Revised
7/2013
Program
Cover Document --- MAT 127: Calculus A
I. Basic Course Information
MAT
127: Calculus A is a freshmen/sophomore level introductory course to
single-variable calculus. It will be scheduled for three lecture periods: two
periods of 80 minutes length and one one-hour meeting period. Its prerequisite
is a passing grade on the departmental MAT 127 placement exam or completion of
MAT 096.
II. Learning Goals
The primary emphasis of
Calculus A is on learning the first-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 limits and derivatives, the rules of differentiation, applications of
differentiation, an introduction to integrals, and applications of integrals. 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 department’s program goals.
In Calculus A, 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 A, together with
its continuation MAT 128: Calculus B, 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
incoming students comprise the majority of enrolled students, an important goal
of Calculus A 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 B will be used
to assess the success of Calculus A in achieving its learning goals and its
contribution to the fulfillment of the department’s 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 department’s 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 127: Calculus A
Introduction: A typical syllabus for Calculus A follows
this sheet. All syllabi for Calculus A should include the points listed below and
cover the items on the Calculus A Topics list with the
indicated depth.
I. Basic
information on course and instructor
A.
Purpose
statement: Calculus A, the first 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 A should also
inspire mathematical curiosity and interest in its students.
B.
Course
description: This course provides students with a solid grounding in
single-variable calculus. The course is designed for students in the
mathematical and physical sciences. Topics include functions and limits,
derivatives and differentiation rules, applications of derivatives, and an
introduction to integrals and their applications.
C.
Course
prerequisites: MAT 096 or having passed the departmental MAT 127 placement
exam.
II. Learning
goals
This course aims to develop student proficiency in the
understanding of calculus concepts through numerical, graphical and functional
analyses.
The course should be taught in a manner that develops and exhibits the
following mathematical practices. Students of calculus should:
·
Engage with the
subject matter as they progressively grow in mathematical maturity and
expertise throughout the sequence.
·
Be able to
address and demonstrate understanding of rigorous problems in multiple
representations;
·
Be able to
explain and justify their solutions (including discussing them with other
students),;
·
Be able to apply
mathematics to real-life applications;
·
Be able to
express their thought process and solution strategies in clear, written form.
·
Be exposed to the
purposeful and appropriate use of technology in the course.
A.
Learning goals: MAT
127 will focus on an in depth understanding of:
a. Concept and evaluation of limits
b. Concept of continuity
c. Notions of (average and instantaneous) rate of change
and computing rates of change
d. Notion of the derivative and methods of differentiation
e. Solving problems involving applications of limits and
derivatives including related rates, sketching, and optimization
f. Fundamental Theorem of Calculus and Applications of
Fundamental Theorem of Calculus
g. Notions of integration (Riemann sums, area under the
curve)
Students will gain experience in communicating calculus topics utilizing
mathematical language. The course will
cover the topics listed on the attached “Calculus A
Topics List” with the indicated emphases.
B. Learning Outcomes: Through solving rigorous problems
and presenting written solutions clearly on formal and informal assessments,
students will be able to demonstrate mastery of concepts involving:
a. Notion of Limits and evaluation of limits (one-sided,
two-sided, infinite)
b. Continuous functions
a. Rate of change of functions
b. Derivatives and differentiation techniques of various
functions.
c. Applications of limits and derivatives
d. Fundamental Theorem of Calculus and its Applications
e. Notions of Integration including Riemann sums, and
area under the curve
f. Students should be able to demonstrate understanding
of multiple (numerical, graphical and algebraic )
representations of a calculus problem.
g. Students should be able to justify their thought
processes and solutions in clear, written form.
h. Students should be able to solve a real-life problem
involving calculus.
III. Student
assessment
A.
Assessment plan: Students
will receive regular feedback on their work through the assignment of homework,
quizzes, student presentations and examinations. The department has created a
set of computerized homework problems (currently using the WeBWorKsystem)
that all professors are required to assign to students. The amount of weight given to these homework
problems is at the discretion of the instructor. The 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 continual
regular feedback on their work.
Calculus
A Topics List
All listed topics are to be
covered. Topics in bold should be covered in depth. The chapter numbers
correspond with Stewart’s Calculus: Early
Transcendentals book.
Chapter 2: Limits and
Derivatives
2.1 Average rate of change, Tangent and
velocity problems
2.2 Numerical/Graphical
approach to limits, One sided limits, Infinite limits
2.3 Calculating limits using the limit laws,
Algebraic manipulations to evaluate limits, Squeeze Theorem
2.5 Continuity, Intermediate
Value Theorem
2.6 Limits at infinity,
Horizontal asymptotes
2.7 Derivatives and rates of
change, Definition of derivative at a point
2.8 Derivative as a function, Differentiability
and continuity, Higher derivatives
Chapter 3: Differentiation
Rules
3.1 Basic
differentiation rules, Derivatives of polynomials and exponential
functions
3.2 Product
and quotient rules
3.3 Derivatives
of trigonometric functions, Trigonometric limits
3.4 Chain
rule, Derivatives of general exponential functions
3.5 Implicit differentiation, Derivative of
inverse functions (not done explicitly in Stewart), Derivatives of inverse
trigonometric functions
3.6 Derivatives
of general logarithmic functions, Logarithmic differentiation
3.9 Related
rates
3.10 Linear approximation
Chapter 4: Applications of
Differentiation
4.1 Maximum and Miminum
values including local and absolute extrema, Critical
points, Optimizing on a closed interval
4.2 Rolle’s Theorem, Mean Value Theorem
4.3 How
derivatives affect the shape of a graph including monotonicity of a function, concavity and inflection points, second
derivative test for critical points
4.4 Indeterminate forms and L’Hospital’s
Rule
4.5 Summary
of curve sketching, aAymptotes
4.7 Optimization
problems
4.9 Antiderivatives
Chapter 5: Integrals
5.1 Summations, Areas and Distances
5.2 Definite
integral including approximating areas by rectangles using endpoints or
midpoints, General Riemann sums, Definite
integral as signed area, Properties of the definite integral
5.3 Fundamental
Theorem of Calculus, part I and part II
5.4 Indefinite integrals and the net change
theorem
5.5 Substitution
rule, including change of variables for definite integrals
Chapter 6: Applications of
Integration
6.1 Areas between curves, including
integration along the y–axis
6.2 Volume as the integral of cross-sectional area
6.4 Average
value of an integrable function over closed interval.
MAT 127: Calculus A
Course Syllabus
|
Instructor: Prof. T. Hagedorn |
Office:
Science Complex 207 |
|
Phone:
x-3053, Email: hagedorn@tcnj.edu |
Office
Hours: M: 4-5:30, Th: 1:30-3, and by appointment.
|
|
Books: Calculus, Seventh Edition, by J. Stewart. Prerequisites:
None |
|
Course Goals: MAT 127: Calculus A is a first-year level course
studying differential calculus and providing an introduction to integral
calculus. The goals of the course are to provide a complete overview of the
derivative and its uses, to define both the indefinite and definite integral
and provide an introduction to its uses. Additional goals of the course are to
familiarize you with many of the real-life applications of calculus as well as
teach you some of the mathematical theory involved. Most of the course will
have a computational flavor to it, but we also have proofs and more abstract
material at times.
Course Content Overview: Calculus
is an incredibly useful subject utilized throughout physics, chemistry,
biology, and even in subjects like economics and business. Much of calculus was
invented in order to solve problems arising in these discipline (for example,
Newton was motivated by problems from physics such as how to model the motion
of a falling apple), so we will be working with many real-world problems
throughout the course. The two main subjects in our course are differential and
integral calculus. These two subjects have different flavors but have many
connections that one exploits in applications of calculus to the real world.
Differential calculus answers the question “How fast is thing A moving/changing?” Some things, like a car traveling on a
highway at the speed limit, will travel at a constant speed like 65 mph. But
when one wants to slow down, or even go faster, the car’s speed is changing. “How
does one calculate the car’s speed when it is accelerating/decelerating?”
goes to the heart of the type of questions we try to answer in differential
calculus. Once one can answer it, we can plot functions, determine the
flight-plan of a spacecraft, and figure out how hard and at what angle Mike
Piazza has to hit a baseball for it to clear the rear fence. We can also then
relate how fast a weirdly shaped bathtub (such as a bathtub for two) may fill
up with water to the flow of the water into the tub. Exciting stuff, yes?
Integral calculus tries to answer the different question, “How much area is
under that curve?” Until now, you have only seen how to calculate the area of simple
shapes like rectangles and triangles. You may know the formula for the area of
a circle, but do you understand why that formula is correct? Integral calculus
will explain why 
is the
correct formula for the area of a circle. The methods of integral
calculus can then be used to calculate volumes of solids, lengths of curves,
and surface areas of solids. All these questions are important in real life
when you want to construct something and you need to know how much wood,
string, paint, or cloth to buy in order to build your project. Most
unexpectedly, we will be able to make use of differential calculus to help
answer many of these questions.
Course and Performance
Goals: The goals of the course are
for you to learn calculus and appreciate its applications, to develop your
ability to tackle complicated problems arising in real-life, to develop your
ability to read, write, and understand proofs, and to further develop your
ability to reason logically. To assist you in accomplishing these goals, we
will provide you with a number of different environments in which you can do
calculus. You will make mistakes. Much of the material in this course will be
new to you and may take time and practice to master. The key to doing well and
learning in this course is to do many exercises, to make mistakes and then to
learn from them so that you do not continue making these mistakes in the
future. In addition to lectures, we will have in-class individual and group
work, weekly homework, out-of-class computer graded homework and quizzes to
give you ample opportunities to test and develop your knowledge.
Assessment: The course grade will be an assessment of how well you
understand the material of the course. Since mastery of much of the material
will take time, the course grade will be heavily weighted towards the
examinations at the end of the course and will be calculated using the
following formula: Final Examination 40%, First In-class Exam 10%, Second
In-Class Exam 20%, Homework 20%, Quizzes 10%. In
addition, good/poor class participation can raise/lower your course grade by up
to two-thirds of a letter grade. The professor reserves the right to change or
modify this formula.
Examinations will mostly
consist of questions based upon material presented in-class, and problems done
either in-class, on homework, or on quizzes. As a result, it is strongly
encouraged that you attend every class and that you take detailed notes. Your
notes should accurately represent everything done during class. In particular,
all material written on the blackboard should be faithfully and legibly
recorded in your notes. One particularly effective study suggestion is to
develop the habit of rewriting your notes for a class before the next class. If
you should need to miss a class, please ensure that you get the notes from
another student in the class. There will be no attendance policy for the
lectures (there will be one for the computer labs), but in the past there has
been a strong correlation between regular attendance and a good grade in this
course
Homework: Weekly homework will be assigned and you will be
expected to work on it in groups of two or three (turning in one copy for the
whole group). The homework should represent the efforts of the group and not
just one member of the group. We will form initial groups during the first week
of class and change them occasionally during the course of the semester.
Computer Homework: This semester, we will be using a computerized
homework system Webwork. Most weeks, we will have
some homework problems that must be done online at the Webwork
site. Further information will be provided in class.
Absence Policy: There will be no makeup quizzes. Makeup exams will
only be given in extraordinary circumstances and only when the request has been
made at least one week in advance of the exam.
Additional Help: The Office of Academic Assistance in Forcina Hall runs a mathematics help desk and has tutors in
Calculus. They have both drop-in sessions and scheduled sessions.
Schedule: During the semester, we will cover Chapters 2-5 of
Stewart. Should you be unsure of the material in Chapter 1, please take
advantage of the Tutoring center listed above to review that material. On
average we will cover one chapter every 3 weeks. The first in-class exam will
be cover the material (that we cover) upto Chapter 3.
and will be given around the sixth week of class. The
second in-class exam will cover most of the material from Chapters 4 and the
beginning part of Chapter 5 and will be given around the 10th week
of the course. The final examination will be cumulative.