4.4c 2nd Fundamental Theorem of Calculus. Second Fundamental Theorem: 1. Derivative of an integral.

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Presentation transcript:

4.4c 2nd Fundamental Theorem of Calculus

Second Fundamental Theorem: 1. Derivative of an integral.

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

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

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:

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. The long way: Second 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 derivative of an integral is equal to the product of the function evaluated at the upper limit and the derivative of the upper limit.

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.

Lesson 4.4c sketches.gsp length of interval

Mean Value Theorem (for definite integrals) guarantees this point! If f is continuous on then at some point c in, Lesson 4.4c sketches.gsp Solve for x where f(x) = 3/2

Courage is the price that Life exacts for granting peace. 4.4c Homework: worksheet and Function average: #45, 47, 51, 53, 55, 63, Motion Problems: #71,83, 87, 89, 91