6-4 Day 1 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

5.2 Definite Integrals Quick Review Quick Review Solutions.

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

Example We can also evaluate a definite integral by interpretation of definite integral. Ex. Find by interpretation of definite integral. Sol. By the interpretation.

Georg Friedrich Bernhard Riemann

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

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)

Section 4.3 – Riemann Sums and Definite Integrals

5.2 Definite Integrals.

Areas & Definite Integrals TS: Explicitly assessing information and drawing conclusions.

4-4: The Fundamental Theorems Definition: If f is continuous on [ a,b ] and F is an antiderivative of f on [ a,b ], then: The Fundamental Theorem:

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.

4.4 The Fundamental Theorem of Calculus

4.4 The Fundamental Theorem of Calculus and The Second Fundamental Theorem of Calculus.

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.

 (Part 2 of the FTC in your book)  If f is continuous on [a, b] and F is an antiderivative of f on [a, b], then **F(b) – F(a) is often denoted as This.

5.4 Fundamental Theorem of Calculus Quick Review.

Chapter 5: The Definite Integral Section 5.2: Definite Integrals

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

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

Calculus Date: 3/7/2014 ID Check Obj: SWBAT connect Differential and Integral Calculus Do Now: pg 307 #37 B #23 HW Requests: SM pg 156; pg 295 #11-17 odds,

Fundamental Theorem of Calculus

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

Chapter 6 Integration Section 5 The Fundamental Theorem of Calculus (Day 1)

Section 4.3 Day 2 Riemann Sums & Definite Integrals AP Calculus BC.

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

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

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

SECTION 4-3-B Area under the Curve. Def: The area under a curve bounded by f(x) and the x-axis and the lines x = a and x = b is given by Where and n is.

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.

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

Chapter Five Integration.

Chapter 5 AP Calculus BC.

Indefinite Integrals or Antiderivatives

Chapter 5 The Definite Integral. Chapter 5 The Definite Integral.

4.4 The Fundamental Theorem of Calculus

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

4.4 The Fundamental Theorem of Calculus

4.4 The Fundamental Theorem of Calculus

ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department

ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department

Ch. 6 – The Definite Integral

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

4.9 – Antiderivatives.

Integration & Area Under a Curve

5.3 – The Definite Integral and the Fundamental Theorem of Calculus

Fundamental Theorem of Calculus

7.2 Definite Integration.

Fundamental Theorem of Calculus

4.3 Day 1 – Day 1 Review Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening and is recorded in the table. Estimate the.

The Fundamental Theorem of Calculus

Sec 5.3: THE FUNDAMENTAL THEOREM OF CALCULUS

The Fundamental Theorem of Calculus

Integration: Evaluation and Applications

Warm Up 1. Find 2 6 2

§ 6.3 Definite Integrals and the Fundamental Theorem.

4.4 The Fundamental Theorem of Calculus

§ 6.3 Definite Integrals and the Fundamental Theorem.

AP Calculus December 1, 2016 Mrs. Agnew

Warm Up 1. Find 2 6 2

Fundamental Theorem of Calculus

More Definite Integrals

Integration Techniques

6.4 Fundamental Theorem of Calculus

Fundamental Theorem of Calculus

6-2 definite integrals.

Presentation transcript:

6-4 Day 1 Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus has 2 parts *The Fundamental Theorem of Calculus has 2 parts. One part relates derivatives and integrals and the other establishes how we evaluate integrals by hand. Fundamental Theorem of Calculus: Part 1 Every continuous function f is deriv of some other function Every continuous function has an antiderivative Integration & differentiation are inverses of each other

Ex 1) Using the Fundamental Theorem of Calculus, find: a) b) Ex 2) Find (Use chain rule!)

Ex 3) Find a) b) (variable lower limits)

Ex 4) Find a function y = f (x) with derivative that satisfies the condition f (3) = 5. Since y (3) = 0 We need to add 5 *Note: We can graph the integral of a function by using fnInt. Just put x as the upper limit.

Fundamental Theorem of Calculus: Part 2 (Integral Evaluation Theorem) *says any definite integral of any continuous function can be calculated without taking limits or Riemann sums – so long as antiderivative can be found Ex 5) Evaluate (can check with fnInt)

Ex 6) Find the area of the region between the curve y = 4 – x2 , 0  x  3, and the x-axis.

To find Total Area Analytically Partition using zeros Integrate each subinterval Add up absolute values To Find Total Area Numerically Use fnInt with absolute value Ex 7) Find the area of the region between the curve y = x cos 2x and the x-axis over the interval –3  x  3 Use Math  9: fnInt (abs (x cos 2x), x, –3, 3) abs value Math  Num  1

homework Pg. 294 # 2, 6, 11, 18, 21, 23, 27, 31, 32, 49 Pg. 306 # 1–49 (mult of 1+3n)