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Antiderivatives An antiderivative of f(x) is any function F(x) such that F’(x) = f(x)

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Example Find antiderivatives of f(x) = x 2

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Example Find antiderivatives of f(x) = 2x

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Example Find antiderivatives of f(x) = 1/x

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Theorem If F(x) is an antiderivative of f(x) then F(x) + C is an antiderivative of f(x) for any constant C

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Antiderivatives Graphically Match the function to its antiderivative f(x) F(x) 1) 2) 3) 4) D C B A

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

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First Fundamental Theorem: 1. Derivative of an integral.

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

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

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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:

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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:

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

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Example If find F’(x)

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

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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.

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Example If f(x) =find f’(x)

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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.

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HW: p. 287/37-42

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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) To evaluate an integral, take the anti-derivatives and subtract.

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Antiderivatives Antiderivatives are also called indefinite integrals They are sometimes written Note that there are no limits on the integral Do not confuse with definite integrals!

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Common Antiderivatives

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Evaluate the integral using FTC2

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Rewrite then evaluate the integral using FTC2

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Evaluate the integral involving trigonometric functions using FTC2

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Special Example: absolute value

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Area using Integrals Find the zeros of the function over the interval [a,b] integrate over each subinterval add the absolute value of the integrals

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Example: Find the area using integrals

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Using the GC to find the integral hit MATH then 9 fnInt( will come up on the screen type in the function, comma, x, comma, -a, comma, b) then hit ENTER Ex:

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Examples: Use GC

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Area using GC To find the area under the curve f(x) from [a,b] type fnInt(abs(f(x)),x,a,b) Example: Find the area under the curve y = xcos 2 x on [-3, 3]

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HW: FTC 2 wksheet

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