Section 3.9 Antiderivatives

Slides:

Advertisements

Similar presentations

6 Integration Antiderivatives and the Rules of Integration

Advertisements

In this handout, 4. 7 Antiderivatives 5

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

Derivative as a function Math 1231: Single-Variable Calculus.

Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus.

Antiderivatives Definition A function F(x) is called an antiderivative of f(x) if F ′(x) = f (x). Examples: What’s the antiderivative of f(x) = 1/x ?

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,

Section 5.3 – The Definite Integral

Section 5.3: Evaluating Definite Integrals Practice HW from Stewart Textbook (not to hand in) p. 374 # 1-27 odd, odd.

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

Exam 1 Math 1231: Single-Variable Calculus. Question 1: Limits.

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

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

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

Today in Calculus Go over homework Derivatives by limit definition Power rule and constant rules for derivatives Homework.

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

Section 4.1 Areas and Distances Math 1231: Single-Variable Calculus.

Miss Battaglia AP Calculus AB/BC. Definition of Antiderivative A function F is an antiderivative of f on an interval I if F’(x)=f(x) for all x in I. Representation.

Antiderivatives. Antiderivatives Definition A function F is called an antiderivative of f if F ′(x) = f (x) for all x on an interval I. Theorem.

CHAPTER 4 SECTION 4.4 THE FUNDAMENTAL THEOREM OF CALCULUS.

Section 6.4 Second 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.

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.

SECTION 4-4 A Second Fundamental Theorem of Calculus.

F UNDAMENTAL T HEOREM OF CALCULUS 4-B. Fundamental Theorem of Calculus If f(x) is continuous at every point [a, b] And F(x) is the antiderivative of f(x)

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

Section 4.2 Definite Integral Math 1231: Single-Variable Calculus.

Section 3.1 Maximum and Minimum Values Math 1231: Single-Variable Calculus.

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.

The Fundamental Theorem of Calculus

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 – The Fundamental Theorem of Calculus

Clicker Question 1 Are you here? – A. Yes – B. No – C. Not sure.

Rate of Change and Derivative Math 1231: Single-Variable Calculus.

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

CHAPTER Continuity Fundamental Theorem of Calculus In this lecture you will learn the most important relation between derivatives and areas (definite.

Section 6.2* The Natural Logarithmic Function. THE NATURAL LOGARITHMIC FUNCTION.

5-7: The 1 st Fundamental Theorem & Definite Integrals Objectives: Understand and apply the 1 st Fundamental Theorem ©2003 Roy L. Gover

Theorems Lisa Brady Mrs. Pellissier Calculus AP 28 November 2008.

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

Antiderivatives Section 4.8. Lesson Objectives Students Will… ›Find the antiderivative of a function from its derivative f(x). ›Apply antiderivatives.

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

Section 17.4 Integration LAST ONE!!! Yah Buddy!.  A physicist who knows the velocity of a particle might wish to know its position at a given time. 

Indefinite Integrals or Antiderivatives

Mean Value Theorem 5.4.

6 Integration Antiderivatives and the Rules of Integration

3.3: Increasing/Decreasing Functions and the First Derivative Test

4.4 The Fundamental Theorem of Calculus

Applications of Derivatives

The Fundamental Theorems of Calculus

§4.9 Antiderivatives There are two branches in calculus:

Lesson 18 Finding Definite and Indefinite Integrals

Unit 6 – Fundamentals of Calculus Section 6

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

Lesson 18 Finding Definite and Indefinite Integrals

Section 4.3 – Area and Definite Integrals

The Fundamental Theorem of Calculus (FTC)

Warm Up Before you start pg 342.

AP Calculus November 29-30, 2016 Mrs. Agnew

Definite Integrals and Antiderivatives

Definite Integrals & Antiderivatives

Copyright © Cengage Learning. All rights reserved.

Presentation transcript:

Section 3.9 Antiderivatives Math 1231: Single-Variable Calculus

Antiderivative: Definition Definition A function F(x) is called an antiderivative of f on an interval I if F ‘ (x) = f(x) for all x in I. f(x) antiderivative F(x) derivative

General antiderivative Theorem If F is an antiderivative of f on an interval I, then the general antiderivative of f on I is F(x) + C, where C is an arbitrary constant. Any two antiderivatives of f(x) only differ by a constant.

Antiderivative Table Function Particular antiderivative General antiderivative f(x) F(x) F(x) + C f(x) + g(x) F(x) + G(x) F(x) + G(x) + C c*f(x) c*F(x) c*F(x) + C xn (n≠1) xn+1/(n+1) xn+1/(n+1) + C sin(x) - cos(x) - cos(x) + C cos(x) sin(x) + C sec2(x) tan(x) tan(x) + C sec(x)tan(x) sec(x) sec(x) + C

Examples Example Find the general antiderivative of the function f(x) = 3cos(t) – 4sin(t). Example Find the general antiderivative of the function f(x) = (1+ t + t2)/sqrt(t). Example Find f given f ’’ (x) = (2/3) x2/3. Example Find f given f ’ (x) = x*sqrt(x) and f(1) = 2.