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5.4 Fundamental Theorem of Calculus

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It is difficult to overestimate the power of the equation: It says that every continuous function f is the derivative of some other function, namely. It says that every continuous function has an antiderivative. It says that the processes of integration and differentiation are inverses of one another.

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Applying the Fundamental Theorem Find by using the Fundamental Theorem.

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The Fundamental Theorem with the Chain Rule Find dy/dx if

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Variable Lower Limits of Integration Find dy/dx.

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Constructing a Function with a Given Derivative and Value Find a function y = f(x) with derivative that satisfies the condition f(3) = 5. Since y(3) = 0, we have only to add 5 to this function to construct one with derivative tan x whose value at x = 3 is 5:

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The second part of the Fundamental Theorem of Calculus shows how to evaluate definite integrals directly from antiderivatives.

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Evaluating an Integral Evaluate using an antiderivative.

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Finding Area Using Antiderivatives Find the area of the region between the curve y = 4 – x², 0≤ x ≤ 3, and the x-axis.

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Homework!!!!! Textbook – p. 302 – 303 # 1 – 26, 41 – 44.

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