# Representation of Specific Antiderivatives

## The Two Fundamental Theorems of Calculus

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Statement and explanation of the First and Second Fundamental Theorems, with examples.

Using the Fundamental Theorem to evaluate an integral.

Eight questions which involve functions defined with integrals, and these functions must be described, evaluated, or used to prove some statement.

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Fundamental Theorem of Calculus is presented and justified.

Using the Fundamental Theorem to evaluate the derivative of a function which is defined in terms of an integral.

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Statement and explanation of the two forms for the Fundamental Theorem of Calculus.

## The Second Fundamental Theorem: Continuous Functions Have Antiderivatives

Explanation of the implications and applications of the Second Fundamental Theorem, including an example.

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## Defining New Functions

Using the Second Fundamental Theorem to define the natural logarithm function and the error function, including diagrams and examples.

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## Proof of the Two Fundamental Theorems

An intuitive argument justifying the Second Fundamental Theorem and a complete proof of the First Fundamental Theorem, inluding remarks about these theorems as they relate to definite and indefinite integrals.

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## Functions Defined By Integrals

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Three part question finding bounds on the values of a function defined in terms of an integral and finding where the function is concave up or concave down.

Four part question which involves finding the derivative, critical points, and an estimation of a function which is defined in terms of an integral.

Four part question which involves finding critical points and values of a function which is defined in terms of an integral.

Finding the first and second derivatives of a function defined in terms of an integral, and expressing another integral in terms of that function.