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

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Presentation on theme: "The Fundamental Theorem of Calculus This One’s a Two-Parter! FTC Part Two First Then FTC Part One."— Presentation transcript:

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 FTC: Part One…An Example Find Find Find

6 FTC: Part One…With a Twist…or a Kink! 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! FTC: Part One…Look at the Kinkage! So… because of the…CHAIN RULE!!! 2xcos(x 2 )ZERO!!!

8 Some More Kinky Problems… Some More Kinky Problems… Find Find Find

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

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

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