Section 5.4a FUNDAMENTAL THEOREM OF CALCULUS. Deriving the Theorem Let Apply the definition of the derivative: Rule for Integrals!

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Section 5.4a FUNDAMENTAL THEOREM OF CALCULUS

Deriving the Theorem Let Apply the definition of the derivative: Rule for Integrals!

Deriving the Theorem This is average value of f from x to x + h. Assuming that f is continuous, it takes on its average value at least once in the interval… For some c between x and x + h Back to the proof…

Deriving the Theorem What happens to c as h goes to zero??? As x + h gets closer to x, it forces c to approach x… Since f is continuous, this means that f(c) approaches f(x): Putting it all together…

Deriving the Theorem Definition of Derivative Rule for Integrals For some c between x and x + h Because f is continuous

The Fundamental Theorem of Calculus, Part 1 If is continuous on [a, b], then the function has a derivative at every point x in [a, b], and …the definite integral of a continuous function is a differentiable function of its upper limit of integration…

The Fundamental Theorem of Calculus, Part 1 Every continuous function is the derivative of some other Every continuous function is the derivative of some other function. function. The processes of integration and differentiation are The processes of integration and differentiation are inverses of one another. inverses of one another. Every continuous function has an antiderivative. Every continuous function has an antiderivative. A Powerful Theorem Indeed!!!

The Fundamental Theorem of Calculus, Part 1 Evaluate each of the following, using the FTC.

The Fundamental Theorem of Calculus, Part 1 Find dy/dx if The upper limit of integration is not x  y is a composite of: and Apply the Chain Rule to find dy/dx:

The Fundamental Theorem of Calculus, Part 1 Find dy/dx if

The Fundamental Theorem of Calculus, Part 1 Find dy/dx if Chain Rule

Deriving More of the Theorem Let If F is any antiderivative of f, then F(x) = G(x) + C for some constant C. Let’s evaluate F (b) – F(a): 0

The Fundamental Theorem of Calculus, Part 2 If is continuous at every point of [a, b], and if F is any antiderivative of on [a, b], then This part of the Fundamental Theorem is also called the Integral Evaluation Theorem.

The Fundamental Theorem of Calculus, Part 2 Any definite integral of any continuous function can be calculated without taking limits, without calculating Riemann sums, and often without major effort  all we need is an antiderivative of !!! Another Very Powerful Theorem!!!

The Fundamental Theorem of Calculus, Part 2 The usual notation for F(b) – F(a) is A “note” on notation: or depending on whether F has one or more terms…

The Fundamental Theorem of Calculus, Part 2 Evaluate the given integral using an antiderivative. Antiderivative: How can we support this answer numerically???

The Fundamental Theorem of Calculus, Part 2 Evaluate the given integral using an antiderivative. Antiderivative:

The Fundamental Theorem of Calculus, Part 2 Evaluate the given integral using an antiderivative. Antiderivative:

The Fundamental Theorem of Calculus, Part 2 Evaluate the given integral using an antiderivative. Antiderivative:

The Fundamental Theorem of Calculus, Part 2 Evaluate the given integral using an antiderivative. Antiderivative:

The Fundamental Theorem of Calculus, Part 2 Evaluate the given integral using an antiderivative. Antiderivative: