Harvard University,FAS
Spring 2001

Mathematics Xb
Spring 2001

Introduction to Functions
and Calculus II

Course Head: Robin Gottlieb

Office: SciCtr 429
Email: gottlieb@math
MathXb applied: a Biological system
MathXb, Spring 2001 Course Announcement

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Follow the online instructions.
Goals of the Course:
Math X aims to provide you with a strong and solid understanding of precalculus and calculus material. We want you to leave the course with a solid set of skills and a versatile conceptual framework so you are well-equipped for future studies, whether in mathematics, the sciences, economics, or other disciplines. In order to achieve this goal we will emphasize multiple approaches to problem solving and stress understanding.

How will Math Xb relate to Math Xa ?
Math Xb is a continuation of Math Xa. This semester we will continue to discuss differential calculus, study integral calculus, work with trigonometric functions and introduce you to infinite series and differential equations. (This amounts to completing the material from Math 1a and giving you a leg-up on the material from Math 1b.) Not only will we study new topics but we will be aiming for a higher level of proficiency and problem solving skills while combining the new topics with those studied in Math Xa. We see the course as being a coherent whole and will certainly include problems using skills Œfrom' Math Xa in problem sets and examinations in Math Xb. As we stated in the syllabus for Xa, we plan to use the entire year to reinforce the most important ideas and skills of both calculus and precalculus; the span of one year gives us enough time to work on eliminating weak spots and to become familiar with and to draw on your strengths.

Format of the Course:
Math X will be taught in small classes in order to provide an environment where students are active participants and dialogue is promoted between the teacher and the class and among students. Small class size will also allow us to tailor the classes to your needs and to offer individualized attention. There will be twice-weekly laboratories designed to focus on problem solving and conceptual understanding. Sometimes new material will be introduced in lab. If you think of mathematics as a language, you can think of the labs as language labs where you work towards fluency; if you think of mathematics as a science, you can think of the labs as science labs where you work on problem solving. Labs will be coordinated with classes, allowing you to get new perspectives on topics presented in class by working on carefully chosen problems in a group setting.

Course Resources:
You are encouraged to talk with your Section Leader, your Lab Leader, and your Course Assistant about the course. We would all be happy to meet with you. Another resource is the Math Question Center (QC) where help is available from 8pm to 10pm Sunday night through Thursday night. The Math Question Center is located in Loker Common. Not only can you get help from Math Question Center staff, but you can find other people in this course to discuss problem sets with. Questions specific to the operations of the MQC should be directed to John Mackey (jfm@math.harvard.edu).


Instructor Office Email Phone
Robin Gottlieb SC 429 gottlieb@math 5-7882
Dale Winter SC 506 amanita@math 5-9063
There will be a course text for Math Xb, just as for Math Xa. You will purchase it from Gnomen Copy. We will give you the first chapter in class and let you know when the rest is ready.

Robin Gottlieb is the coursehead - which means that all complaints about the course and the text go to her. (She is eager to meet you, but she likes writing this sentence about complaints in the third person! )

The only way to learn math is by doing math.

Daily homework assignments: Problems will generally be assigned each class and are due at the next class. Assignments will be marked and returned by the following meeting. Some of the problems will look different from problems discussed in class. This is not an accident. We want you to actively think about the material, to be able to apply it in unfamiliar settings, and to interpret it in different ways. Therefore we will not give you a recipe for solving every problem. Your job is to accept this as a challenge - a challenge that we plan to help you meet.
Problems are an integral part of the course, and it is virtually impossible to do well on the exams without working through the homework problems in a thoughtful manner. Don't just crank through the computations and write them down ... think about the meaning of the computations you are performing and the answers you get. The main point is not to come up with specific answers to the particular problems you're working on, but to develop an understanding of what you're doing so that you can apply your reasoning to a wide range of similar mathematical situations. It is unlikely that later on in life you will see the exact same math problems you're working on now - so learn the material in such a way that it is a portable tool.
Solutions to homework will be made available on this course Web site. Lab leaders are in charge of the solutions. We encourage you to form study groups with other students in your class so you can discuss the work with one another. Early in the semester your Section Leader will provide names and contact information for everyone in your section in order to help facilitate discussion. This is most useful after you have worked independently. Although we encourage you to talk with your classmates, work must be written up individually. Homework must be turned in on time. Assuming all your homeworks are turned in on time, then we will drop your lowest five homework grades when computing your homework average. Suppose due to an emergency one of your homeworks is not submitted on time. You can correct it yourself, using the solutions posted on the Web, and submit a completed, corrected assignment to be checked off and then your lowest four homeworks will be dropped - and so one, up to a count of five.

Coursewide Exams:

Exam Date Time Location
1st exam: Tuesday, March 13 7:30 pm Sci. Cent. A
2nd exam: Tuesday, April 17 7:30 pm Sci. Cent. A
Final: Tuesday, May 22 - -
Grading Policy:

midterm score: Take the higher of the following two options
  • 45% 1st exam + 45% 2nd exam + 10% lab score (*)
  • 40% 1st exam + 50% 2nd exam + 10% lab score (*)
exam score: Take the higher of the following two options
  • 20% midterm score + 80% final exam
  • 65% midterm score + 35% final exam
course score: Take the higher of the following two options
  • 90% exam score + 10% homework score
  • 80% exam score + 20% homework score

The basic notion here is that students should be able to present his/her scores in a way that sets off the work of highest quality.
(*) Lab scores will be based on a combination of attendance and scores on lab quizzes and gateway tests. Gateway tests will be given periodically in lab or on the Web, depending upon the status of the technology. Questions on gateways will be relatively short and direct. You will either pass a gateway test or you will take variations on the test until you have passed. We will tell you in advance what will be covered. Passing the gateways gives you a 70 point cushion on your lab score. Please try not to save passing the gateways for a reading period activity! Passing the gateways gives you a 70 point cushion on your lab score. Quizzes and\or lab attendance contribute to the other 30 points. If your final exam score is substantially higher than all your other scores, your section leader may weigh the final exam as even more than 80% of your exam score.
Notice that using this grading scheme the final examination can count as little as 32% or as much as 72% of your final course score depending on how it compares to your other grades. The final and homework can collectively be worth as much as 84% of your grade.

We are looking forward to continuing a good year with you in Mathematics X!

  • Unit I: Differentiation
    • Revisiting the Chain Rule
    • Logarithmic Differentiation
    • Implicit Differentiation
    • Implicit Differentiation in Context (alias Related Rates of Change)
  • Unit II: An Interesting Limit and an Introduction to Differential Equations
    • What is a Differential Equation and What is a Solution to One?
    • Solving Differential Equations of the form dy/dt = k y
  • Unit III: Geometric Sums and Series
    • Geometric Sums and Series
    • General Study of Series
    • Applications of Geometric Sums and Series (biological and economic applications emphasized)
  • Unit IV: Trigonometry
      Circle Trigonometry - periodicity
    • Triangle Trigonometry
    • Differentiation of Trigonometric Functions
    • Inverse Trigonometric Functions and Their Derivatives
  • Unit V: Integration Introduction
    • Areas and Net Change in Amount
    • Defining the Definite Integral
    • The Area Function
    • The Fundamental Theorem of Calculus
  • Unit VI: Integration
    • Indefinite Integrals - using Substitution
    • Numerical Methods of Integration
    • Applications of Integration
  • Unit VII: Differential Equations
    • Words to Mathematical Models (biological and economic applications emphasized)
    • Solving Differential Equations
    • Qualitative Analysis of Differential Equations

Last update, 1/29/2001, mathxb@fas.harvard.edu