Calculus for Business: 3 hrs. Fundamental aspects of calculus are covered with applications in business and economics. The topics covered include limits, differentiation, integration, and some multivariate calculus. (PR: MTH 123 or equivalent, or Mathematics ACT or at least 27)

Course Objectives

1. To provide the student with an understanding of the fundamental concepts in differential and integral calculus, with a particular emphasis on their applications in business and economics.

2. To further develop the student's problem solving skills, so that they may apply the techniques learned in this course in a variety of situations.

Course Contents

All material in this course should be presented with an emphasis of its application, particularly to problems in business and economics.

The Derivative

Rate of change and slope as a motivation for both the concept of a limit and the concept of a derivative. Limits and their evaluation. The derivative as a limit of difference quotients and its interpretation as a slope and weight of change. Techniques of differentiation, including the power rule, product and quotient rules, and the power form of the chain rule. Marginal analysis and the derivative as a representation of marginal quantities.

Graphing and Optimization

Understanding of the relationship between the graph of a function and its first and second derivatives. Relative extrema and concavity. Curve sketching. Maximizing and minimizing functions, with particular attention to optimization problems with applications in business and economics.

Derivatives of Exponential and Logarithmic Functions

The constant e and its realization from considering continuously compounded interest. Differentiation of Logarithmic and Exponential functions, combination of new derivative rules with the product and quotient rules, and the general form of the chain rule.

Integration

Antiderivatives and indefinite integrals as the reverse process of differentiation. Integration by substitution as an inverse to the chain rule. (First order separable) differential equations and the exponential growth law. Area and definite integrals, and the interpretation in terms of marginal analysis and total change of a function. Formalization of these latter ideas and the definite integral as a limit of a sum. Fundamental Theorem of Calculus.

Applications of Integration

Area between curves. Continuous income streams. Consumers and Producers Surplus.

Multivariate Calculus

Functions of two or more variables. First and higher order partial derivatives. Maxima and Minima of functions of two variables using the first and second partial derivatives. Lagrange multipliers and constrained optimization.
 
 

Textbook:    Applied Mathematics for Managerial, Life, and Social Sciences ( 3rd Edition by Tan)
 

Graphing Calculators

Graphing calculator policy is left to individual instructors. Instructors who choose to incorporate graphing calculators in the course are encouraged to ensure that all students complete the course with sufficient "by hand" skills.

Assessments

Students will be evaluated on their fulfillment of the course objectives by (approximately) three in-class exams and one final exam, which is recommended to be comprehensive. Instructors may also include graded homework, quizzes, projects, grades for attendance and class participation, or other means of assessment.
 

Week	Dates Fall 2005	Sections covered and suggested topics
1		9.1  average rate of change and average velocity        instantaneous rate of change and velocity        concept of a limit: numerical and graphic        examples of how a limit can fail to exist        algebraic properties of limits        finding  (limit f(x) as x approaches c) by             -  plugging in x = c             -  techniques to find limits 0/0-type limits in rational expressions             -  when a radical appears, multiplying by a conjugate in numerator and denominator         finding limit of a rational function as  (limit f(x) as x approaches infinity) 9.2   notation for right hand and left hand limits         a function fails to have a limit if right hand and left hand limits do not agree         definition of continuity at a point on an open interval:               analyzing the graph at a discontinuity point to decide which conditions from the definition fail         determining whether or not a piecewise function is continuous                                                                                                            (continued)
2		9.2   algebraic properties of continuous functions         determining where a rational function has discontinuities         Intermediate Value Theorem 9.3  limit of slope of secant line is equal to the slope of the tangent line formal definition of derivative
3		9.3  applications of derivatives              - slope of  tangent line, also finding the equation of the tangent line using slope-point form              - instantaneous rate of change      how a derivative can fail to exist; sharp edges, vertical tangent lines, and discontinuity points  9.4  rules for finding derivatives              - constant funcion rule, power rule, constant times a function rule, sum/difference rule       more applications of derivatives              - velocity, rate of change
Week
4		9.5  product rule and quotient rule       finding higher derivatives and using notation for higher derivatives Exam 1
5		9.6  Chain rule, including the general power rule         problems which combine product, quotient and chain rules 9.7  the constant "e"         derivatives of  e x and ln x        derivatives involving products and quotients of e x and ln x       Chain rule for exponential and logarithmic functions
6		9.8  marginal cost vs. the exact cost of producing one additional item         computing marginal cost, marginal revenue and marginal profit       average cost, average revenue, and average profit 10.1  using  f ' "sign charts" to find intervals where f is increasing or decreasing          finding critical values (i.e. critical points) for a function f                                                                                                  (continued)
7		10.1  local (i.e. relative) extrema vs. absolute extrema          critical values are canidates for where local maxima and minima occur          1st derivative "sign chart" test for local extrema 10.2   how concavity is related to the 2nd derivative          points of inflection          using 2nd derivative sign charts to indentify concavity and points of inflection           the 2nd derivative test for local maxima and local minima          curve sketching using 1st and 2nd derivative sign charts
8		10.3  sketching graphs of polynomials using 1st and 2nd derivative sign charts          finding horizontal and vertical asymptotes for rational functions          sketching graphs of rational functions using 1st and 2nd derivativer sign charts
9		Exam 2 10.4  finding absolute extrema on closed intervals          related max-min word problems
10		10.5  understanding when a local maximum or minimum is absolute          applying the 2nd derivative test for local extrema to find absolute extrema on open intervals          knowing when the 2nd derivative test is inconclusive           related max-min word problems 11.1  antiderivatives          indefinite intergral: using the notation; the arbitrary constant          rules for computing indefinite integrals of powers, a constant times a function, sums and differences           antiderivatives of  e to the power of  x  and  1/x        finding a general solution to a simple 1st order differential equation and finding a particular solution to an initial value problem        business applications
Week		Sections covered and suggested topics
11		11.2 differential of a function           using the method of u-substitution to compute indefinite integrals:                 when and how to do it 11.3    summing up the areas of rectangles to approximate the area under a graph - approximations using 4 rectangles - using right or left hand endpoints to determine the heights of the             rectangles            definition of a definite integral
12		11.4    using the Fundamental Theorem of Calculus to evaluate definite integrals            computing area under a graph by evaluating definite integrals 11.5    special properties of definite integrals            definite integral problems which involve u-substitution            application of definite integrals - finding a total change by integrating the rate of change - computing average value of a function on an interval            business applications
13		Exam 3 11.6   using definite integrals to compute area between a curve and the x-axis           computing the area between two curves
14		11.7  consumers and producers surplus 12.1  evaluating and graphing functions of two ( or more ) independent variables 12.2definition of partial derivatives with respect to x , with respect to y           how to compute partial derivatives           2nd order and mixed partial derivatives
15		12.3 local maximum, local minimum and saddle points in graph of a function of                two variables           finding critical points for functions of two variables           2nd derivative Test for Local Extrema for functions of two variables