Spring 2003 Syllabus for MTH1125

Instructor/office/contact info/office hours:

Dr. David B. Massey Office: 529 Nightingale Hall Phone: 373-5527

E-mail: dmassey@neu.edu Web site: www.massey.math.neu.edu

Office hours: Monday, Tuesday 11:45am-12:50pm; Wednesday 2:50-5:10pm

other times by appointment

Prerequisites:

One of MTH1121, MTH1124, MTH1141, MTH 1724, MTH1741

Required Materials:

Text: Thomas’ Calculus, Early Transcendentals, 10^th Edition, by Finney, Weir, and Giordano

Calculator: scientific, graphing calculator (recommend TI83 or higher)

Course Web Page:

www.math.neu.edu/undergrad/mth1125/

Various course materials can be found here.

Course Objectives:

This course has two main goals: to have students understand the concepts of differential equations and infinite series, and to enable students to display that understanding through a variety of applications. Specific, measurable, manifestations of your understanding that will be tested during the quarter include your ability to:

• translate word problems into differential equations, with or without initial conditions;

• solve certain kinds of differential equations via separation of variables;

• approximate solutions of differential equations via Euler’s method;

• graph and describe the qualitative nature of solutions to autonomous differential equations;

• determine whether a series converges or not, and what it converges to, when given formulas for the partial

sums;

• find formulas for the partial sums of geometric and telescoping series;

• identify a power series and the center of a power series;

• identify the domain of a power series as the interval of convergence, and to describe the interval of

convergence;

• estimate power series by Taylor polynomials for x-values near the center;

• manipulate power series, e.g., differentiate, integrate, substitute into, add, subtract, and multiply power

series;

• memorize power series for e^x, sin(x), cos(x), 1/(1-x), ln(1+x), and (1+x)^p, and memorize the intervals on

which the functions equal the given series;

• memorize and be able to derive Euler’s formula;

• estimate functions involving the above six functions using Taylor polynomials;

• approximate solutions to differential equations using power series.

Algebra/Calculus Help and Tutoring:

There are many resources for improving your algebra and Calculus skills. The best one is to go over any problems with your instructor. Other resources: walk-in tutoring in Cahners Hall and from Engineering tutors in 222 Snell Engineering, tutoring by appointment (sign up in the Media Center in the library), and study aids in the library (Schaum’s Outlines are great).

Most students find infinite series to be the single hardest topic in first- and second-year Calculus. IT IS ESSENTIAL THAT YOU KEEP UP WITH THE MATERIAL, AND – IF YOU ARE CONFUSED – IMMEDIATELY GO OVER THE MATERIAL WITH YOUR INSTRUCTOR OR A TUTOR.

Attendance:

It is essential that you attend class regularly. The easiest way for you to learn the material, and to know what material has been covered, is to come to class each day. Students are responsible for finding out what material has been covered or what announcements have been made on days that they miss class.

Excused Absences or Late Work:

In order to turn in assignments late or to take make-up quizzes/tests, students must bring written proof of some emergency situation; notes from doctors or nurses, documents verifying court appearances, receipts from having a car towed are all examples of valid documentation. Notes from family members are not acceptable. If a situation is of a personal nature, discuss the matter with your academic advisor; an e-mail message from your advisor saying that they believe that you should be allowed to make-up work is acceptable.

Homework and quizzes:

Homework will be assigned daily, but will not be collected. It is essential that you at least attempt the homework before the next class meeting. We will go over a limited number of homework problems near the beginning of each class. If you have many questions on the homework, you should come to my office hours to go over the material. To ensure that students are keeping up with the homework, there will be a short (10-20 minute) quiz at the beginning of class each Thursday. Material from the immediately preceding class will not be tested on the quiz (since we will not have gone over that homework beforehand).

There will be a 65-minute midterm exam on Thursday, May 1.

Grading:

The final exam will count as 40% of your grade. Quizzes will count as 35% of your grade; I will drop your worst quiz grade. The midterm exam will count as the remaining 25% of your grade.

The midterm and final exams will be graded on a 100-point scale with the A-range from 100-90, B-range from 89-80, C-range from 79-70, D-range from 69-60, and F-range from 59-0. Quizzes and labs will be graded on a 10-point scale, with the grade ranges correspondingly divided by 10. It is possible that a given test/quiz will be scaled; in such a case, the numbers themselves will be raised, with the letter grades still corresponding to the same numbers as above (e.g. a raw grade of 60 might be scaled to a 70, and thus be a C-).

Cheating Policy:

Cheating is an insult to honest students – it will not be tolerated. The University’s cheating policy and related disciplinary actions are detailed in the Student Handbook; the Handbook also includes a description of what is considered cheating by the University. Cheating in this class includes (but is not limited to): looking at the papers of others during a quiz or test, talking to other students during a quiz/test, looking at notes during a quiz/test (unless it is specifically announced that you may), copying other students’ work outside of class, and obtaining help from others on take-home tests.

In this class, working together on homework assignments is NOT considered cheating; however, you MUST write up your homework individually. Please be aware that this policy on working together outside of class varies greatly from one course to the next; the policy on what is allowed, that has been described in this paragraph, may well be considered cheating in your other classes.

The use of advanced calculators is NOT considered cheating in this course. Be aware, however, that other courses may well have a policy barring such calculators. Also, your instructor reserves the right to decide on-the-spot between what constitutes a “calculator” and what constitutes a full-fledged “computer”.

All incidents of cheating will be reported to the Office of Judicial Affairs.

If you have any questions as to what constitutes cheating, please ask me.

Additional Contacts:

If you have concerns/ problems in the course, and are not comfortable discussing them with your instructor, please contact either of the following:

Course Coordinator: Prof. David Massey, dmassey@neu.edu, 529 NI, 373-5527

Vice-chairman of Mathematics: Prof. Donald King, donking@neu.edu, 447 LA, 373-5679

Topics and homework:

Week 0.5 and Week 1: Translating word problems into differential equations (with or without initial data):

For this week, we will concentrate on turning word problems into mathematics problems, but we will not learn how to solve the mathematics problems until next week. In the problems listed here, regardless of what the instructions in the book say, all that we want to do is to write down the appropriate differential equation or initial value problem. The key idea here is that the phrase “rate of change” means “derivative”. It is also frequently important to translate “is proportional to” as “equals a constant times”.

§3.4 16, 17, 20 a,b; §4.1 57, 58; §5.4 17, 20, 21 a, 22 §6.3 25, 26

Weeks 2 and 3: Solving separable differential equations:

Now, we will learn how to solve some differential equations by separation of variables. A number of these homework problems involve finishing the word problems from the first week.

§3.4 16, 17, 20 a,b; §4.1 57, 58; §5.4 1-6, 8, 16, 17, 19, 20, 21, 22, 24

Friday, April 11 Last day to drop without a W grade

Week 4: Approximating solutions to differential equations by using Euler’s method:

Euler’s (pronounced “Oiler’s) method is a numerical method for approximating solutions at specific x-values.

§6.4 1-6

Week 5: Graphical solutions to autonomous differential equations

In this section, we will use 1^st derivative information in order to analyze the qualitative nature of solutions to autonomous 1^st order differential equations. Regardless of the instructions in the book, you should not worry about calculating 2^nd derivative information.

§3.4 1-3, 9, 11, 15, 17

Thursday, May 1 Midterm exam

Week 6: What does the sum on an infinite set of numbers mean?

An infinite sum is the limit of the partial sums.

§8.3 1, 3, 4, 7-9, 17, 19, 20, 22-24, 26-29, 33, 35, 37, 38, 41, 42, 47

Week 7: Definition of power series centered at a; manipulating power series; approximating power series near the center

Power series are a new way of defining a function. A power series is essentially a polynomial with an infinite number of terms. Power series are easy to manipulate and approximate. Regardless of the instructions in the book, we do not want you to calculate radii or intervals of convergence of power series.

§8.6 41, 42 Other problems to be created by instructor

Friday, May 16 Last day to drop with a W grade

Week 8: Some basic power series

Essentially a continuation from Week 7, after you have memorized power series that equal e^x, sin(x), cos(x), 1/(1-x), ln(1+x), and (1+x)^p, and for what x-values each function equals its power series. You will also be told a general procedure for producing something called a Taylor series centered at a for a given function f(x). If f(x) equals a power series centered at a (for all x’s in an open interval around a), then there is no choice – the series that equals f(x) must be the Taylor series centered at a. You will not be asked to calculate Taylor polynomials or series from scratch. In the homework problems, you should use your knowledge of the six power series that you have memorized.

§8.7 1, 7, 22, 24, 25, 26 §8.8 1, 2, 7, 12 Other problems to be created by instructor

Monday, May 26 Memorial Day – University closed

Week 9: Approximating solutions to differential equations using power series

In each problem, find the first 4 non-zero terms of the power series (centered at the initial x-value) which satisfies the initial value problem.

§8.8 15, 19, 21, 27 Other problems to be created by instructor

Monday, June 2 – Friday, June 6 FINAL EXAMS