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Schink 1/19 Course Outline Title: Calculus for Business and Social Sciences Catalog Number: MA140 Credit Hours: 4 Lab. Hours: 0 Prerequisite: MA139 College Algebra Catalog Description: This course is for those whose programs do not require the Calculus sequence. Topics include an intuitive study of limits, and the analytic geometry, differentiation and integration of polynomial, rational, exponential, logarithmic, and power functions. Applications are selected from business, economics, and the social sciences. COURSE TEACHING GOALS FOR ALL TOPICS: GOAL A: Use mathematical processes to acquire and convey knowledge. GOAL B: Systematically solve problems and interpret data or information. SUNY Learning Outcomes The student will develop well reasoned arguments. The student will identify, analyze, and evaluate arguments as they occur in their own and others work. The student will demonstrate the ability to interpret and draw inferences from mathematical models such as formulas, graphs, tables, and schematics. The student will demonstrate the ability to represent mathematical information symbolically, visually, numerically, and verbally. The student will demonstrate the ability to employ quantitative methods such as arithmetic, algebra, geometry, or statistics to solve problems. The student will demonstrate the ability to estimate and check mathematical results for reasonableness. TOPIC 1. FUNCTIONS, LIMITS, AND THE DERIVATIVE Functions and their graphs, limits, continuity, tangent line, and the derivative are discussed. Emphasis is placed on an intuitive understanding of concepts. Topic Goal: To help students acquire an intuitive understanding of the definition of function, the graph , limit and continuity of a function, tangent line to the graph of a function, and the derivative of a function. Student Outcomes: The student will: Identify functions using the definition and the vertical line test Identify the independent and dependent variables, domain, and range for functions Sketch the graphs of functions including piecewise defined functions Demonstrate an understanding of the algebra of functions, including the composition of functions Demonstrate an understanding of the use of functions as mathematical models Determine whether functions have limits including one-sided limits Find limits of functions Determine whether functions are continuous Demonstrate an understanding of the properties of limits, limits at infinity, and indeterminate forms 1.10 Use the limit definition to find the derivative of a function 1.11 Interpret the derivative in terms of tangent lines and instantaneous rates of change TOPIC 2. DIFFERENTIATION The basic rules for differentiation such as derivatives of sums, products, quotients, powers, and the chain rule are discussed. Applications of derivatives to marginal functions in economics and as rate of change are presented. Topic Goal A: To use rules to find the derivative for sums, powers, products, and quotients of function and to use the chain rule to find the derivative. Topic Goal B: To apply the concept of derivative to marginal functions and rate of change. Student Outcomes: The student will: Apply rules of differentiation to find the derivatives of sums, powers, products and quotients Use the chain rule to find derivatives of the composition of functions, especially powers of functions Calculate and interpret the derivative as marginal functions and rate of change Calculate and interpret higher order derivatives in terms of acceleration Demonstrate an understanding of implicit differentiation Find the differential of a function and use it to approximate the change in a function TOPIC 3 APPLICATIONS OF THE DERIVATIVE The first and second derivatives are used to sketch the graphs of functions. Also, the derivative is used to solve problems that require maximizing profit and minimizing costs. Topic Goal A: To use the first and second derivatives, in sketching the graph of a function, to determine when a function is increasing or decreasing, and concave upward or downward. Topic Goal B: To apply derivatives to find maximum and minimum values of real-life functions such as profit, cost, and height and velocity. Student Outcomes: The student will: Use the derivative to determine the intervals where functions are increasing or decreasing Use the second derivative to determine the intervals where the graphs of functions are concave upward or downward Determine the maximum and minimum values of functions and sketch the graphs of functions Find extreme values of functions to maximize or minimize costs, revenues, profit, areas, volumes, heights, velocities, etc. TOPIC 4. EXPONENTIAL AND LOGARITHMIC FUNCTIONS The exponential and logarithmic functions are introduced and their importance in applications is discussed. Formulas for obtaining derivatives are obtained. Applications to compound interest, growth and other real-life problems are presented. Topic Goal A: To develop the students understanding of the exponential and logarithmic function and their uses. Topic Goal B: To develop formulas to obtain the derivative for functions involving the exponential and logarithmic functions. Topic Goal C: To use exponential and logarithmic functions as mathematical models for real-life applications. Student Outcomes: The student will: Demonstrate an understanding of the definitions of the exponential and logarithmic functions Sketch the graphs of exponential and logarithmic functions Use exponential functions to solve problems involving compound interest Use rules to obtain derivatives involving exponential and logarithmic functions Solve problems with exponential functions as mathematical models for growth, radioactive decay, population growth, etc. TOPIC 5. INTEGRATION The concept of antiderivative and rules for integration are presented. The definite integral and the application to area are introduced. The Fundamental Theorem of Calculus for evaluating definite integrals is developed. Applications to area and in business and economics are considered. Topic Goal A: To develop the students understanding of antiderivatives and obtain rules to find indefinite integrals. Topic Goal B: To develop the students understanding of the definite integral and its role in finding the area under a curve. Topic Goal C: To present the fundamental theorem and help students in its use to evaluate definite integrals Topic Goal D: To apply the definite integral to find areas, and values for an income stream or annuity Student Outcomes: The student will: Demonstrate an understanding of the antiderivative Use rules for integration to find indefinite integrals of sums, powers, exponential functions Demonstrate an understanding of the technique of integration by substitution to evaluate indefinite integrals Find approximations to areas under curves using Riemann sums Demonstrate an understanding of the definite integral and its relation to area under a curve Use the Fundamental Theorem of Calculus to evaluate definite integrals and find areas Apply the definite integral to various applications from the fields of Business and Economics Teaching Guide Title: Calculus for Business and Social Sciences Catalog Number: MA140 Credit Hours: 4 Lab. Hours: 0 Prerequisite: MA139 College Algebra Catalog Description: This course is for those whose programs do not require the Calculus sequence. Topics include an intuitive study of limits, and the analytic geometry, differentiation and integration of polynomial, rational, exponential, logarithmic, and power functions. Applications are selected from business, economics, and the social sciences. Text: Calculus For Business, Economics, and the Social and Life Sciences, Brief Ninth Edition, L. Hoffmann and G. Bradley, McGraw Hill, New York, NY, 2007 Note: A TI-83, TI-83+, TI-84, TI-84+ or equivalent graphing calculator is recommended. Calculators with Computer Algebra Systems are not allowed. Some Preliminary Comments: Calculus for Business and Social Sciences is intended to give the student a strictly intuitive, non-rigorous introduction to the basic concepts and operations of the calculus of functions of a single variable. No attempt should be made to put the material on a sound theoretical basis. Instead, the emphasis should be placed on first making the operations intuitively plausible and then applying the material to situations in the managerial, life and social sciences. (Needless to say, "traditional" applications such as curve sketching, area under a curve, etc., are also considered.) Topics Chapter 1 Functions, Graphs, and Limits 12 Hours Functions The Graph of a Function Linear Functions Functional Models Limits One-Sided Limits and Continuity Chapter 2 Differentiation: Basic Concepts 13 Hours 2.1 The Derivative 2.2 Techniques of Differentiation 2.3 Product and Quotient Rule; Higher-Order Derivatives 2.4 The Chain Rule 2.5 Marginal Analysis and Approximations Using Increments 2.6 Implicit Differentiation and Related Rates Chapter 3 Additional Applications of the Derivative 10 Hours 3.1 Increasing and Decreasing Functions; Relative Extrema 3.2 Concavity and Points of Inflection 3.3 Curve Sketching 3.4 Optimization 3.5 Additional Applied Optimization Chapter 4 Exponential and Logarithmic Functions 8 Hours 4.1 Exponential Functions 4.2 Logarithmic Functions 4.3 Differentiation of Logarithmic and Exponential Functions 4.4 Additional Exponential Models Chapter 5 Integration 13 Hours 5.1 Antidifferentiation: The Indefinite Integral 5.2 Integration by Substitution 5.3 The Definite Integral and the Fundamental Theorem of Calculus 5.4 Applying the Definite Integral: Area Between Curves and Average Value 5.5 Additional Applications to Business and Economics 5.6 Additional Applications to Life and Social Sciences NOTE: The teaching guide allows 4 hours for the in-class assessment of student learning. 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