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.

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Presentation transcript:

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 choice. Quote from CALCULUS by Ross L. Finney and George B. Thomas, Jr., ©1990.

If f is continuous on [a,b], then 1. Derivative of an integral. Fundamental Theorem of Calculus (FTC) Part One

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

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

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

What’s the significance? Every continuous f is the derivative of some other function, namely Every continuous function has an antiderivative. The processes of integration and differentiation are inverses of each other!

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. Example 1

You try! Find dy/dx

The upper limit of integration does not match the derivative, but we could use the chain rule. Example 2

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. Example 3

Neither limit of integration is a constant. It does not matter what constant we use! We split the integral into two parts. Example 4

(Limits are reversed.) (Chain rule is used.)

Homework 6.4A Derivatives Quiz coming soon!