the interval,

Ex: Evaluate

Ex: Find the derivative of

Notes Break! Work on the following problems from your book: p. 294 – 295 #66, 69 – 89 EOO

Distance VS Displacement Distance: total amount traveled Displacement: distance from starting point

**Remember: The derivative of the position function is the velocity function. The antiderivative of the velocity function is the position function.

Displacement: Distance:

Homework Finish problems from p. 294 – 295 #66, 69 – 89 EOO