MATH 900-3B: How to Think "The Calculus Way": July 14th to August 1st, 2008
Professor Steven Miller (sjmiller AT math.brown.edu), Kassar House, Room 210, 863-1123
Webpage with many useful links and explanations and examples: http://www.math.brown.edu/UTRA/
Review sheet of all derivatives and rules
DESCRIPTION: Calculus, as the most important subject in mathematics, has far-reaching consequences and applications in many sciences. However, the main ideas involved in understanding calculus are often challenging, and many students find the gap between their pre-calculus and calculus courses too difficult to overcome. Strong emphasis will be placed on following all strains of thought back to first principles and constantly re-examining one's own thought processes. The session will contain many of the main topics covered in a first semester college calculus course: limits, derivatives, applications of the derivative, and a small amount of integration theory. But more than this, the course will require students to work and think more intensely about mathematical ideas; techniques of proof will be covered and used frequently, and literature discussions will reinforce mathematical concepts, force verbal expressions of symbolic language, and place the mathematical experience in a cultural context. Students' performance will be evaluated through homeworks, an independent project on a calculus-related topic outside the scope of the course, and a final exam. This course is thus intended to serve as a preparation for students planning to take high-school or college calculus and who are already familiar with the basics of algebra, trigonometry, and pre-calculus. Students will also learn problem-solving techniques, how to write up homework solutions, how to read a calculus book, and how to think "the calculus way." Course homepage: http://www.math.brown.edu/~sjmiller/900-3B/index.htm.
IMPORTANT: It is essential that you have a solid grasp of high school algebra for this course. Over the years I have found many students' understanding of calculus is hindered by gaps in their ability to do basic algebra. As we will use algebra throughout the course, you must be familiar and comfortable with it at the start of the class. You should review your algebra. Any textbook is fine; one particularly good (and cheap!) one is Schaum's Easy Outlines: Precalculus (Based on Schaum's Outline of Precalculus by Fred Safier, Abridgement Editor: Kimberly S. Kirkpatrick). You should know the material from Chapters 1 2, 3, 4, 5, 6, and 8. You can also find algebra review online (see, for example, http://www.math.uakron.edu/~dpstory/mpt_home.html).
GENERAL: There will be daily homework assignments; Homework problems here. Homework is to be submitted on time, neatly written and stapled (if not, the homework may not be graded). Working in groups is encouraged; everyone must submit their own assignment. If you work in groups, please write the name of everyone in your group on the first page.
TEXTBOOK:
The Calculus Lifesaver:
All the Tools You Need to Excel at Calculus, Adrian Banner,
ISBN13: 978-0-691-13088-0.
Webpage with many useful links and explanations and examples:
http://www.math.brown.edu/UTRA/
TENTATIVE SYLLABUS:
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Week One:
Monday: Introduction, Review, Definition of the Derivative
Tuesday: Limits, Derivatives of Simple Functions
Wednesday: Rules of Differentiation
Thursday: Rules of Differentiation and Applications (Tangent Line)
Friday: Implicit Differentiation, Maximization and Minimization
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Week Two:
Monday: The Chain Rule
Tuesday: : Derivatives of trigonometric functions
Wednesday: Derivatives of logarithmic and exponential functions
Thursday: Curve sketching
Friday: L’Hospital’s Rule and Newton’s Method
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Week Three:
Monday: Areas, distances and integratals
Tuesday: The definite integral
Wednesday: The fundamental theorem of calculus
Thursday: Applications of integration
Friday: Review
HANDOUTS: