I. Basic
Course Information
MAT
128: Calculus B will be scheduled for
three lecture periods: two periods of 80 minutes length and one one-hour
meeting period.
II. Learning Goals
The primary emphasis of
Calculus B is on learning the second-half of single-variable calculus. Its subject matter is part of the foundation
that many higher-level courses in mathematics, science, and engineering are
built upon.
The major topics covered
are techniques of integration, applications of integrals, an introduction to
differential equations, and sequences and series. Upon completion of the course, we expect
students to show competence with the ideas of calculus and its calculations, to
understand how to apply calculus to solve real-world problems, to exhibit an
improved ability to describe a real-world problem mathematically, to have an
increased mathematical maturity, and to have an improved ability to read, write,
and understand mathematics. These
performance goals are in agreement with the program goals of the MATA, MATT,
and MATC majors.
In Calculus B, students will gain exposure to
both the theoretical and applied aspects of calculus. By working on many real-life problems, students
will gain an appreciation for the practical applications of calculus. Simultaneously, their mathematical maturity will
be built up through the presentation of theory and the expectation of a higher
level of reasoning than has previously been demanded in their mathematics courses. They are also exposed to the concepts and
techniques of problem solving through individual and group work on the
exercises.
Calculus B, together with
its predecessor MAT 127: Calculus A, serves as a bridge course between the high-school
and college mathematical curricula. College level mathematics, science, and
engineering courses demand a higher level of quantitative reasoning than that
demanded in the high school curriculum. As
many of our students enter with AP credit and will be taking Calculus B as
their first mathematics course at the College, an important goal of Calculus B is
to raise the level of their mathematical reasoning skills to the collegiate
level.
III. Student Assessment
Students
will receive regular feedback on their work through the assignment of homework,
quizzes, student presentations and examinations. Through this feedback, students will be able
to see and correct their misunderstandings and improve their performance. Student performance on these assessment instruments
and the performance of students in their future courses such as Calculus C will
be used to assess the success of Calculus B in achieving its learning goals and
its contribution to the fulfillment of the MATA, MATT, and MATC program goals. Peer reviews and student evaluations will
also be used to evaluate the course.
IV. Learning Activities
Learning activities will consist of a combination of lectures, group
work, student presentations, and computer assignments. The specific choice will depend upon the
individual instructor. Outside of class,
students are expected to do a significant amount of individual and group
homework to achieve the learning goals. These
learning activities are typical of the learning activities in the MATA, MATT, MATC programs. By
giving students a multitude of ways to learn and do mathematics, the learning
activities promote a deeper understanding of the concepts of calculus and
contribute to the learning goals of these programs.
Course
Syllabus Guide -- MAT 128: Calculus B
Introduction:
A typical syllabus
for Calculus B follows this sheet. Any
syllabus for Calculus B should include the points listed below (the required
course requirement sections) and use the recommended topics list as the basis
for decisions on the course content.
I. Basic
information on course and instructor
A.
Purpose statement: Calculus B, the second half of the
single-variable calculus experience, is a foundational course for the
mathematics, science, and engineering curricula. It introduces students both to calculus and
the higher expectations of college-level mathematics courses. Calculus B should also inspire mathematical
curiosity and interest in its students.
B.
Course
description: A second course in calculus covering integral calculus and series. The course will cover both the theoretical
and applied aspects of Calculus.
C.
Course
prerequisites: Calculus A.
II. Learning
goals
A.
Content goals: The
choice of topics covered and their emphases should be based upon the attached
recommended topics list.
B.
Performance
goals: At the completion of the course, students should show competence with
the ideas of calculus and its calculations, to understand how to apply calculus
to solve real-world problems, to exhibit an improved ability to describe a
real-world problem mathematically, to have an increased mathematical maturity, and
to have an improved ability to read, write, and understand mathematics.
III. Student
assessment
A.
Assessment plan: Students will receive regular feedback on
their work through the assignment of homework, quizzes, student presentations
and examinations. A syllabus should
clearly describe the schedule for these assessment tools and how they will be
used to calculate grades.
B.
Rationale: Students need to be able to use calculus
correctly in their future courses. Through the use of regular feedback from
homework, quizzes, student presentations and examinations, students will be
able to see and correct their misunderstandings and improve their
performance.
C.
Methods and
criteria: We will use the assessment of
homework, quizzes, student presentations, and examinations to evaluate student
accomplishment of the course learning goals.
These assessment tools are similar to the manner in which students will need
to display their knowledge of calculus in the future and are an appropriate way
to assess the accomplishment of course learning goals.
IV. Learning
activities
A.
Summary of
learning activities: Learning activities
will consist of a combination of lectures, group work, student presentations,
and computer assignments. The specific
choice will depend upon the individual instructor. Outside of class, students are expected to do
a significant amount of individual and group homework to achieve the learning
goals.
B.
Calendar or
outline: A guide to the organization of
the course, a schedule of assessment tools, and a plan for the coverage of
topics should be provided to the students.
As an approximate guide, 1-2 topics on the recommended list can be
covered in a class period. Homework, quizzes,
and examinations should be spaced at appropriate intervals throughout the
semester.
C.
Rationale By giving students a multitude of ways to
learn and do mathematics, the learning activities promote a deeper
understanding of the concepts of calculus and contribute to the learning goals
of these programs. A regular spacing of
assessment tools insures that students receive continual regular feedback on
their work.
Recommended
Topics List
A. Techniques of Integration (10 lectures)
1. Integration by Parts
2. Trigonometric
Integrals
3. Trigonometric
Substitution
4. Integration
of Rational Functions by Partial Fractions
5. Strategy
for Integration
6. Integration
using Tables and Computer Algebra Systems (optional)
7. Approximate
Integrals (optional)
8. Improper
Integrals
B. Further Applications of Integration (3
lectures)
1. Arc Length
2. Area
of a Surface of Revolution
3. Applications
to Physics and Engineering
4. Applications
to Economics and Biology
5. Probability
C. Differential Equations (5 lectures)
1. Modeling with Differential Equations
2. Direction
Fields and Euler’s Method
3. Separable
Equations
4. Exponential
Growth and Decay
5. The
Logistic Equation
6. Linear
Equations
7. Predator-Prey
Systems
D. Parametric Equations and Polar
Coordinates (7 lectures)
1. Curves Defined by Parametric Equations
2. Tangents
and Areas
3. Arc
Length and Surface Area
4. Polar
Coordinates
5. Areas
and Lengths in Polar Coordinates
6. Conic
Sections
7. Conic
Sections in Polar Coordinates
E. Infinite Sequences and Series (10
lectures)
1. Sequences
2. Series
3. The
Integral Test and Estimates of Sums
4. The
Comparison Tests
5. Alternating
Series
6. Absolute
Convergence and the Ratio and Root Tests
7. Strategy
for Testing Series
8. Power
Series
9. Representations
of Functions as Power Series
10. The Binomial Series
12. Applications
of