Instructor: Geoff Hagopian
Main Text: Calculus, Early Transcendentals, 8th ed., by Stewart ISBN10: 1-285-74155-2 ISBN13: 978-1-285-74155-0
This is a first course centered on the study of rates of change and values that accumulate at variable rates, especially as relates to doing computations with infinite and infinitesimal quantities. The concept of limiting values is used develop the idea of instantaneous rates of change and methods for computing derivative functions, especially for polynomial, rational, algebraic, trigonometric, exponential and logarithmic functions. Finally, the concept of the integral is connected with the concept of the derivative through the fundamental theorem of calculus.
Math/Science Study Center in Math 4 --computers and tutor.
MESA club in MSTC.
You are required to access computer algebra systems and have a scientific calculator of some sort, though this is mostly for the purposes of study. Calculators are not allowed on many exams.
Teaching and Learning:
According to George Polya, we can articulate three major principles of learning (which governs teaching):
1. Active Learning. It has been said by many people in many ways that learning should be active, not merely passive or receptive; merely by reading books or listening to lectures or looking at moving pictures without adding some action of your own mind you can hardly learn anything and certainly you cannot learn much.
There is another often expressed (and closely related) opinion: The best way to learn anyting is to discover it by yourself. Lichtenberg (an eighteenth century German physicist, better known a s a writer of aphorisms) adds an interesting point:
"What you have been obliged to discover by yourself leaves a path in your mind which you can use again whn the need arises."
Less colorful but perhaps more widely applicable, is the following statement:
"For efficient learning, the learner should discover by himself as large a fraction of the material to be learned as feasible under the given circumstances."
This is the principle of active learning (Arbetisprinzip, in German) It is a very old principle: it underlies the idea of "Socratic Method."
2. Best Motivation. Learning should be active, we have said. Yet the learner will not act if he has no motive to act. He must be induced to act by some stimulus, by the hope of some reward, for instance. The interest of the material to be learned should be the best stimulus to learning and the pleasure of intensive mental activity should be the best reward for such activity. Yet, where we cannot obtain the best we should try to get the second best, or the third best, and less intrinsic motives of learning should not be forgotten.
For efficient learning, the learner should be interested in the material to be learned and find pleasure in the activity of learning. Yet, beside these best motives for learning, there are other motives too, some of them desirable. (Punishment for not learning may be the least desirable motive.)
Let us call this statement the principle of best motivation.
3. Consecutive phases. Let us start from an often quoted sentence of Kant: "Thus all human cognition begins with intuitions, proceeds from thence to conceptions, and ends with ideas." The English translation uses the terms "cognition, intuition, idea." I am not able (who is able?) to tell in what exact sense Kant intended to use these terms. Yet I beg your permission to present of Kant's dictum:
Learning begins with action and perception, proceeds from thence to words and concepts, and should end in desirable mental habits.
To begin with, please, take the terms of this sentence in some sense that you can illustrate concretely on the basis of your own experience. (to induce you to think about your personal experience is one of the desired effects.) "Learning" should remind you of a classroom with yourself in it as student or teacher. "Action and perception" should suggest manipulating and seeing concrete things such as pebbles, or apples, or Cuisenaire rods; or ruler and compasses; orinsturments in a laboratory; and so on.
These principles proceed from a certain general outlook, from a certain philosophy, and you may have a different philosophy. Now, in teaching as in several other things, it does not matter much what your philosophy is or is not. It matters more whetner you have a philosophy or not. And it matters very much whether you try to live up t your philosophy or not . The only priniciples of teaching which I thoroughly dislike are those to which people pay only lip service.
Much of the course is centered around applying these definitions and theorem by solving problems. The basic outline for general problem solving devised by Polya is a four step program:
1. Understand the problem
2. Devise a plan for solving the problem
3. Carry out the plan
4. Look back
Homework, Tests, Projects and Grades:
Homework assignments will be announced in class. There will be at least 4 in-class chapter tests. Also, there may be some take-home projects (time permitting) and there will be a cumulative final exam. Your grade in the course will be based on a weighted average of these chapter tests, project and final exam scores. Home study is essential to do well on these. The broad formula is that you should spend at least 2 hours studying outside of class (on your own or in a group) for every 1 hour of class. To be recognized for this work you'll need to either keep a very neat homework notebook and use the free MyOpenMath.com homework system for this class. The ID = 20169 and the key = ada.
Grades will be weighted with 10% for homework and quizzes, 65% for the in-class chapter tests and 25% for the final exam.
Generally, calculators and other electronic devices are not allowed during exams. Anyone using such devices in violation of this policy may be dropped from the course.
Course Student Learning Outcomes:
1. Demonstrate fluency and depth of understanding in fundamental skills and knowledge in arithmetic, algebra, geometry, analytic geometry, and trigonometry.
2. Define the concept of limit of a function as the behavior of a function when the input variable gets arbitrarily close to a certain value or its magnitude becomes arbitrarily small or arbitrarily large.
3. Define the concept of the derivative of a function as the limiting behavior of the rate of change of a function’s value withrespect to the input variable.
4. Use of a variety of methods of finding derivatives of functions, including the definition of the derivative.
5. Use the concept of derivative (in conjunction with general skills from arithmetic, algebra, geometry, analytic geometry, and trigonometry) to model and solve application problems that involve, directly or indirectly, rates of change.
6. Demonstrate critical and logical thinking, by frequent use of deductive reasoning in mathematics, in the context of differential calculus.Course Objectives: Upon completion of this course, students will be able to:
a. Compute the limit of a function at a real number;
b. Determine if a function is continuous at a real number;
c. Construct the derivative of a function as a limit;
d. Construct the equation of a tangent line to a function;
e. Compute derivatives using differentiation formulas;
f. Use differentiation to solve applications such as related rate problems and optimization problems;
g. Use implicit differentiation;
h. Graph functions using methods of calculus;
i. Evaluate a definite integral as a limit;
j. Evaluate integrals using the Fundamental Theorem of Calculus; and
k. Apply integration to find area.