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6034 Fundamental Theorem of Calculus (Part 2). AB CALCULUS. The Indefinite Integral (Antiderivative) finds a Family of Functions whose derivative is given. Given an Initial Condition we find the Particular Function. The Definite Integral as a Particular Function :.
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6034 Fundamental Theorem of Calculus (Part 2) AB CALCULUS
The Indefinite Integral (Antiderivative) finds a Family of Functions whose derivative is given. Given an Initial Condition we find the Particular Function
The Definite Integral as a Particular Function: Evaluate the definite integral. Evaluate at 1 Evaluate the Definite Integral for each of these points. The Definite Integral is actually finding points on the Accumulation graph.
Since A(x) is a function, what then is the rate of change of that function? Take derivative In words, integration and differentiation are inverse operations
2nd Fundamental Theorem of Calculus Given: , we want to find Note: a is a constant, u is a function of x; and the order matters! “a” is a constant 2nd Fundamental Theorem of Calculus: If fis continuous on an open interval, I, containing a point, a, then for every x in I :
Demonstration: < function x only > find In Words: Sub in the function u and multiply by derivative of u
Example: Find and verify: this Not this =
Example: Find without Integrating:
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)
THE COMPOSITE FUNCTION If , (a composite function) then In Words: Sub u in for t and multiply by u’
Demonstration: < The composite function > Find: Verify In Words: Sub in for t and multiply by the derivative of
Example : Find without Integrating: If , solve for
Example: Rewriting the Integral Find without integrating: Show middle step
Example: Rewriting the Integral - Two variable limits: Find withoutIntegrating: break into two parts . . . . . chose any number in domain of for a and rewrite into required form .
Last Update: • 1/25/11 • Worksheet