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4009 Fundamental Theorem of Calculus (Part 2) BC CALCULUS

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The Indefinite Integral (Antiderivative) finds a Family of Functions whose derivative is given. Given an Initial Condition we find the Particular Function

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The Definite Integral as a Particular Function: Evaluate at Evaluate the Definite Integral for each of these points. The Definite Integral is actually finding points on the Accumulation graph. Evaluate the definite integral.

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Since A(x) is a function, what then is the rate of change of that function? In words, integration and differentiation are inverse operations

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2 nd Fundamental Theorem of Calculus Given:, we want to find Note: a is a constant, u is a function of x ; and the order matters! 2 nd Fundamental Theorem of Calculus: If f is continuous on an open interval, I, containing a point, a, then for every x in I :

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Demonstration: find In Words:

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Example: Find and verify:

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Example: Find without Integrating:

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THE COMPOSITE FUNCTION If g(x) is given instead of x: = In words: Substitute in g(x) for t and then multiply by the derivative of g(x)…exactly the chain rule (derivative of the outside * derivative of the inside)

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THE COMPOSITE FUNCTION If, (a composite function) then In Words:

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Demonstration: Find: In Words:

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Example : Find without Integrating: If, solve for

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Example: Rewriting the Integral Find without integrating: Show middle step

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Example: Rewriting the Integral - Two variable limits: Find without Integrating: break into two parts..... chose any number in domain of for a and rewrite into required form.

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Last Update: 1/25/11 Worksheet

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