Presentation on theme: "4009 Fundamental Theorem of Calculus (Part 2) BC CALCULUS."— Presentation transcript:
4009 Fundamental Theorem of Calculus (Part 2) BC 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 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.
Since A(x) is a function, what then is the rate of change of that function? In words, integration and differentiation are inverse operations
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 :
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: