Slide 5- 1 Quick Review. Slide 5- 2 Quick Review Solutions.

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Presentation transcript:

Slide 5- 1 Quick Review

Slide 5- 2 Quick Review Solutions

Slide 5- 3 What you’ll learn about Fundamental Theorem, Part 1 Graphing the Function Fundamental Theorem, Part 2 Area Connection Analyzing Antiderivatives Graphically … and why The Fundamental Theorem of Calculus is a Triumph of Mathematical Discovery and the key to solving many problems.

The Fundamental Theorem of Calculus, Part 1 If f is continuous on, then the function has a derivative at every point in, and

Slide 5- 5 The Fundamental Theorem of Calculus

First Fundamental Theorem: 1. Derivative of an integral.

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

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

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

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:

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

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

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.

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.

The Fundamental Theorem of Calculus, Part 2 If f is continuous at every point of, and if F is any antiderivative of f on, then (Also called the Integral Evaluation Theorem) We already know this! To evaluate an integral, take the anti-derivatives and subtract. 

Slide The Fundamental Theorem of Calculus, Part 2

Slide Example Evaluating an Integral

Slide How to Find Total Area Analytically Now do example problem 41 on page 303.