Leonhard Euler 1707 – 1783 Leonhard Euler was a Swiss mathematician who made enormous contributions to a wide range of mathematics and physics including.

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

5.4 Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998 Morro Rock, California.

CALCULUS II Chapter 5. Definite Integral Example.

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.

Derivatives of polynomials Derivative of a constant function We have proved the power rule We can prove.

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,

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.

5.3 Definite Integrals and Antiderivatives Greg Kelly, Hanford High School, Richland, Washington.

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.

CALCULUS II Chapter 5.

5.3 Definite Integrals and Antiderivatives. 0 0.

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

Warmup 1) 2). 5.4: Fundamental Theorem of Calculus.

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.

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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Fundamental Theorem of Calculus Section 5.4.

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

First Fundamental Theorem. If you were being sent to a desert island and could take only one equation with you, might well be your choice. Here is a calculus.

Part I of the Fundamental Theorem says that we can use antiderivatives to compute definite integrals. Part II turns this relationship around: It tells.

CHAPTER 4 SECTION 4.4 THE FUNDAMENTAL THEOREM OF CALCULUS.

5.4 Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998 Morro Rock, California.

SECTION 4-4 A Second Fundamental Theorem of Calculus.

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

The Seven Bridges of Konigsberg (circa 1735) In Konigsberg, Germany, a river ran through the city such that in its centre was an island, and after passing.

5.4 Second 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 choice.

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

Leonhard Euler 1707 – 1783 Leonhard Euler 1707 – 1783 Leonhard Euler was a Swiss mathematician who made enormous contributions to a wide range of mathematics.

Second Fundamental Theorem of Calculus

Integral Review Megan Bryant 4/28/2015. Bernhard Riemann  Bernhard Riemann ( ) was an influential mathematician who studied under Gauss at the.

Pierre-Simon Laplace 1749 – 1827 Pierre-Simon Laplace 1749 – 1827 Laplace proved the stability of the solar system. In analysis Laplace introduced the.

The Fundamental Theorem of Calculus Area and The Definite Integral OBJECTIVES  Evaluate a definite integral.  Find the area under a curve over a given.

Slide 5- 1 Quick Review. Slide 5- 2 Quick Review Solutions.

MAT 1235 Calculus II 4.3 Part I The Fundamental Theorem of Calculus

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

Fundamental Theorem of Calculus: Makes a connection between Indefinite Integrals (Antiderivatives) and Definite Integrals (“Area”) Historically, indefinite.

Chapter Five Integration.

Indefinite Integrals or Antiderivatives

5.3 The Fundamental Theorem of Calculus

Calculus Section 3.6 Use the Chain Rule to differentiate functions

Lecture 22 Definite Integrals.

MTH1170 The Fundamental Theorem of Calculus

Exponential Functions

Review Calculus.

6-4 Day 1 Fundamental Theorem of Calculus

Ch. 6 – The Definite Integral

Copyright © Cengage Learning. All rights reserved.

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

Lesson 3: Definite Integrals and Antiderivatives

The Fundamental Theorem of Calculus (FTC)

5.3 Definite Integrals and Antiderivatives

1. Reversing the limits changes the sign. 2.

Section 5.3 Definite Integrals and Antiderivatives

5.4 First Fundamental Theorem

Sec 5.3: THE FUNDAMENTAL THEOREM OF CALCULUS

The Fundamental Theorem of Calculus

5.4 First Fundamental Theorem

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Integration by Substitution

Copyright © Cengage Learning. All rights reserved.

7.2 Antidifferentiation by Substitution

5.4 Fundamental Theorem of Calculus

Fundamental Theorem of Calculus

6.4 Fundamental Theorem of Calculus

Fundamental Theorem of Calculus

5.3 Definite Integrals and Antiderivatives MLK JR Birthplace

Presentation transcript:

Leonhard Euler 1707 – 1783 Leonhard Euler was a Swiss mathematician who made enormous contributions to a wide range of mathematics and physics including analytic geometry, trigonometry, geometry, calculus and number theory.

Leonhard Euler 1707 - 1783

Leonhard Euler’s Work Quantity: Euler’s collected works, the Opera Omnia, run to over 25,000 pages. In 1910 Gustav Eneström catalogued Euler’s 866 books and papers in a document that itself ran to 388 pages. After his death, Euler published … … 228 papers.

Quality: Mathematical terms (from MATHWORLD Wolfram): Euler – 96 entries Gauss – 70 entries Cauchy – 33 entries Bach – 0 entries

The number e (1748)

Euler’s Identity (1748)

Another look at FTC Part 1

t f (t) g(x) x

Use the First Fundamental Theorem of Calculus to find the derivative of the function:

First Fundamental Theorem: 1. Derivative of an integral.

First Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration.

First Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

First Fundamental Theorem: New variable. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

The long way: First Fundamental Theorem: 1. Derivative of an integral. Derivative matches upper limit of integration. Lower limit of integration is a constant.

1. Derivative of an integral. Derivative matches upper limit of integration. Lower limit of integration is a constant.

The upper limit of integration does not match the derivative, but we could use the chain rule.

The lower limit of integration is not a constant, but the upper limit is. We can change the sign of the integral and reverse the limits.

Neither limit of integration is a constant. We split the integral into two parts. It does not matter what constant we use! How about zero! (Limits are reversed.) (Chain Rule.)

or CHAIN RULE

We end this section by bringing together the two parts of the Fundamental Theorem. We noted that Part 1 can be rewritten as which says that if f is integrated and then the result is differentiated, we arrive back at the original function f. Part 2 is a formula for computing the definite integral, the area under the curve.