Math 120 Calculus I
Instructors
Course Description
This is the first course in a three-semester calculus
sequence designed for students majoring in a field that requires the tools of
calculus. Besides the computational aspects of calculus, we will develop the concepts
of calculus with some rigor.
The subject in this first course, Math 120, is differential calculus.
Topics include functions, limits, continuity, differentiation
of algebraic and trigonometric functions, mean value theorem,
and applications of derivatives.
A detailed syllabus is available for Math 120.
See also
Clark
University Academic Catalog for more information on the Calculus sequence
(Math 120/121/122) and the Honors Calculus sequence (Math 124/125).
Prerequisites
In order to register for Math 120, you need to either
- pass the math placement test, available
on-line, anytime,
- have passed the Advanced Placement test in Calculus, 4 or 5 required, or
- pass Math 119, Precalculus, or Math 114, Discrete Mathematics,
with an appropriate grade
The placement test
is available on-line, to be taken anytime. Information on how to take
it is at
http://aleph0.clarku.edu/text_2003/courses/placement_test.shtml
Textbook
Our text for this course is Calculus, One and Several Variables,
by Salas, Hille, Etgen, 10th edition. We will discuss the first four
chapters in Math 120. This same book will also be used in Math 121 and
Math 122.
The 9th edition will do. Part of section 3.4 in the 9th edition was moved
to section 4.9 in the 10th, and all of section 3.8 was moved to section 4.10.
Also, the exercise numbers have been changed. The changes are noted in the
practice problems.
Tests and Exams
There will be three tests during the semester and a final examination
during finals week in December.
The three tests will each be given in a three-hour block in the
evening. The tests are common for all students taking Math 120 and
will be based on material discussed in the lecture or presented in the
text prior to the date of the exam. Although each exam is designed as
a one-hour test, you may take up to three hours to complete it. You
may arrive anytime between 6:00 and 7:00 and finish before 9:00.
The exams will be held in Johnson Auditorium, Sackler Science Center.
They will be closed book, closed notebook. The use of calculators will not be
allowed. In the event that you have a legitimate, documented excuse
for missing a test, you must contact your instructor prior to the
scheduled test time. A makeup may be rescheduled at the instructor's
convenience.
- Exam 1. Thursday, October 2, 6-9 pm, Johnson Auditorium
- Exam 2. Thursday, October 30, 6-9 pm, Johnson Auditorium
- Exam 3. Thursday, December 4, 6-9 pm, Johnson Auditorium
- Final Exam.
| Date |
Course Title |
Building |
Room |
Start/End |
| 12/16/08 |
Calculus I - Sect. 1 |
Biophysics |
BP326 |
04:00 PM / 06:00 PM |
| 12/19/08 |
Calculus I - Sect. 2 |
Biophysics |
BP326 |
06:30 PM / 08:30 PM |
| 12/16/08 |
Calculus I - Sect. 3 |
Biophysics |
BP220 |
08:00 AM / 10:00 AM |
| 12/19/08 |
Calculus I - Sect. 4 |
Biophysics |
BP326 |
08:00 AM / 10:00 AM |
The final is a comprehensive exam, given during the
final exam period.
Homework and quizzes
Practice problems will be assigned daily from the text to help
you master the concepts discussed in class. Although the problems will not be
collected regularly, it is expected that you will keep up to date on the problems.
Periodically, a few specific problems may be assigned, collected,
and graded. Homework is due in class on the assigned day. No late assignments will
be accepted.
Short 15-20 minute quizzes may be given periodically throughout the semester without warning.
You are responsible for being in class.
Tutors
Tutors by experienced upperclassmen will be available in the evenings, 8:00 to 10:00, on Sundays through Thursdays
(but not Fridays or Saturdays) in the seminar room BP312.
Course grade
The course grade is based on the three midterm exams, the homework and
quizzes, and the final exam. The precise weightings for these components is as
follows: 20% for each midterm exam, 35% for the final exam, and 5% for
homework and quizzes.
Syllabus
We will follow the order of topics in the text, Salas, Helle, & Etgen's Calculus,
but we'll stress some topics and pass over some others.
- Review and Preview
Calculus is about the relation between a quantity and its rate of change.
For an example, if the quantity is the distance travelled at a given time, then its rate
of change is velocity. If the velocity is constant, then calculus is not required: the
distance travelled is the product of the elapsed time and the velocity. But when the
velocity is not constant, then this formula doesn't apply. Nonetheless, the distance
and velocity are intimately related. If the distance travelled at all times is known,
then the velocity at any given time can be determined; and if the velocity at all times
is known, then the distance travelled at any given time can be determined. These two
operations are called differentiation and integration.
Much of calculus involves analyzing and developing these concepts and their
applications.
The review only contains things that you should already know, but
it helps to see them again just before you use them. Concepts under review are real
numbers, functions, intervals and inequalities, graphs of functions, absolute value,
piecewise defined functions, symmetry and even and odd functions, operations on graphs,
and composition of functions. A variety of functions are reviewed including linear
functions (and with them slopes of lines), power functions, polynomials, rational
functions, trigonometric, exponential, and logarithmic functions. Notations under
review include functional notation and substitution, interval notation, unions,
intersections, set notation, and algebra of functions.
These are discussed in chapter 1 of the text, but we will not go over the chapter
section by section.
- Limits and Continuity
We first must clarify the concept of derivative. In some ways it is intuitively clear
that a travelling body has a velocity, or more generally, any changing quantity has a
rate of change. But just what is the rate of change? The answer is the rate of
change at an instant is the limit of the average rates of change near that instant.
The concept of limit is much more subtle than it first appears. We will discuss it
in some detail and develop a formal defintion of a limit and a formal notation to go
along with it.
Key concepts associated to the concept of limit are tangent lines, limit laws,
continuity, the pinching theorem, left- and right-limits, trigonometric limits,
the intermediate value theorem (IVT), and the exterme value theorem (EVT).
Chapter 2 Limits and Continuity
2.1 The Idea of Limit
2.2 Definition of Limit
2.3 Some Limit Theorems, Additional information on Infinite Limits
2.4 Continuity
2.5 The Pinching Theorem; Trigonometric Limits
2.6 Two Basic Properties of Continuous Functions
- Derivatives
With a solid defintion of limit, we can proceed to define a derivative (instantaneous
rate of change) as the limit of average rates of change, and then develop the
properties of derivatives. There are a number of rules for differentiation (finding
derivatives), mostly easily learned, although the chain rule, for some reason, seems
to be more difficult to master. There are a couple of different notations for
derivatives that everyone uses.
It is assumed that you know the trig functions, sine, cosine, etc., and we will
find and use their derivatives.
Further topics in differentiation include implicit differentiation, and higher
derivatives.
Chapter 3 Differentiation
3.1 The Derivative
3.2 Some Differentiation Formulas
3.3 The d/dx Notation; Derivatives of Higher Order
3.4 The Derivative as a Rate of Change
3.5 The Chain Rule
3.6 Differentiating the Trigonometric Functions
3.7 Implicit Differentiation; Rational Powers
- The Mean Value Theorem and Curve Sketching
The purpose of curve sketching is no so much to draw the graph of the
function, but to get a better understanding of the relation between a
function and its derivative. For instance, if the derivative is positive,
then the function is increasing; at a maximum or a minimum of a function,
the derivative is zero. We will prove these (obvious) statements using a
theorem called the mean value theorem. We'll also see what second derivatives
have to do with the graph of a function.
The applications of derivatives are numerous. Besides classical applications in
physics and the natural sciences, there are applications in the social sciences, for
instance, marginal profits are just derivatives of profits.
Chapter 4 The Mean-Value Theorem and Applications
4.1 The Mean-Value Theorem
4.2 Increasing and Decreasing Functions
4.3 Local Extreme Values
4.4 Endpoint and Absolute Extreme Values
4.5 Some Max-Min Problems
4.6 Concavity and Points of Inflection
4.7 Vertical and Horizontal Asymptotes
4.8 Curve Sketching
4.9 Velocity and Acceleration; Speed
4.10 Related Rates of change per Unit Time
Practice Problems
You should do all the problems listed below as they're assigned. There are two lists.
The first refers to the 10th edition of the text; the second to the 9th edition. Nearly
all of the problems are identical, but there are a very few that are different. The biggest
change between the editions is that the white space was reduced in the new edition. That is,
the text and formulas are more cramped in the new edition.
List of problems for the 10th edition
Review:
Chapter 1. Precalculus Review
1.2 p.10 25, 27, 29, 31, 33, 35, 36, 37
1.3 p.16 5, 7, 9, 11, 13, 15, 21, 23, 27, 31
1.4 p.23 3, 5, 9, 11, 17, 20, 21, 23, 31, 33, 37, 39
1.5 p.30 3, 7, 11, 17, 23, 25, 27, 37, 45, 47
1.6 p.39 13, 21, 23, 24, 25, 26, 29, 30, 33, 34
1.7 p.45 3, 5, 11, 19, 21, 25, 27, 45, 46, 52
New material:
Chapter 2. Limits and Continuity
2.1 p.61 1, 3, 7, 11, 13, 15, 19, 21, 51
2.2 p.71 21, 22, 35, 36, 37, 38
2.3 p.79 1, 5, 7, 9, 13, 16, 19, 31, 37, 41, 43
2.4 p.88 1, 3, 5, 11, 15, 21, 23, 31, 47
2.5 p.96 3, 7, 13, 15, 17, 21, 25, 27, 34
2.6 p.100 3, 7, 9, 13, 14, 15, 16, 29
Chapter 3. The Derivative
3.1 p.112 1, 5, 9, 11, 15, 17, 21, 22, 27, 35, 45
3.2 p.122 7, 9, 13, 17, 21, 25, 27, 31, 33, 35, 41, 43
3.3 p.128 1, 3, 5, 7, 9, 13, 17, 23, 25, 27, 31, 39, 41, 49
3.4 p.132 3, 4, 7, 9, 11, 12
3.5 p.138 7, 9, 11, 13, 15, 17, 19, 21, 25, 31, 35, 47
3.6 p.145 5, 9, 13, 15, 17, 19, 21, 23, 33, 35, 43, 49
3.7 p.150 3, 9, 19, 21, 25, 29, 35, 43, 47
Chapter 4. Applications
4.1 p.158 1, 3, 5, 13, 19, 23, 25, 38
4.2 p.165 1, 7, 11, 15, 19, 23, 25, 35, 40, 41, 51
4.3 p.173 1, 3, 7, 11, 23, 26, 33, 42
4.4 p.180 1, 3, 5, 7, 11, 15, 22, 23, 27, 31, 35, 41
4.5 p.187 5, 7, 10, 12, 17, 25, 30, 35, 43, 45
4.6 p.193 1, 2, 5, 7, 9, 11, 13, 15, 21, 27, 33, 35
4.7 p.200 1, 2, 3, 5, 7, 9, 13, 35, 37, 43, 45
4.8 p.208 1, 3, 9, 11, 13, 21, 23, 27, 29, 31, 45, 53, 55
4.9 p.215 3, 8, 11, 12, 13, 14, 39, 47
4.10 p.185 3, 7, 13, 17, 19, 23, 29
List of problems for the 9th edition
Review:
Chapter 1. Precalculus Review
1.2 p.10 25, 27, 29, 31, 33, 35, 36, 37
1.3 p.18 5, 7, 9, 11, 13, 15, 21, 23, 27, 31
1.4 p.27 3, 5, 9, 11, 17, 20, 21, 23, 31, 33, 37, 39
1.5 p.34 3, 7, 11, 17, 23, 25, 27, 37, 45, 47
1.6 p.45 13, 21, 23, 24, 25, 26, 31, 32, 33, 34
1.7 p.52 3, 5, 11, 19, 21, 25, 27, 45, 47, 48
New material:
Chapter 2. Limits and Continuity
2.1 p.69 1, 3, 7, 11, 13, 15, 19, 21, 53
2.2 p.81 21, 22, 35, 36, 37, 38
2.3 p.90 1, 5, 7, 9, 13, 16, 19, 31, 37, 41, 43
2.4 p.100 1, 3, 5, 11, 15, 21, 23, 31, 47
2.5 p.109 3, 7, 13, 15, 17, 21, 25, 27, 34
2.6 p.115 3, 7, 9, 12, 19, 23, 24, 25, 26, 35, 39
Chapter 3. The Derivative
3.1 p.129 1, 5, 7, 11, 15, 17, 21, 27, 35, 39, 51
3.2 p.141 7, 9, 13, 17, 21, 25, 27, 31, 33, 35, 41, 43
3.3 p.147 1, 3, 5, 7, 9, 13, 17, 23, 25, 27, 31, 39, 41, 49
3.4 p.157 3, 4, 7, 9, 11, 12, 19, 23, 27, 28, 29, 30, 41, 45, 53
(problems 19-53 were moved to section 4.9 in the 10th edition)
3.5 p.165 7, 9, 11, 13, 15, 17, 19, 21, 25, 31, 35, 47
3.6 p.172 5, 9, 13, 15, 17, 19, 21, 23, 33, 35, 43, 49
3.7 p.179 3, 9, 19, 21, 25, 29, 35, 43, 47
3.8 p.185 3, 7, 11, 15, 19, 21, 25, 31
(section 3.8 was moved to section 4.10 in the 10th edition)
Chapter 4. Applications
4.1 p.202 1, 3, 5, 13, 17, 21, 23, 36
4.2 p.210 1, 7, 11, 15, 19, 23, 25, 35, 40, 41, 55
4.3 p.218 1, 3, 7, 11, 23, 26, 31, 41
4.4 p.226 1, 3, 5, 7, 11, 15, 22, 23, 27, 31, 35, 41
4.5 p.234 5, 7, 10, 12, 17, 23, 28, 33, 41, 43
4.6 p.241 1, 2, 3, 5, 7, 9, 11, 13, 19, 25, 27, 31, 33
4.7 p.249 1, 2, 3, 5, 7, 9, 13, 35, 37, 43, 45, 47
4.8 p.258 1, 3, 9, 11, 13, 21, 23, 27, 29, 31, 45, 53, 55
Previous Tests
External Links
66 Challenge problems for differentiation
AVI movies that illustrate limits (by H. Servatius)
David Joyce's Guiude to Studying calculus
Clark University's Math Problem Solving Team
MathArchives links for
Calculus Resources On-line. Also
Visual Calculus
List of links at the Math Forum @ Drexel for
Calculus of a single variable
Ask Dr. Math: answers questions about
Calculus