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Warm Up

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6.4 Fundamental Theorem of Calculus

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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.

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

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

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

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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

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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!

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

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You try! Find dy/dx

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

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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

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

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

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Homework 6.4A Derivatives Quiz coming soon!

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