国际通识课程 30.Syllabus of Calculus I

SYLLABUS OF CACULUS I

Instructor: Zhou Jia

Office Phone: 021-51278083

Email:1110260#xdsisu.edu.cn（#替换为@）

Office:821

COURSE OUTLINE  

I. COURSE NUMBER AND TITLE  

50MAT103: Calculus (3 credits)  

II.  COURSE DESCRIPTION

This course covers the fundamental topics of functions and derivatives with emphasis on methods, optimization, and applications in business and economics. It is especially directed toward biology, business and social science majors for whom calculus can be a valuable and useful tool for solving problems.   

III.  EFFECTIVE DATE  

Spring 2016

IV.  COURSE OBJECTIVES  

At the completion of this course, the student will be able to:   

•  Compute derivatives of products, quotients, powers, and compositions of polynomial, exponential and logarithmic functions.   

•  Use his or her understanding of the derivative as a rate of change to set up an appropriate mathematical model to solve typical calculus word problems.   

•  Use techniques of differential calculus to solve optimization problems involving a differentiable real-valued function of a single variable.   

•  Use the techniques of differentiable calculus to sketch a graph of a function that accurately highlights important features, including asymptotes, local extreme values and inflection points.   

•  Understand the elementary trigonometric functions from a function-theoretic perspective and apply that perspective to solve problems.   

V. COURSE CONTENT  

CHAPTER 1 Functions, Graphs, and Limits 1.0 week

1.1 Functions  

1.2 The Graph of a Function  

1.3 Linear Functions  

CHAPTER 1 Functions, Graphs, and Limits (continued) 1.0 week

1.4 Functional Models  

1.5 Limits  

1.6 One-Sided Limits and Continuity

CHAPTER 2 Differentiation: Basic Concepts 1.0 week

2.1 The Derivative  

2.2 Techniques of Differentiation

CHAPTER 2 Differentiation: Basic Concepts (continued) 1.0 week

2.3 Product and Quotient Rules; Higher-Order Derivatives

2.4 The Chain Rule

CHAPTER 2 Differentiation: Basic Concepts (continued) 1.0 week

2.5 Marginal Analysis and Approximations Using Increments

2.6 Implicit Differentiation and Related Rates

CHAPTER 3 Additional Applications of the Derivative 1.0 week

3.1 Increasing and Decreasing Functions; Relative Extrema  

3.2 Concavity and Points of Inflection  

3.3 Curve Sketching  

CHAPTER 3 Additional Applications of the Derivative (continued) 1.0 week

3.4 Optimization; Elasticity of Demand  

3.5 Additional Applied Optimization

CHAPTER 4 Exponential and Logarithmic Functions 1.0 week

4.1 Exponential Functions; Continuous Compounding  

4.2 Logarithmic Functions  

4.3 Differentiation of Exponential and Logarithmic Functions  

4.4 Applications; Exponential Models

CHAPTER 5 Integration 1.0 week

5.1 Antidifferentiation: The Indefinite Integral  

5.2 Integration by Substitution  

5.3 The Definite Integral and the Fundamental Theorem of Calculus  

CHAPTER 5 Integration  (continued) 1.0 week

5.4 Applying Definite Integration: Area Between Curves and Average Value  

5.5 Additional Applications to Business and Economics  

5.6 Additional Applications to the Life and Social Sciences

CHAPTER 6 Additional Topics in Integration 1.0 week

6.1 Integration by Parts; Integral Tables  

6.2 Introduction to Differential Equations  

CHAPTER 6 Additional Topics in Integration (continued) 1.0 week

6.3 Improper Integrals; Continuous Probability  

6.4 Numerical Integration

CHAPTER 7 Calculus of Several Variables 1.0 week

7.1 Functions of Several Variables  

7.2 Partial Derivatives

7.3 Optimizing Functions of Two Variables  

CHAPTER 7 Calculus of Several Variables (continued) 1.0 week

7.4 The Method of Least-Squares  

7.5 Constrained Optimization: The Method of Lagrange Multipliers  

7.6 Double Integrals  

VI. PREREQUISITE KNOWLEDGE  

Students should have a strong understanding of college algebra.  

VII. TEACHING METHODS AND MATERIALS  

Methods: Taught by lecture with class discussion based on assigned homework.  

Text:    Calculus for Business, Economics, and the Social and Life Sciences (11^th Edition)

ISBN: 978-007-131071-0

VIII. GRADING  

Participation               10%  

Assignments       3        20%  

Midterms          1          20%  

Final             1          50%

COURSE CONTENTS AND REQUIREMENTS

I.Functions,Graphs,andLimits

Analysisofgraphs.

With the aid of technology, graphs of functions are often easy to produce . The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function .

Limits of functions (including one-sided limits)

•  An intuitive understanding of the limiting process .

•  Calculating limits using algebra .

•  Estimating limits from graphs or tables of data .

•  Asymptotic and unbounded behavior

•  Understanding asymptotes in terms of graphical behavior  .

•  Describing asymptotic behavior in terms of limits involving infinity .

•  Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth) .

Continuity as a property of functions

•  An intuitive understanding of continuity . (The function values can be made as close as desired by taking sufficiently close values of the domain .)

•  Understanding continuity in terms of limits .

•  Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem) .

* Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form .

II. Derivatives

Conceptofthederivative

•  Derivative presented graphically, numerically, and analytically .

•  Derivative interpreted as an instantaneous rate of change .

•  Derivative defined as the limit of the difference quotient  .

•  Relationship between differentiability and continuity .

•  Derivative at a point

•  Slope of a curve at a point . Examples are emphasized, including points at which  

there are vertical tangents and points at which there are no tangents .

•  Tangent line to a curve at a point and local linear approximation .

•  Instantaneous rate of change as the limit of average rate of change .

•  Approximate rate of change from graphs and tables of values .

Derivativeasafunction

•  Corresponding characteristics of graphs of ƒ and ƒ∙ .

•  Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ∙ .

•  The Mean Value Theorem and its geometric interpretation  .

•  Equations involving derivatives . Verbal descriptions are translated into equations involving derivatives and vice versa .

Secondderivatives

•  Corresponding characteristics of the graphs of ƒ, ƒ∙, and ƒ  ∙ .

•  Relationship between the concavity of ƒ and the sign of ƒ  ∙ .

•  Points of inflection as places where concavity changes .

Applicationsofderivatives

•  Analysis of curves, including the notions of monotonicity and concavity .

+  Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration .

•  Optimization, both absolute (global) and relative (local) extrema .

•  Modeling rates of change, including related rates problems.

•  Use of implicit differentiation to find the derivative of an inverse function .

•  Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration .

•  Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations .

+  Numerical solution of differential equations using Euler’s method .

+  L’Hospital’s Rule, including its use in determining limits and convergence of  

improper integrals and series  .

Computationofderivatives

•  Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions .

•  Derivative rules for sums, products, and quotients of functions .

•  Chain rule and implicit differentiation .

+  Derivatives of parametric, polar, and vector functions .

III.  Integrals

Interpretationsandpropertiesofdefiniteintegrals

•  Definite integral as a limit of Riemann sums .

•  Definite integral of the rate of change of a quantity over an interval interpreted  

as the change of the quantity over the interval:

•  Basic properties of definite integrals (examples include additivity and linearity) .

* Applicationsofintegrals.

Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations . Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems . Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral . To provide a common foundation, specific applications should include finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, the length of a curve (including a curve given in parametric form), and accumulated change from a rate of change .

Fundamental Theorem ofCalculus

•  Use of the Fundamental Theorem to evaluate definite integrals .

•  Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined .

Techniquesofantidifferentiation

•  Antiderivatives following directly from derivatives of basic functions .

+  Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only) .

+  Improper integrals (as limits of definite integrals) .

Applicationsofantidifferentiation

•  Finding specific antiderivatives using initial conditions, including applications to motion along a line .

•  Solving separable differential equations and using them in modeling (including the study of the equation and exponential growth) .

+  Solving logistic differential equations and using them in modeling .

Numericalapproximationstodefiniteintegrals.

Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values .

*IV.  PolynomialApproximationsandSeries

* Conceptofseries.

A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums .  

Technology can be used to explore convergence and divergence .

* Seriesofconstants

+  Motivating examples, including decimal expansion .

+  Geometric series with applications .

+  The harmonic series .

+  Alternating series with error bound .

+  Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series .

+  The ratio test for convergence and divergence .

+  Comparing series to test for convergence or divergence .

* Taylorseries

+  Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve) .

+  Maclaurin series and the general Taylor series centered at x = a .

+  Maclaurin series for the functions sinx, cosx.

+  Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series .

+  Functions defined by power series .

+  Radius and interval of convergence of power series .

+  Lagrange error bound for Taylor polynomials