国际通识课程 01.Syllabus of Advanced Mathematics

来源：国际交流学院发布时间：2018-06-10


SYLLABUS OF ADVANCED MATHEMATICS


Instructor: ZHANG CHAO

Office Phone:51278249

Email:hezile.keer@163.com



REQUIRED TEXT

1. Thomas' Calculus, 11th Edition, George B. Thomas, Addison Wesley;


DESCRIPTION

 This course provides a standard introduction to differential and integral calculus and covers topics ranging from functions and limits to derivatives and their applications to definite and indefinite integrals and the fundamental theorem of calculus and their applications, including differential equations.


OBJECTIVES

The students should gain a thorough understanding of the basic concepts of Calculus of one and multi variables. The main goals of this course are to introduce the ideas andtechniques ofcalculus and to explore the applications or uses of these ideasand techniques. Throughout wewill emphasize conceptual understanding, whichleads to a better view of how these ideas and techniques were developed.


COURSE ASSESSMENT

        • Class Attendance: 20%. Responsible attendance is expected of eachindividual enrolled in this course.

  • Group/individual presentation & Class discussion: 10%. Students will be assigned presentations on the course topics, and are encouraged to actively participate in the class discussion.

  • Examinations: 70%. There will be 2 examinations,including the Midterm and Final examinations during this semester. Midterm Exam will be worth 40%, while the Final will be worth 40%. The questions will come from the assigned reading material, the lecture notes, as well as classdiscussions and exercises. Test questions will be presented as multiple choice, true/false, and short essay.



COURSE CONTENTS



Topics

Class Hours


Week

1

Preliminaries.
Lines.
Functions and Graphs.
Exponential Functions.
Inverse Functions and Logarithms.
Trigonometric Functions and Their Inverses.
Parametric Equations.
Modeling Change.



 5



Week

2-3

Limits and Continuity.
Rates of Change and Limits.
Finding Limits and One-Sided Limits.
Limits Involving Infinity.
Continuity.
Tangent Lines.


 10



Week

4-5

Derivatives.
The Derivative as a Function.
The Derivative as a Rate of Change.
Derivatives of Products, Quotients, and Negative Powers.
Derivatives of Trigonometric Functions.
The Chain Rule.
Implicit Differentiation.
Related Rates.


 10



Week

6-8

Applications of Derivatives.
Extreme Values of Functions.
The Mean Value Theorem and Differential Equations.
The Shape of a Graph.
Graphical Solutions of Autonomous Differential Equations.
Modeling and Optimization.
Linearization and Differentials.
Newton's Method.


 10



Week

9

  • Mid-Term Exam


 3



Week

9-11

Integration.
Indefinite Integrals, Differential Equations, and Modeling.
Integral Rules; Integration by Substitution.
Estimating with Finite Sums.
Riemann Sums and Definite Integrals.
The Mean Value and Fundamental Theorems.
Substitution in Definite Integrals.
Numerical Integration.


 10



Week

11-13

Applications of Integrals.
Volumes by Slicing and Rotation About an Axis.
Modeling Volume Using Cylindrical Shells.
Lengths of Plane Curves.
Springs, Pumping and Lifting.
Fluid Forces.
Moments and Centers of Mass.



 15




Second Semester

Week 1-4


Multivariable Functions and Their Derivatives.
Functions of Several Variables.
Limits and Continuity in Higher Dimensions.
Partial Derivatives.
The Chain Rule.
Directional Derivatives, Gradient Vectors, and Tangent Planes.
Linearization and Differentials.
Extreme Values and Saddle Points.
Lagrange Multipliers.
Partial Derivatives with Constrained Variables.
Taylor's Formula for Two Variables.



 20



Week

5-10 and Midterm

  • Infinite Series.
    Limits of Sequences of Numbers.
    Subsequences, Bounded Sequences, and Picard's Method.
    Infinite Series.
    Series of Nonnegative Terms.
    Alternating Series, Absolute and Conditional Convergence.
    Power Series.
    Taylor and Maclaurin Series.
    Applications of Power Series.
    Fourier Series.
    Fourier Cosine and Sine Series.


 23



Week 10-15

Differential Equations.
Derivatives of Inverse Trigonometric Functions; Integrals.
First Order Separable Differential Equations.
Linear First Order Differential Equations.
Euler's Method; Population Models.




 25