This section contains documents created from scanned original files, which are inaccessible to screen reader software. A "#" symbol is used to denote such documents.
Lecture Notes
PDF
Section 1, Page 1 to page 2
Statement and explanation of the First and Second Fundamental Theorems, with examples.
Prof. David Jerison
None
Back to Top
PDF
Section 1, Page 2 to page 3
Explanation of the implications and applications of the Second Fundamental Theorem, including an example.
Prof. David Jerison
The Fundamental Theorems of Calculus (page 1 of this file)
Back to Top
PDF
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.
Prof. David Jerison
The Fundamental Theorems of Calculus (page 1 of this file)
Back to Top
PDF
Section 3, Page 5 to page 6
Fundamental Theorem of Calculus is presented and justified.
Prof. Jason Starr
Riemann Integrals (section 4 of lecture 14) and Derivatives (section 2 of lecture 1)
Back to Top
PDF
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.
Prof. David Jerison
The Fundamental Theorems of Calculus (page 1 of this file)
Back to Top
Online Textbook Chapter
Document
Statement and explanation of the two forms for the Fundamental Theorem of Calculus.
Prof. Daniel J. Kleitman
Anti-derivatives (OT19.1) and Riemann Sums (OT20.2)
Back to Top
Practice Problem
PDF
Problem 1 (page 1)
Using the Fundamental Theorem to evaluate the derivative of a function which is defined in terms of an integral.
Prof. Jason Starr
None
Solution (PDF) Pages 2 to 3
Back to Top
Exam Questions
PDF
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.
Prof. David Jerison
None
Solution (PDF)# Page 1
Back to Top
PDF
Problem 5 (page 1)
Using the Fundamental Theorem to evaluate an integral.
Prof. David Jerison
None
Solution (PDF)# Page 1
Back to Top
PDF
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.
Prof. David Jerison
None
Solution (PDF)# Page 1
Back to Top
PDF
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.
Prof. David Jerison
None
Solution (PDF)# Page 2
Back to Top
PDF
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.
Prof. David Jerison
None
Solution (PDF)# Page 1
Back to Top
PDF - 2.2 MB
Problem 3D-1 (page 23) to problem 3D-8 (page 23)
Eight questions which involve functions defined with integrals, and these functions must be described, evaluated, or used to prove some statement.
Prof. David Jerison
Back to Top
