The Syllabus of Calculus 教学纲要

The Syllabus of Calculus（教学纲要） Chapter 1 TheDerivative 1. Content: Rate of change and slope; limits; The derivative; Derivativesof constants. power forms, sums, products and quotients; Chain rule; Marginal analysis in business and economics. 2. Focal point and difficult point: Focal point: (a).limit evaluation; (b).The existence of the derivative. (c).power rule, derivatives of sums, differences, products and quotients. (d)Chain rule and combining rules of differentiation. (e).Marginal cast, revenue and profit difficult point: (a) The applications of derivative; (b) The evaluation of derivative. 3. Profundity: (a) Master the technique of derivative’s evaluation.

(b) Understand the definition of derivative.

4. Class hour’s distribution . Instruct 7 class hours. Review 2 class hours

5. Group activity: Minimal average cost. Chapter 2 .Graphing and optimization 1.Content: Continuity and graphs; first derivative , second derivative and graphs; Optimization : absolute Maxim and minima. 2.Focal point and difficult point: Focal point: (a).The definition and properties of Continuity (b).Increasing and decreasing functions; (c).Local extreme and first-derivative test (d).Concavity inflection points and second-derivative test. (e).Absolute maxim and minima. difficult point: (a) The definition and properties of continuity; (b) Analyzing graphs (c)Limits at infinity 3.Profundity: (a) Grasp the definition and properties of continuity; (b) Sketch the graph of by using the graphing strategy

values of a continuous function on a closed interval 4.Class hour’s distribution . Instruct 5 class hours. Review 2 class hours 6. Group activity: (1) Maximizing profit (2) Minimizing construction costs. Chapter 3 Additional derivative topics 1. Content: The constant and continuous compound interest; Derivatives of Logarithmic and Exponential Functions; Chain rule ; Implicit Differentiation; Related Rates. 2. Focal point and difficult point: Focal point: (a). continuous compound interest; (b).Derivative formulas for and (c). Generalized derivative rules (d)Implicit differentiation. (e).Related Rates problems difficult point: (a) Growth of exponential and logarithmic functions (b) Composite functions and Chain rule. (c)Implicit differentiation. (d) Related Rates problems 3. Profundity:

(a) Grasp the derivative formulas for and

(b) Master the generalized derivative rules

(d) Understand the suggestions for Solving. Related Rates problems 4. Class hour’s distribution . Instruct 5 class hours. Review 2 class hours 5 . Group activity: (a) Elasticity of demand; (b) Point of diminishing returns. Chapter 4 Integration 1.Content: Anti-derivatives and indefinite integrals; Integration by Substitution; Differential equations; Geometric-Numeric introduction to the definite integral. Fundamental theorem of calculus. 2.Focal point and difficult point: Focal point: (a). Anti-derivatives and indefinite integrals: algebraic; exponential and Logarithmic forms. (b). Reversing the chain rule (c).Integration by substitution (d)Population growth, radioactive decay and learning (e).Rate area and total change difficult point: (a) Reversing the chain rule (b)Integration by substitution (c) Definite integral as a limit of a sum (d) Fundamental theorem of calculus. 3. Profundity: (a)Master anti-derivatives and indefinite integrals: algebraic; exponential and Logarithmic forms. (b) Master the method of integration by substitution (c) Understand the definite integral as a limit of a sum (d) Grasp the fundamental theorem of calculus. 4. Class hour’s distribution . Instruct 5 class hours. Review 2 class hours 5. Group activity: (a) Simpson’s rule; (b) Bell-shaped curves Chapter 5 Additional Integration Topics 1.Content: Area between curves; Integration by parts; Applications in business and economics 2.Focal point and difficult point: Focal point: (a). Area between two curves (b).consumers’ and producers’ surplus; (c). Integration by Parts (d) Integration using tables difficult point: (a) Area between two curves (b) Integration by Parts 3.Profundity: (a) Master the Calculation of Area between two curves (b) Master the method of integration by Parts (c) Understand the integration’s applications in business and economic 4.Class hour’s distribution . Instruct 4class hours. Review 2 class hours 5.Group activity: (a) Analysis of income. Concentration form raw data; (b) Grain exchange Chapter 6 Multivariable Calculus 1.Content: Functions of several variables; Partial derivatives Maxim and minima using Lagrange multipliers; Method of least squares ; Double integrals over rectangular regions. 2.Focal point and difficult point: Focal point: (a).Functions of two or more independent variables (b).Partial derivatives and second-order partial derivatives (c).Second-derivative test for local extreme (d).Maxim and minima using language multipliers (e).Least Squares approximation; (f)Definition of the double integral difficult point: (a) Partial derivatives; (b)Maxim and minima using language multipliers (c)Method of least Squares (d)Double integrals over rectangular regions 3.Profundity: (a) Master the definition and method of partial derivatives (b)Master the method of obtaining maxim and minima using language multipliers (c) Understand the method of least squares 4.Class hour’s distribution . Instruct 6 class hours. Review 2 class hours 5.Group activity: (a) city planning (b) Numerical integration of multivariable 教学进度表 Calculus I

Chapter 1:Limits and Continuity (12 hours ) 1.1 The Idea of Limit 1.2 Definition of Limit 1.3 Some Limit Theorems, Additional information on Infinite Limits 1.4 Continuity 1.5 The Pinching Theorem; Trigonometric Limits 1.6 Two Basic Properties of Continuous Functions Chapter 2: Derivatives (12 hours ) 2.1 The Derivative 2.2 Some Differentiation Formulas 2.3 The d/dx Notation; Derivatives of Higher Order 2.4 The Derivative as a Rate of Change 2.5 The Chain Rule 2.6 Differentiating the Trigonometric Functions 2.7 Implicit Differentiation; Rational Powers Chapter 3: Applications of Derivatives (12 hours , 2 hours

(pratice)) 3.1. Extreme values of functions 3.2. The mean-value theorem and differential equations 3.3 The Mean-Value Theorem and Applications 3.4. The Shape and a graph 3.5. Graphical solution of autonomous differential equations 3.6. Modeling and optimization 3.7. Linearization and differentials 3.8. Newton's methods Chapter 4: Integration (14 hours) 4.1 Indefinite integrals, differential equations, and modeling 4.2 Integral rules, integration by substitution 4.3 Estimate with finite sums 4.4 Riemann Sums and definite integrals 4.5. The mean value and fudametal theorem 4.6 Substitute in definite integrals 4.7 Numerical integration Chapter 5: Applications of integrals: (4 hours for practice) 5.1 Volumes byslicing and rotation about am axis 5.2 Modeling volume using cylindical shells 5.3 Lengths of plane curves 5.4 Spring, pumping, and lifting 5.5 Fluid forces 5.6 Moments and centers of mass Chapter 6 Transcendental functions and differential equation (14 hours) 6.1 Lograithms 6.2 exponential functions 6.3 derivatives of inverse trigomometric function: intgrals 6.4 Firs-order differential equation 6.5 Linear first-order differential equation 6.6 Euler’s methods: population models 6.7 Hyberbolic function Chapter 7 Integration Techniques, L’Hoptial Rule, and

Improper Integrals (14 hours ) 7.1 Basic integration formulas 7.2 Integration by parts 7.3 Partial functions 7.4Trigonometric Substitutes 7.5 Integral table, computer algebra systems, and Monte Carlo integration (In practice 2 hours) 7.6 L’Hoptial Rule 7.7 Improper Integrals Calculus II Chapter 8: Infinite Series (14 hours) 8.1 Limits of sequences of numbers 8.2 Subsequences, bounded sequences, and Picard’s methods 8.3 Infinite series 8.4 Series of Nonnegative terms 8.5 Alternating series, absolute and conditional; convergence 8.6 Power series 8.7 Taylor series 8.8 Applications of power series Chapter 9: Vectors in the plane and polar functions (14 hours) 9.1 Vector in the plane 9.2 Dot Products 9.3 Vector-valued function 9.4 Modeling projectile Motion 9.5 Polar coordinate and Graphs 9.6 Calculus of polar curves Chapter 10: Vectors and Motion in Space (12 hours) 10.1 Cartesian Coordinate and vectors in space 10.2 Dot and Cross products 10.3 Lines and plane in space 10.4 Cylinders and quadratic surfaces 10.5 vector-valued functions and space curves 10.6 Arc length and the unit tangent vector T 10.7 The TNB frame: Tangential and Normal components of acceleration 10.8 Planetary motion and satellites Chapter 11: Multivariable functions and their derivatives

(12 hours) 11.1 functions of several variables 11.2 limits and continuity in higher dimension 11.3 Partial derivatives 11.4 Chain rule 11.5 Directional derivative, 11.6 Linearization and differentials 11.7 extreme values and saddle points Chapter 12: Multiple Integrals (14 hours) 12.1 Double integral 12.2 areas, moments and centers of Mass 12.3 Double integral in polar coordinates Chapter 13: Integration in Vector fields (12 hours) 13.1 Linear integrals 13.2 Vector fields, 13.3 Path independence 13.4 Green’s theorem 13.5 Surface area and surface integrals

课程内容

课程内容 /2.pdf 课程内容 preparation-chapter 1.pdf 课程内容 preparation-chapter 2.pdf 课程内容 preparation-chapter 3.pdf 课程内容 preparation-chapter 4.pdf 课程内容 preparation-chapter 6.pdf 课程内容 tcu11_02_01.pdf 课程内容 tcu11_02_02.pdf 课程内容 tcu11_06_01.pdf 课程内容 tcu11_06_02.pdf 课程内容 tcu11_06_03.pdf 课程内容 tcu11_06_04.pdf 课程内容 tcu11_06_05.pdf 课程内容 Thomas_Ch7.pdf 教学教案 1, 教学教案 2

习题集 1：

习题 1 cont.pdf 习题 1 curves1.pdf 习题 1 curves2.pdf 习题 1 defn.pdf 习题 1 higher.pdf 习题 1 diff.pdf 习题集 2：

习题 2 tan.pdf 习题 2 alseries.pdf 习题 2 defn.pdf 习题 2 integral.pdf 习题 2 limits.pdf 习题 2 poseries.pdf 习题 2 sequence.pdf

课程考核试卷

课程考核试卷 1114.pdf 课程考核试卷 1115.pdf 课程考核试卷 1116.pdf

参考文献

1.Thomas' Calculus, 第十版, 科学出版社, 影印版 2004

2. Calculus-James-Stewar, Brooks/Cole, (2008)

3. Calculus, Tom M. Apostol, John Wiley & Sones. Inc.

(paperback) 2005

4. 高等数学 (上册,下册) 同济大学数学教研室编写 , 高等教育出

版社出版, 2004 5. 高等数学(上册, 下册), 杨则燊，边馥萍主编, 天津大学出版

社,2005 .

6. 高等数学(上册, 下册), 上海大学数学系主编, 上海大学出版社,

2006.

7. http://webcast.berkelay.

8. Web: Thomas' Cauculus