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课程编号

19299004

开课学院

College of Science

开课系

Department of Mathematics

课程名称

中 文

 

课程类别

a required course

英 文

Calculus

课程学时

总学时

理论教学

实验教学

上 机

课程设计

176

176

 

 

1.有    2. 无√

课程简介:
简要描述课程的性质及专业地位,培养目标(理论、能力和技能)
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 .

 

Part 1、Functions,Limits and Continuity

1. The main content
(1). Functions and Their Graphs.
(2). Exponential Functions, Trigonometric Functions and Their Inverses.
(3). Parametric Equations .
(4). Rates of change and limits.
(5). the precise definition of a limit.
(6). Limit Theorems.
(7). Limits at Infinity, Infinite Limits.
(8). Continuity of Functions.
(9). Tangent lines ,velocities and other rates of change.

2. Focal and difficult points
Focal points:Concept of  function, limit . Limit theorems and continuity of functions.
Difficult point:the precise definition of a limit.

3. The hours required:15 hr.
(1).Functions:Lectures 4 hr.
(2).Limits:Lectures 6 hr.
(3).Continuity:Lectures 3 hr.
(4). Recitations :2 hr.

4. Exercises:40~60 items. 

Part 2 、Derivatives and Their Applications

1. The main content
(1). The derivative as a rate of change.
(2). Rules for Finding Derivatives.
(3). Derivatives of Trigonometric Functions.
(4). differentiation formulas.
(5). The Chain Rule.
(6). Parametric Equations.
(7). Higher-Order Derivatives.
(8). Implicit Differentiation.
(9). Related rates.
(10).Extreme values of functions.
(11). The Mean Value Theorem.
(12). The Shape of a Graph .
(13). Modeling and Optimization.
(14). linearization and differentials.
(15). Transcendental Functions and Their Derivatives: (a). Logarithms,(b). Exponential Functions, (c). Derivatives of Inverse Trigonometric Functions ,(d). hyperbolic functions.
(16). L'hospital's rule.

2. Focal and difficult points
Focal points:Concept of derivative and differentials. Derivatives of sums, differences, products and quotients. The Chain Rule. The Mean Value Theorem. L'hospital's rule. Calculation of Extreme values of functions.
Difficult points:The Chain Rule. Taylor’s theorem .

3. The hours required:32 hr.
(1). Derivatives:Lectures 10 hr,Recitations 2 hr.
(2). The Mean Value Theorem:Lectures 3 hr,Recitations 1 hr.
(3). Applications of Derivatives:Lectures 7 hr,Recitations 1 hr.
(4). Transcendental Functions and Their Derivatives: Lectures 5 hr,Recitations 1 hr.
(5). L'hospital's rule: Lectures 2 hr.

4. Exercises:150~170  items.

 
Part 3、Integration and its Applications

1. The main content
(1).Indefinite integrals.
(2). Integral rules.
(3). Riemann sums and Definite Integrals.
(4). The Mean Value and Fundamental Theorems.
(5). Substitution Rule.
(6). applications of integration: (a). Volumes by slicing and rotation about axis. (b).Lengths of Plane Curves .
(7).Integration Techniques: (a). Basic integration formulas,(b).Integration by parts,(c). Trigonometric Substitutions,(d). Integral tables.
(8). Improper integrals.

2. Focal and difficult points
Focal points:Concept of Indefinite integrals and Definite integrals .Substitution Rule. Integration by parts. Fundamental theorems.
Difficult points:Substitution Rule. Definition of definite integrals. Proof for fundamental theorems for calculus .

3. The hours required:33 hr.
(1).Indefinite integrals:Lectures 4 hr,,Recitations 1 hr.
(2).Definite Integrals:Lectures 8 hr,,Recitations 2 hr.
(3).applications of integration:Lectures 4 hr,Recitations 1 hr.
(4). Integration Techniques: Lectures 8 hr,,Recitations 2 hr.
(5). Improper integrals : Lectures 3 hr.

4. Exercises: 150~170 items.

Part 4、Infinite Series

1. The main content
(1) . Limits of sequences of numbers.
(2) . Infinite series.
(3) . Series of nonnegative terms.
(4) . Alternating series, absolute and conditional convergence.
(5). Power series.
(6). Taylor and Maclaurin series.
(7). Fourier series.
(8). Fourier cosine and sine series.

2. Focal and difficult point
Focal points: convergence and divergence of sequence and Infinite series.The Ratio Test. Alternating Series .Absolute and Conditional Convergence .Finding Taylor Polynomials for a function and Fourier series.
Difficult points:Proof for The Ratio Test. Finding Taylor polynomials for a function by direct method. Estimating the remainder of a Taylor polynomial.

3. The hours required:21 hr.
(1).Sequences: Lectures 3 hr,Recitations 1 hr.
(2). Infinite series:Lectures 10 hr,Recitations 2 hr.
(3).Fourier series:Lectures 4 hr,Recitations 1 hr.

4.Exercises:100~120 item.

Part 5、Vectors in the Plane and Space

1. The main content
(1). Vectors in the Plane and Polar Functions: (a) .Vectors in the plane ,(b). Dot product, (c) .Vector-valued functions,(d). Polar coordinates and graphs,(e). Calculus of polar curves.
(2). Vectors and motion in space: (a). Vectors in space,(b). Dot and cross product,
(c). Lines and Planes in space,(d). Cylinders and Quadric Surfaces,(e).Vector valued functions and space curves,(f). Arc length and the unit tangent vector T.

2. Focal and difficult points
Focal points: Concept of vector.Component form for a vector. Dot and cross product. Polar coordinates and graphs, Finding Equations of Planes in space .Finding bector equations for a line in space. Concept of surface equation.
Difficult point: Concept of cross product. Graphing the region enclosed by the several surfaces.

3. The hours required:19 hr.
(1). Vectors in the Plane: Lectures 6 hr,Recitations 1 hr.
(2). Vectors in space: Lectures 10 hr,Recitations 2 hr.

4. Exercises60~80 items

Part 6、Multivariable Functions and Their Derivatives

1. The main content
(1). Functions of several variables
(2). Limits and continuity in higher dimensions
(3). Partial derivatives
(4). The chain rule
(5). Directional derivatives, gradient vectors, and tangent planes
(6). Linearization and differentials
(7). Extreme values and Saddle points
(8). Lagrange multipliers

2. Focal and difficult points
Focal points: Concept of Partial derivatives and Total differentials. Chain rule for composite functions in Higher Dimensions. Implicit Differentiation.
Difficult points:Concept of Total differentials. Chain rule.

3. The hours required:12 hr.
Lectures 10 hr,Recitations 2 hr.

4.Exercises:80~90 items

Part 7、Multiple integrals

1. The main content
(1). Double integrals.
(2). Areas, and centers of mass
(3). Double integrals in polar form
(4). Triple integrals in rectangular coordinates
(5). Masses and moments in three dimensions
(6). Triple integrals in cylindrical and spherical coordinates
(7). Line integrals
(8). Vector fields, work, circulation, and flux
(9). Path independence, potential functions, and conservative fields
(10). Green’s theorem in the plane
(11). Surface area and surface integrals
(12). Parametrized surfaces
(13). Stokes’theorem
(14). Divergence theorem and unified theory

2. Focal and difficult points
Focal points: Evaluating Multiple Integrals. Calculation of Line integrals and surface integrals. Green’s theorem .
Difficult points:Evaluating Triple integrals in spherical coordinates. Calculation of surface integral for flux.

3. The hours required:32 hr.
(1).Multiple Integrals and Their Applications:Lectures 10 hr,Recitations 2 hr.
(2).Line and Surface Integrals, Field Theory: Lectures 16 hr,Recitations 4 hr.

4.Exercises100~120 items

Part 8、 Differential Equations

1. The main content
(1). Introduction to Differential Equations.
(2). First-Order Separable Differential Equations.
(3). Linear First-Order Differential Equations.
(4). Second-Order,Linear Diff. with constant coefficients.

2.Focal and difficult points
Focal points: Concept of Differential Equations. First-Order Separable Differential Equations. Linear First-Order Differential Equations. Second-Order,Linear Differential with constant coefficients.
Difficult point:The theory for solution of Linear Differential Equations.

3. The hours required:12 hr.
Lectures 10 hr,Recitations 2 hr.

4.Exercises70~80 items

The final grade will be based on the following:

  1. Final exam 70%.
  2. Midterm 20% .
  3. Exercises 10%.

Comprehensive 2-hour final exam.
Examination: No bringing any reference. and textbook.

Preliminaries

Mathematics in high school

Textbook

Thomas’Caluculus

Reference

Varberg D.,Purcell E.J., Rigdon S. E., Calculus, 8th edition, Prentice Hall, Inc.
Calculus by James Stewart, published by Brooks/cole publishing company in 1995.

Web site of This Course

http://gc.nuaa.edu.cn/math(高数精品课程)

teaching method

Lectures and Recitations

specialty

Aero. and Astro. Engineering etc. 

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