Concepts of Calculus
Course Syllabus
Overview. Reform efforts in calculus have reshaped the nature of that course. The success of
these new courses depends on teachers who understand the subject from conceptual, theoretical,
and practical points of view. The typical “Real Analysis” course proves properties of real
numbers fundamental to the theory of calculus but often fall short of making an explicit
connection to the actual material that would comprise a typical high school calculus course. This
course will bridge that gap for teachers of mathematics who would like to teach calculus or just to
better understand how other mathematical ideas lead to calculus concepts.
Textbooks:
Reform Calculus Book
Traditional Calculus Book
Advanced Calculus
Ostebee/Zorn Calculus (2nd Edition)
Stewart Calculus, Thomas, Calculus, or any other traditional
calculus text
Schaum’s Outline of Advanced Calculus
Schedule. The course will be broken into three parts: Topics from Advanced Calculus, Concepts
from “Reform Calculus” and Calculus from a Traditional Viewpoint. The following outline is an
example of how the goals of the course could be met:
I. Definition of Limit
a. Review limits by reading the limit chapters in traditional and reform books.
b. Do problems from traditional calculus book
c. Prove the Limit Laws
d. Discuss how to explain limits to calculus students in a way that gives it meaning
e. Read the advanced calculus book on limits and continuity. Do problems from the
book.
II. Derivatives
a. Read introductory chapter in the reform book and the traditional book, and
compare. Do problems from both.
b. Discuss the definition of derivative. Do some derivatives using the definition
and limits.
c. Prove the derivative laws. Read chapter on derivatives in the advanced calculus
book, and do problems from the book.
III. Mean Value Theorem
a. Prove the Mean Value Theorem. Do problems from traditional text.
b. Discuss the way the Mean Value Theorem comes up in reform programs, and the
role of this result in calculus.
c. Read about Mean Value Theorem in the advanced calculus book, and do
problems from the book.
d. Make a proof for the Mean Value Theorem that calculus students would
understand and find reasonable.
IV. Integration
a. Read the introductory integration chapters in the traditional and reform books.
Do problems from each, and discuss the difference.
b. Review discrete summation formulas and how they lead to integration rules
c. Derive the integration rules from the appropriate derivative rules.
d. Read the integral chapter in the advanced calculus book, and do problems from
the book.
V. Differential equations
a. Discuss the role of differential equations in calculus
b. Do problems from calculus books on differential equations, especially those that
emphasize graphical and numerical concepts.
VI. Sequences and Series (as time permits – this topic is not always included in a high school
course)
a. Discuss the role of sequences and series in calculus, particularly the difference in
emphasis between high school calculus and advanced calculus.
b. Do problems from calculus books on sequences and series, especially those on
power series, since these can be directly applied to previous material.
c. Do problems from Advanced Calculus book on sequences and series.
Grades. Students are expected to demonstrate an aptitude for traditional and reform calculus, an
understanding of major theoretical results that make calculus “work,” and a plan for how to
present material to students to meet the objectives of a high school calculus course.
Students will be expected to provide responses to Blackboard discussion questions posed
throughout the semester. The purpose of these questions, in general, is to have you reflect on the
connections between the course material and the middle/high school curriculum.
There will be homework problem sets assigned each day that will be collected on a regular basis.
They will involve a combination of problems from the reform calculus textbook, the traditional
calculus textbook, as well as advanced calculus material.
Two exams will be given throughout the session.
In addition, you will be responsible for developing a calculus curriculum project that could be
used in a high school calculus course or a project that uses calculus concepts that could be used in
a precalculus course.
The evaluation for the course grade will include assessment in all three of these areas. For
example, the grades might be broken down as follows:
 30% Out-of-class calculus problem sets
 20% Blackboard responses
 30% Two in-class exams on calculus theory (i.e, “Advanced Calculus”)
 20% Calculus curriculum project