Math 1a - Introduction to CalculusInformation and Syllabus, Spring 2001Course Head: Tom Graber, Science Center 426g, 495-8797, graber@mathGoals: Math 1a is a first semester calculus course covering differentiation, anintroduction to integration, and applications. It emphasizes understandingas much as computation. When you leave Math 1a we want you to take withyou the ideas that will enable you to use the concepts of calculus later, bothin mathematics and in other fields.Prerequisites: Some of you will have had calculus before, some of you willhave not. However, those of you who haven’t need not be alarmed - in thepast students without a calculus background have done as well as or betterthan those with this background. Doing well in Math 1a does require asolid background in precalculus, as demonstrated by an 18 on part 1 of theHarvard Math Placement Test. For those of you who are not comfortablewith high-school algebra, basic trigonometry and the like, we recommend thesequence Math Xa-Xb.Classes and Problem Sessions: Class will meet three hours per week. Youwill be assigned to a problem session which meets once a week for 1 hourand is led by a course assistant (CA). Course Assistants grade homeworkassignments, attend classes, and hold weekly problem sessions. The problemsessions are an integral part of the course and will be devoted primarilyto working problems and amplifying the material. The pace of the course isquite fast, so these sessions should be particularly valuable to you in learningthe material. You are strongly urged to attend.Homework: Homework exercises are an integral part of the course. It’sunlikely that you’ll understand the material and do well on the exams withoutworking through the homework problems in a thoughtful manner. Don’tjust crank through computations and write down answers - think about theproblems posed, your strategy, the meaning of your computations, and theanswers you get. We encourage you to form study groups with other studentsin the class so that you can discuss the work with each other. (Although allwork submitted must be written up individually.)Problems will generally be assigned in each class and are due at the nextclass. Assignments will be graded by your course assistant and will usuallybe returned at the following meeting.Finally, a somewhat harsh-sounding rule: we will not accept late homework.It just doesn’t work to do so - new homework is given out every class, andit’s important to keep current, since each homework assignment is relevantto that class and the next. We understand that occasionally circumstancesmake it impossible to do a particular assignment, and up to three missedassignments will be ignored, but beyond that it will begin to affect yourgrade.Text: Calculus, Brief Edition (Sixth Edition) by Howard Anton; John Wiley1999Exams: There will be two in class midterms and a final exam. The midtermswill be on February 27th and April 10th. Grading: The weights of thevarious parts of the course are as follows (subject to minor modification) -Each midterm : 20% Homework: 20% Final Exam: 40%Calculators: The use of a graphing calculator can prove helpful in under-standing many topics in the course, from limits and successive approximationto graphing. However, we encourage you to not rely too heavily on a graph-ing calculator as you work through your homework problems. In particular,calculators are not required for the course, and won’t be allowed during ex-ams.Topics to be covered: (in approximate order)Basics - functions, domain, range, graphs, lines, slope,linear functions, etc.Introduction to rates of change via examplesVelocity as a rate of changeDefinition of the derivative - calculating numerically and algebraicallyTangent lines and linear approximation (part one)Second derivativesLimits and successive approximationContinuity and differentiabilityTechniques of differentiationDerivatives of the trigonometric functionsDerivatives of exponential functionsRates of change and rectilinear motionComposite functions and the chain ruleImplicit differentiationInverse functions and their derivativesRelated ratesDifferentials and linear approximation (part two)Newton’s MethodExtreme values of a continuous functionThe Mean Value TheoremFirst Derivative TestConcavity and the second derivative testInfinite limits and asymptotesOptimizationL’Hopital’s RuleAntiderivativesArea as the limit of a sumRiemann Sums and the definite integralThe Fundamental Theorem of CalculusThe Mean Value Theorem for integrals; average valueNumerical Integration: Trapezoid Rule, Midpoint Rule, and Simpson’s RuleExponential and logarithmic FunctionsThe inverse trigonometric functionsAssorted
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