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Higher Mathematics Course Syllabus

日期:2009-01-12浏览次数:587

Objective: The objective is to master ideas and methods of calculus so that the students can use these to solve the questions they meet in studying other courses and in life. It is also very important to improve quality of students in all aspects to analyse questions and solve problems.
The main content:
1. Limits and Derivatives : 1)the tangent and velocity problem, 2)the limit of a function, 3)calculating limits using the limit laws, 4)the precise definition of a limit, 5)continuity, 6) limits at infinite, horizontal asymptotes, 7) tangents, velocities, and other rates of change, 8) derivatives, 9) the derivative as a function.
2. Differentiation Rules: 1) Derivatives of polynomials and exponential functions, 2) the product and quotient rules, 3) rates of change in the natural and social sciences, 4) derivatives of trigonometric functions, 5) the chain rule, 6) implicit differentiation, 7) higher derivatives, 8) derivatives of logarithmic functions, 9) hyperbolic functions, 10) related rates, 11) linear approximations and differentials.
3. Applications of Differentiation: 1) maximum and minimum values, 2) the mean value theorem, 3) how derivatives affect the shape of a graph, 4) indeterminate forms and L’Hospital’s Rule, 5) summary of curve sketching, 6) graphing with Calculus and Calculators, 7) optimization problems, 8) applications to business and economics, 9) Newton’s method, 10) antiderivatives.
4. Integrals: 1) areas and distances, 2) the definite integral, 3) the fundamental theorem of calculus, 4) indefinite integrals and the Net Change Theorem, 5) the substitution, 6) the logarithm defined as an integral.
5. Applications of integration: 1) areas between curves, 2) volumes, 3) volumes by cylindrical shells, 4) work, 5) average value of a function.
6. Techniques of Integration: 1) integration by parts, 2) trigonometric integrals, 3) trigonometric substitution, 4) integration of rational functions by partial fractions, 5) strategy for integration, 6) integration using tables and computer algebra systems, 7) approximate integration, 8) improper integrals.
7. Further applications of integration: 1) arc length, 2) area of a surface of revolution, 3) applications to physics and engineering, 4) applications to economics and biology, 5) probability.
8. Differential Equations: 1) Modeling with differential equations, 2) direction fields and Euler’s Method, 3) separable equations, 4) exponential growth and decay, 5) the logistic equation, 6) linear equations, 7) Predator-Prey systems.
9. Parametric Equations and Polar Coordinates: 1) curves defined by parametric equations, 2) calculus with parametric curves, 3) polar coordinates, 4) areas and lengths in polar coordinates, 5) conic sections, 6) conic sections in polar coordinates.
10. Infinite sequences and series: 1) sequences, 2) series, 3) the integral test and estimates of sum, 4) comparison tests, 5) alternating series, 6) absolute convergence and the ratio and root test, 7) strategy of for testing series, 8) power series, 9) representation of functions as power series, 10) Taylor and Maclaurin series, 11) The binomial series, 12) Applications of Taylor polynomials.
11. Vectors and the Geometry of Space: 1) three-dimensional coordinate systems, 2) vectors, 3) the dot product, 4) the cross product, 5) equations of lines and planes, 6) cylinders and quadric surfaces, 7) cylindrical and spherical coordinates.
12. Vector Functions: 1) vector functions and space curves, 2) derivatives and integrals of vector functions, 3) arc length and curvature, 4) motion in space: velocity and acceleration.
13. Partial Derivatives: 1) functions of several variables, 2) limits and continuity, 3) partial derivatives, 4) tangent planes and linear approximations, 5) the chain rule, 6) directional derivatives and the gradient vector, 7) maximum and minimum values, 8) Langrange multipliers.
14. Multiple Integrals: 1) double integral over rectangle, 2) iterated integrals, 3) double integral over general regions, 4) double integrals in polar coordinates, 5) applications of double integrals, 6) surface area, 7) triple integrals, 8) triple integrals in cylindrical and spherical coordinates, 9) change of variables in multiple integrals.
15. Vector Calculus: 1) vector fields, 2) line integrals, 3) the fundamental theorem for line integrals, 4) Green’s theorem, 5) curl and divergence, 6) parametric surfaces and their areas, 7) surface integrals, 8) Stokes’s theorem, 9) the Divergence theorem, 10) summary.
16. Second-Order Differential Equations: 1) second-order linear equations, 2) nonhomogeneous linear equations, 3) applications of second-order differential equations, 4) series solutions.
Recommendation for the textbook: James Stewart, Calculus (Fifth Edition), 2004. 影印版,上下两册,高等教育出版社。