# Calc05_4 - 5.4 First Fundamental Theorem Morro Rock...

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5.4 First Fundamental Theorem Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1998 Morro Rock, California

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If you were being sent to a desert island and could take only one equation with you, ( 29 ( 29 x a d f t dt f x dx = might well be your choice. Here is my favorite calculus textbook quote of all time, from CALCULUS by Ross L. Finney and George B. Thomas, Jr., ©1990.
The Fundamental Theorem of Calculus, Part 1 If f is continuous on , then the function [ ] , a b ( 29 ( 29 x a F x f t dt = has a derivative at every point in , and [ ] , a b ( 29 ( 29 x a dF d f t dt f x dx dx = =

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( 29 ( 29 x a d f t dt f x dx = First Fundamental Theorem: 1. Derivative of an integral.
( 29 ( 29 a x d f t dt x f x d = 2. Derivative matches upper limit of integration. First Fundamental Theorem: 1. Derivative of an integral.

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( 29 ( 29 a x d f t dt f x dx = 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.
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