Blue part is out of 44 Green part is out of 58

Slides:

Advertisements

Similar presentations

Warm up Problem If , find .

Advertisements

7.5 Area Between Two Curves

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

INTEGRALS The Substitution Rule In this section, we will learn: To substitute a new variable in place of an existing expression in a function,

6 Integration Antiderivatives and the Rules of Integration

7.5 Area Between Two Curves Find Area Between 2 Curves Find Consumer Surplus Find Producer Surplus.

Warm-up: 1)If a particle has a velocity function defined by, find its acceleration function. 2)If a particle has an acceleration function defined by, what.

Definite Integrals Finding areas using the Fundamental Theorem of Calculus.

Every slope is a derivative. Velocity = slope of the tangent line to a position vs. time graph Acceleration = slope of the velocity vs. time graph How.

The Integral chapter 5 The Indefinite Integral Substitution The Definite Integral As a Sum The Definite Integral As Area The Definite Integral: The Fundamental.

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

Section 6.2: Integration by Substitution

4.1 The Indefinite Integral. Antiderivative An antiderivative of a function f is a function F such that Ex.An antiderivative of since is.

The Fundamental Theorem of Calculus (4.4) February 4th, 2013.

Math 155 – Calculus 2 Andy Rosen PLEASE SIGN IN.

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. MCS 122 Chapter 5 Review.

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

INTEGRALS The Substitution Rule In this section, we will learn: To substitute a new variable in place of an existing expression in a function,

Math – Antidifferentiation: The Indefinite Integral 1.

The Indefinite Integral

4.1 ANTIDERIVATIVES & INDEFINITE INTEGRATION. Definition of Antiderivative  A function is an antiderivative of f on an interval I if F’(x) = f(x) for.

13.1 Antiderivatives and Indefinite Integrals. The Antiderivative The reverse operation of finding a derivative is called the antiderivative. A function.

INDEFINITE INTEGRALS Indefinite Integral Note1:is traditionally used for an antiderivative of is called an indefinite integral Note2: Example:

4.1 Antiderivatives and Indefinite Integration Definition of Antiderivative: A function F is called an antiderivative of the function f if for every x.

MAT 212 Brief Calculus Section 5.4 The Definite Integral.

In this section, we introduce the idea of the indefinite integral. We also look at the process of variable substitution to find antiderivatives of more.

Antiderivatives and uses of derivatives and antiderivatives Ann Newsome.

Integration Copyright © Cengage Learning. All rights reserved.

ANTIDERIVATIVES Definition: reverse operation of finding a derivative.

5.a – Antiderivatives and The Indefinite Integral.

position time position time tangent!  Derivatives are the slope of a function at a point  Slope of x vs. t  velocity - describes how position changes.

Ch. 6 – The Definite Integral

Integration 4 Copyright © Cengage Learning. All rights reserved.

Warm up Problems More With Integrals It can be helpful to guess and adjust Ex.

Blue part is out of 50 Green part is out of 50  Total of 100 points possible.

To find the area under the curve Warm-Up: Graph. Area under a curve for [0, 3]  The area between the x-axis and the function Warm-up What is the area.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Integration 5 Antiderivatives Substitution Area Definite Integrals Applications.

INTEGRALS 5. If u = g(x), Summary If f is even, f(-x) = f(x), If f is odd, f(-x) = -f(x),

Essential Question: How is a definite integral related to area ?

Chapter 6 The Definite Integral. There are two fundamental problems of calculus 1.Finding the slope of a curve at a point 2.Finding the area of a region.

Introduction to Integrals Unit 4 Day 1. Do Now  Write a function for which dy / dx = 2 x.  Can you think of more than one?

Indefinite Integrals or Antiderivatives

Copyright © Cengage Learning. All rights reserved.

Section 6.2 Constructing Antiderivatives Analytically

Antiderivatives 5.1.

Antidifferentiation and Indefinite Integrals

6 Integration Antiderivatives and the Rules of Integration

Ch. 2 – Limits and Continuity

Techniques of Integration

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Review Calculus.

ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department

ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department

Ch. 6 – The Definite Integral

The Fundamental Theorem of Calculus Part 1 & 2

Transforming functions

Lesson 18 Finding Definite and Indefinite Integrals

Calculus for ENGR2130 Lesson 2 Anti-Derivative or Integration

C2 – Integration – Chapter 11

Use the Table of Integrals to evaluate the integral. {image}

Fundamental Theorem of Calculus Indefinite Integrals

Lesson 18 Finding Definite and Indefinite Integrals

6.1: Antiderivatives and Indefinite Integrals

Clicker Question 1 According to the FTC, what is the definite integral of f (x) = 1/x2 over the interval from 1 to 5? A. 4/5 B. 1/5 C. -4/5 D. -1/x E.

Lesson 5-4b Net Change Theorem.

(4)² 16 3(5) – 2 = 13 3(4) – (1)² 12 – ● (3) – 2 9 – 2 = 7

Sec 4.9: Antiderivatives DEFINITION Example A function is called an

The Indefinite Integral

Section 1 Antiderivatives and Indefinite Integrals

Presentation transcript:

Blue part is out of 44 Green part is out of 58 Total of 102 points possible Grade is out of 100

Antiderivatives F(x) is called an antiderivative of f (x). Ex. Let F(x) be such that F(x) = f (x). Sketch a graph of F(x). F(x) is called an antiderivative of f (x). Notice that antiderivatives are not unique.

Computing Antiderivatives Ex. Find an antiderivative of 3x2. Ex. Find an antiderivative of x5.

The last answer was It could have been or To describe all possible answers, we write  This is called the general antiderivative.

Pract. Find the general antiderivative of x2 – 4. Def. The indefinite integral of f (x), written , is the general antiderivative of f (x). Ex. “find the integral” requires “+ c” “find an antiderivative” doesn’t need “+ c”

vs. Indefinite integral Has no endpoints Is a function General antiderivative Definite integral Has endpoints Is a number Area under the curve

Integral Rules

Ex. ∫ (4x2 – x3) dx Ex.

Pract. Find the following in groups: