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Fundamental Theorems of Calculus 6.4. The First (second?) Fundamental Theorem of Calculus If f is continuous on, then the function has a derivative at.

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Presentation on theme: "Fundamental Theorems of Calculus 6.4. The First (second?) Fundamental Theorem of Calculus If f is continuous on, then the function has a derivative at."— Presentation transcript:

1 Fundamental Theorems of Calculus 6.4

2 The First (second?) Fundamental Theorem of Calculus If f is continuous on, then the function has a derivative at every point in, and

3 First Fundamental Theorem: 1. Derivative of an integral.

4 2. Derivative matches upper limit of integration. First Fundamental Theorem: 1. Derivative of an integral.

5 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. First Fundamental Theorem:

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

7 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. The long way: First Fundamental Theorem:

8 Example Applying the Fundamental Theorem

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

10 Example Variable Lower Limits of Integration

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12 Example The Fundamental Theorem with the Chain Rule

13 Neither limit of integration is a constant. It does not matter what constant we use! (Limits are reversed.) (Chain rule is used.) We split the integral into two parts.

14 Second (first?) FTOC

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17 How to Find Total Area Analytically

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