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Outline Differentiation as finding slope Integration as finding area Integration as inverse of differentiation
Limit “Limit” as finding the slope of a curve –The slope of the curve y=mx+c is m. –How about the slope of the curve y=x^2? –Approximately, the slope at x1 is (y2-y1)/(x1-x2) x1x2 y1 y2
Limit x1x2 y1 y2
Differentiation of x^2 x1x2 y1 y2 Let’s find dy/dx at x=x1
Differentiation of x^n
Differentiating a sum of two functions of x
Area under curve x Y= f(x) ba
Approximation by Summation xba
Integration: Infinitesimal Approximation by Summation xba … Divide into n stripes, and let n goes to infinity
The Fundamental Theorem of Calculus Integration is antiderivative:
Explaining Fundamental Theorem of Calculus Differentiating area gets curve Differentiating curve gets slope x Y= f(x) ba
Follow the link to the slide. Then click on the figure to play the animation. A Figure Figure
6 Integration Antiderivatives and the Rules of Integration
Chapter 5 Integration.
Section 4.3 Indefinite Integrals and Net Change Theorem Math 1231: Single-Variable Calculus.
Definite Integrals Finding areas using the Fundamental Theorem of Calculus.
Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus.
4.6 Numerical Integration -Trapezoidal Rule -Simpson’s Rule
Stuff you MUST know Cold for the AP Calculus Exam.
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.
Section 5.3 – The Definite Integral
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Antiderivatives and Slope Fields Section 6.1.
The Fundamental Theorem of Calculus Lesson Definite Integral Recall that the definite integral was defined as But … finding the limit is not often.
5.c – The Fundamental Theorem of Calculus and Definite Integrals.
Constructing the Antiderivative Solving (Simple) Differential Equations The Fundamental Theorem of Calculus (Part 2) Chapter 6: Calculus~ Hughes-Hallett.
Section 5.4a FUNDAMENTAL THEOREM OF CALCULUS. Deriving the Theorem Let Apply the definition of the derivative: Rule for Integrals!
Today in Calculus Go over homework Derivatives by limit definition Power rule and constant rules for derivatives Homework.
2.6 – Limits involving Infinity, Asymptotes
3.3: Rules of Differentiation Objective: Students will be able to… Apply the Power Rule, Sum and Difference Rule, Quotient and Product Rule for differentiation.
1 When you see… Find the zeros You think…. 2 To find the zeros...
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:
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