Essential Understanding 3.3

Essential Understanding 3.3 Students will understand that the Fundamental Theorem of Calculus, which has two distinct formulations, connects differentiation and integration.

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Learning Objective 3.3A Students will be able to analyze functions defined by an integral.

Essential Knowledge 3.3A1 Students will know that the definite integral can be used to define new functions; for example,

Essential Knowledge 3.3A2 Students will know that a if f is a continuous function on the interval [a, b], then , where x is between a and b.

Essential Knowledge 3.3A3 Students will know that graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as

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Learning Objective 3.3B(a) Students will be able to calculate antiderivatives and (b) evaluate definite integrals.

Essential Knowledge 3.3B1 Students will know that the function defined by , is an antiderivative of f.

Essential Knowledge 3.3B2 Students will know that if f is continuous on the interval [a, b] and F is an antiderivative of f, then

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Essential Knowledge 3.3B3 Students will know that the notation means that F'(x) = f(x), and is called an indefinite integral of the function f.

Essential Knowledge 3.3B4 Students will know that many functions do not have closed form antiderivatives.

Essential Knowledge 3.3B5 Techniques for finding antiderivatives include algebraic manipulation (such as long division and completing the square) and substitution of variables.