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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus. This One’s a Two- Parter ! FTC Part Two First Then FTC Part One. FTC: Part Two…First. If f is continuous on [ a, b ], and if F is any antiderivative of f on [ a, b ], then But we already knew that now didn’t we?!. FTC: Part One.

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The Fundamental Theorem of Calculus

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  1. The Fundamental Theorem of Calculus This One’s a Two-Parter! FTC Part Two First Then FTC Part One

  2. FTC: Part Two…First • If f is continuous on [a, b], and if F is any antiderivative of f on [a, b], then • But we already knew that now didn’t we?!

  3. FTC: Part One • If f is continuous on [a, b], then the function has a derivative at every point x in [a, b], and

  4. FTC: Part One…One More Time • Isn’t ? • And, since F(a) is just a constant w/a derivative of zero, then to take the derivative we just take the derivative of F(x)…which is f(x)!

  5. FTC: Part One…An Example • Find • Find • Find

  6. FTC: Part One…With a Twist…or a Kink! • Apply the second FTC on the integral… • Now take the derivative of (don’t forget the chain rule!)

  7. FTC: Part One…Look at the Kinkage! • So… because of the…CHAIN RULE!!! 2xcos(x2) ZERO!!!

  8. Find Find Find Some More Kinky Problems…

  9. Using Graphs and the FTC • Find h(1).

  10. Using Graphs and the FTC • Is h(0) positive or negative? JYA

  11. Using Graphs and the FTC • Find the value of x for which h(x) is a maximum. Where does f(x) change from positive to negative?

  12. Using Graphs and the FTC This is a graph of f(t) NOT h(x)! • Find the x-values of the inflection points of h(x). Where does the slope of f(x) change?

  13. Assignment • p. 302 #1-19 odd • p. 303 #57 • 2002 AB-2 Handout

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