# Representation of Specific Antiderivatives

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## Lecture Notes

#### The Two Fundamental Theorems of Calculus

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Section 1, Page 1 to page 2

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

Instructor: Prof. David Jerison
Prior Knowledge: None

#### The Second Fundamental Theorem: Continuous Functions Have Antiderivatives

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Section 1, Page 2 to page 3

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

Instructor: Prof. David Jerison
Prior Knowledge: The Fundamental Theorems of Calculus (page 1 of this file)

#### Defining New Functions

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Section 3, Page 4 to page 5

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

Instructor: Prof. David Jerison
Prior Knowledge: The Fundamental Theorems of Calculus (page 1 of this file)

#### The Fundamental Theorem of Calculus(18.01, Fall 2005)

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Section 3, Page 5 to page 6

Fundamental Theorem of Calculus is presented and justified.

Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Integrals (section 4 of lecture 14) and Derivatives (section 2 of lecture 1)

#### Proof of the Two Fundamental Theorems

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Section 4, Page 6 to page 7

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.

Instructor: Prof. David Jerison
Prior Knowledge: The Fundamental Theorems of Calculus (page 1 of this file)

## Online Textbook Chapter

#### The Fundamental Theorem

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

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Anti-derivatives (OT19.1) and Riemann Sums (OT20.2)

## Practice Problem

#### Fundamental Theorem of Calculus

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Problem 1 (page 1)

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

Instructor: Prof. Jason Starr
Prior Knowledge: None

## Exam Questions

#### Functions Defined By Integrals

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Problem 4 (page 1)

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.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Fundamental Theorem of Calculus

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Problem 5 (page 1)

Using the Fundamental Theorem to evaluate an integral.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Functions Defined by Integrals

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Problem 3 (page 1)

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

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Functions Defined by Integrals

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Problem 5 (page 2)

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

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Functions Defined by Integrals

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Problem 10 (page 2)

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

Instructor: Prof. David Jerison
Prior Knowledge: None