Offered: Fall 2016 Instructor: Kevin J. Mitchell Office: Lansing 301.5 Phone: (315) 781-3619 Fax: (315) 781-3860 E-mail: mitchell@hws.edu Office Hrs: M 3:30-4:30, Tu 11:00-1:00, W 12:15-1:15, F 1:30-2:30 all in Lansing 301, or by appointment. Class: Section 130-01: M-W-F 8:00 to 8:55 in Napier 201 Lab: Thursday 10:20 to 11:45 in Gulick 206A Final Exam: Wednesday, Dec. 14, 2016 1:30PM-4:30PM Class: Section 130-02: M-W-F 9:05 to 10:00 in Napier 201 Lab: Thursday 11:55 to 1:20 in Gulick 206A Final Exam: Wednesday, Dec. 14, 2016 8:30AM-11:30AM Text: Briggs, Cochran, & Gillett: Calculus: Early Transcendentals, second ed. Math Intern: Chris Wilson, Lansing 310. Sunday to Thursday: 3:00-6:00, 7:00-10:00pm Website: Course Website and Assignments: http://math.hws.edu/~mitchell/Math130F16/index.html WeBWorK: WeBWorK Home Page for Math 130: http://math.hws.edu/webwork2/Math130-Mitchell/ WeBWorK Instructions and FAQs: http://math.hws.edu/~mitchell/WebWorkFAQs.html WeBWorK Syntax and List of Functions: http://math.hws.edu/~mitchell/WeBWorKSyntax.html
A single concept is crucial to the analysis of both problems: the notion of a limit. A typical limit problem involves trying to make sense of a ratio where both the numerator and denominator are getting very small (both are nearly 0) or both are becoming very large (becoming "infinite"). We will see that the usual laws of algebra must be extended in some way to make sense of these "nonsensical" expressions.
You can get a sense of the somewhat paradoxical notions involved with limits if you think about a familiar rate of change: the speed of a car. For example, if you travel the 210 miles from Albany to Geneva in 3.5 hours, then your average speed for the trip is 210 miles divided by 3.5 hours or 60 miles per hour. It's unlikely that your speed was exactly 60 mph at each moment of the trip (even if you used cruise control). Rather, if you had looked at your speedometer at each instant along the way, sometimes it might have read 65 mph or more and at other times (like when exiting the thruway) it might have read much lower, say 25 or 30 mph.
The problem is what do we mean by speed "at an instant"? We just calculated an average speed by dividing distance by time. But in an instant, no time passes and no distance is traveled! So there is no change and, consequently, no rate of change. Well, what is a speedometer measuring, you may ask. Good question! It is actually measuring average rates of change over very small time intervals. This is one of the reasons why when you first start to move, the speedometer does not change, or why when you stop the speedometer does not immediately go to 0. Intuitively, the smaller the time period, the closer the average rate of change is to the instantaneous rate of change. This is how the notion of a limit (a ratio with a small numerator and denominator) was born. To make all this work out correctly and consistently requires a careful mathematical treatment that we will develop during the term. This development took over two-thousand years from the time of the Greek mathematicians (or even earlier) to the period in the 17th century of Newton and Leibniz. In fourteen weeks we will not be able to give all the details of this work, but we will consider some of the major ideas involved.
It turns out that this notion of rate of change is intimately related to slope. This is not so surprising; after all both are ratios. Slope is rise over run or the change in y over the change in x, and velocity, for example, is distance over time. If we plot position on a graph with horizontal axis x being time and the vertical axis y being distance, then the change in distance over the change in time (velocity) becomes the change in y over the change in x (slope). We will exploit this connection several times during the term.
For example, we will identify instantaneous rates of
change with slopes of curves (not just slopes of straight lines) at specified points (instants). This identification
has a myriad of applications, but here's a simple one. Think about the flight of a ball that you throw up in the air.
Its velocity is 0 when it reaches its highest point and the ball seems to hang in the air momentarily.
"Markspeople" like Annie Oakley and Buffalo Bill would shoot silver dollars that had been tossed in the air.
Though this is quite a feat, they made it easier by shooting at the coin when it "stood still" at the highest
point in its flight.
Similarly, a tennis player will want to hit a serve when the ball is at or near the top of the toss because the ball
is almost still there. More generally, the highest point on a graph (of an appropriate function) will occur when the rate of change or slope
(or "velocity") of the graph is 0. This point can be determined without ever having to graph the function, once
we develop some methods to calculate instantaneous rates of change. This is quite useful. For example, profit
is a function of the price at which an item is sold, so we should be able to determine which price produces the
highest (maximum) profit.
On the theory side by the end of the course you should be able to (1) display an understanding of the notion of a limit of a function and why it occurs in the study of rates of change; (2) display a familiarity with the key steps in the development of the derivative and the necessity of using the the limit concept; and (3) display an understanding of the main theoretical results of the differential calculus, especially the Mean Value Theorem. On the practical side, successful students in the course should be able to (1) classify functions (as having limits, as continuous, or as differentiable); (2) classify the critical points of functions and apply this classification to solve extremization problems; (3) interpret the meaning of the derivative; (4) display a knowledge of basic differentiation techniques, (5) justify through written work the application of theorems and techniques to particular problem-solving situations.
Mathematics is different from other disciplines. This is reflected in a few basic principles that I use in constructing this course (which I have adapted from Roger Askey's article).
Topic | See especially |
Basic Algebra | Appendix A |
Sets of Real Numbers, Intervals, Interval Notation | Appendix A |
Solving Inequalities | Appendix A, Example 2 |
Absolute Value and Distance | Appendix A, Example 3 |
Coordinates, Distance, and Circles | Appendix A |
Equations of Lines | Appendix A |
Functions | Chapter 1.1 |
Representing Functions (especially piecewise functions) | Chapter 1.2 |
Inverse, Exponential, and Log Functions | Chapter 1.3 |
Trig Functions | Chapter 1.4 (especially page 40-41) |
About twice a week, there will be an assignment of problems to hand in for grading. You may discuss graded homework assignments with others. Verbalizing your questions, explaining your mathematical ideas, and listening to others will increase your understanding. BUT you are not free to simply copy others' work or to make your work available in this way to others. You will be heavily penalized for such instances. Make sure you write up your answers on your own. Plagiarism acts against my obligation as an instructor to insist that students do work that will benefit them and help them think and learn independently. If you are confused about the meaning of this policy, see me. I reserve the right to change this policy if it is abused. Note: Other faculty will have different policies about homework. Your work will be due at the beginning of the next class. It should be neat and done in pencil. If more than one page is required, please staple them. No late assignments, please.
There will be regular, graded WebWork computer exercises that review the material and concepts we are currently covering. You will get immediate feedback from these exercises that will allow you to assess your progress. Further, if you submit an incorrect answer, you may return to the problem and work on it again until you get it right. Students typically earn 85-100% on this part of the course. Though every student's problems will be similar, they will not be identical to each other.
I may also use a few announced 10-minute quizzes to check your progress.
There will be three in-class tests and a final exam. The dates are listed in the outline below. Tests will be cumulative but will concentrate on more recent material. It is impossible to construct fair makeup exams in mathematics. For your own protection, my policy is that there are no makeup examinations. If for some extraordinary reason you find you are unable to take an exam, let me know as soon as possible, certainly well before the exam is administered.
Your course grade will be calculated as follows. First I will make a list of your grades:
Extra Credit: You may earn extra credit
by attending a Mathematics and Computer Science Department seminar.
You may recieve a maximum of three such credits.
Week | Topics and Readings |
1 | Syllabus, Introduction, Preview, The Slope Problem and Instaneous Rates, Introduction to Limits Read: Syllabus, Chapter 2.1, begin 2.2. Review Chapter 1.1, 1.2, 1.3, 1.4 & Appendix A on your own as needed. Topics: Functions, polynomials, rational, trig, exponentials, logs, and inverse funtions (see page 10 for definitions.) |
2 | Types of Limits, Evaluating Limits, The Limit Definition (hard). Read 2.2-2.3, and 2.7 (p. 112-117). Review Chapter 1.3 on Logs and Exponentials. |
3 | An Important Trig Limit, Infinite Limits, and Limits at Infinity. Read 2.4-2.6. |
4 | Continuity, Introduction to Derivatives. Derivatives: Why We Needed Limits. Derivative Functions. Read 2.6 and 3.1. |
5 |
Exam 1 Monday, September 26 in Albright Auditorium @ 7:40/8:45 am (Section 1/Section 2).
Basic Derivative Rules and the Derivatives of Exponentials. Read 3.1-3.3. |
6 | The Product and Quotient Rules. The Trig Derivatives. Derivatives as Rates of Change. The Chain Rule. Read 3.4-3.7. Review Inverse Functions in Chapter 1.3. |
7 | The Chain Rule, Implicit Differentiation and Derivatives of Logs, Inverse Functions, and Inverse Trig Functions. Read 3.7-3.10. |
8 | Derivatives of Inverse Trig Functions. Application: Related Rates. Extrema. Read 3.10-3.11, 4.1. |
9 |
Exam 2 Thursday, October 27 in Lab. The Mean Value Theorem. The First Derivative Test. Read 4.6 and 4.2. |
10 | The Second Derivative and Concavity. Graphing. Read 4.2-4.3. |
11 | More Graphing. Optimization. Read 4.3-4.4. |
12 | L'Hopital's Rule. Antiderivatives. Read 4.7 and 4.9. |
13 | Exam 3 Monday, November 21 in Albright Auditorium @ 7:40/8:45 am (Section 1/Section 2). |
14 | Antiderivatives and Applications to Motion. Reversing the Chain Rule. Read 4.9, 5.5. |
15 | If we have time: Looking ahead to Area Under A Curve. |
Final | Final Exam: Wednesday, December 14, 2016. |