# Business Calculus

## Contents

## General Description

- Freshman level Business Calculus course
- Pre-requisite: College Algebra
- Satisfies math/calculus requirement for (most) business majors
- This model assumes that exponential functions and logarithmic functions are covered after differentiation and derivative applications. These sets can be rearranged to meet individual needs.

Possible textbooks include, but are not limited to:

- Marvin L. Bittinger, David J. Ellenbogen, Scott Surgent. Calculus and Its Applications, Tenth Edition. Pearson, 2011.
- Laurence D. Hoffmann, Gerald L. Bradley. Calculus for Business, Economics, and the Social and Life Sciences, Tenth Edition. McGraw Hill, 2010.

Developed at the 2013 PREP Model Course Workshop by:

- Joe Fields (Southern Connecticut State University)
- Tim Flowers (Indiana University of Pennsylvania)

## Problem Sets

### Use of Problem Sets

The problem sets were assembled to allow for personalization by individual faculty. The topics covered are fairly standard in a business calculus course and treated less rigorously than they would be in a traditional calculus course. Faculty can rearrange the topics and delete any sections they do not wish to cover, or wish to assess by other means. The names of the problem sets are meant to be descriptive and the learning objectives will help you evaluate if the set should be included or not.

### Download the problem sets

A copy of the course can be found at modelCourse Business Calculus at the MAA website.

To install a local copy of the course, download the tar file for the model course from GitHub.

The set definition files for the individual assignments are also on GitHub.

### Description of Problem Sets

**Set 01 Review of Function Concepts**

Students will be able to:- Distinguish between the graphs of functions and non-functional relations.
- Demonstrate understanding of the terminology: domain, range, independent variable, dependent variable, inverse function.
- Determine the domain and range of functions given algebraically and graphically. Use interval notation and inequalities to express these sets.
- Be able to compose two functions and decompose a function into simpler components
- Identify vertical and horizontal shifts and reflections of a given function.
- Work with functions that are defined piecewise.
- Translate between graphical and tabular representations of the same function.

**Set 02 Review of Linear, Polynomial and Power Functions**

Students will be able to:- Translate between general form, slope-intercept form and point-slope form for the equations of lines
- Translate between algebraic, graphical and tabular representations of the same linear function.
- Decide whether a pair of lines have 0,1 or infinitely many intersection points.
- Find the intersection point (when it is unique) of two lines and solve business applications problems using this skill.
- Develop a linear model from a verbal description of a scenario and use it in making predictions.
- Demonstrate understanding of the terminology: slope, intercept, polynomial, binomial, monomial, term, coefficient, leading coefficient, degree, (relative) minima, maxima, and extrema, even and odd functions.
- Demonstrate familiarity with the laws of exponents and their applications.
- Interpret the meaning of negative and fractional exponents
- Identify polynomial expressions and distinguish between polynomial and non-polynomial functions from their algebraic and graphical representations.
- Identify from graph the intervals in which a function is increasing or decreasing.
- Determine the qualitative behavior of the graph of a power function based on its coefficient and exponent.

**Set 03 Limits**

Students will be able to:- Evaluate limits from a graphical or algebraic representation of a function or distinguish those cases in which the limit does not exist.
- Use factoring to compute limits of rational functions that initially give indeterminate forms.
- Use properties of limits (linearity, products, quotients and compositions) to calculate limits or determine that the limit does not exist.
- Use a graph or a tabular representation to estimate limits at infinity.
- Evaluate limits at h=0 of the difference quotients of a variety of functions.

**Set 04 The Derivative**

Students will be able to:- compute the average rate of change of a function over an interval.
- Estimate the instantaneous rate of change of a function from its graph.
- Use the definition of the derivative to compute the derivative of simple power functions and polynomials.
- Solve business application problems involving finding the marginal of some function using differentiation.
- Compute the tangent line (a.k.a. best linear approximation) to the graph of a function at a point.

**Set 05 Power Rule and Differentiation**

Students will be able to:- Calculate the derivative of linear and polynomials functions.
- Recognize power functions written in rational or radical form and then implement the power rule to find the derivative of the function.
- Evaluate derivatives at a point.
- Find the slope and equation of a tangent line of given function
- Apply above to cost/revenue/profit application

**Set 06 Product and Quotient Rules**

Students will be able to:- Find the derivative of a function using the product rule.
- Find the derivative of a function using the quotient rule.
- Use the product and quotient rules with unspecified functions.
- Apply the product rule in business contexts (e.g. revenue).

**Set 07 Chain Rule**

Students will be able to:- Find the derivative of functions using the chain rule.
- Find the derivative of functions which require using chain rule plus the product or quotient rule.

**Set 08 Higher Order Derivatives**

Students will be able to:- Use prior derivative rules (power, chain, quotient, etc.) to find higher order derivatives.
- Understand that the first and second derivatives of position (along a straight path) give velocity and acceleration, resp.
- Interpret changes in speed in terms of positive/negative first and second derivatives.
- Identify where object (along a straight path) is speeding up/slowing down.

**Set 09 Marginals Differentials and Estimation**

Students will be able to:- Understand that marginal cost/revenue/profit can be found using a derivative.
- Find marginal cost/revenue/profit function and evaluate function at a point
- Use derivative to estimate cost/revenue/profit of (n+1)st item; compute actual cost/revenue/profit of (n+1)st item
- Use a differential to estimate change in cost/revenue/profit

**Set 10 Implicit Differentiation and Related Rates**

Students will be able to:- Use implicit differentiation to solve for dy/dx.
- Execute implicit differentiation to find value of dy/dx at a point and the slope of a tangent line at a point.
- Apply implicit differentiation to related rates problem in a business context (profit, demand, etc.)

**Set 11 Differentiation and Properties of Curves**

Students will be able to:- Determine intervals of increasing/decreasing by using the derivative
- Identify locations and values of relative maximums and minimums
- Determine intervals of concavity by using the second derivative
- Identify locations of inflection points
- Draw conclusion about signs of derivatives by looking at graph of function
- Apply above to a mathematical model involving cost/revenue/profit

**Set 12 Optimization of Business Models**

Students will be able to:- Use derivatives to find minimum cost and maximum revenue/profit
- Use pricing information to find cost/revenue/profit functions and then optimize (including cost of materials and inventory costs)
- Find extreme values of a cost model on a closed interval

**Set 13 Elasticity of Demand**

Students will be able to:- Compute the elasticity of demand at a given price
- Classify the elasticity of demand as elastic, inelastic, or unitary
- Decide if price should be raised or lowered by using elasticity of demand
- Use elasticity of demand to determine maximum revenue

**Set 14 Review of Exponentials and Logarithms**

Students will be able to:- Identify graphs of exponential and logarithmic functions
- Evaluate and simplify log expressions using log properties
- Use the “change of base” formula to evaluate logs
- Apply exponential models to solve application problems

**Set 15 Derivatives of Natural Exponential and Natural Log**

Students will be able to:- Apply rules to find derivatives of functions involving e^x and ln(x)
- Combine above with prior derivative rules (product, chain, etc.)
- Use log properties to rewrite function before derivative step
- Solve optimization problems which involve log function

**Set 16 Antiderivatives and Indefinite Integrals**

Students will be able to:- Evaluate antiderivatives and indefinite integrals of functions using power rule, e^x rule, and ln(x) rule.
- Solve basic initial value problems, by using antidifferentiation and solving for the missing constants.

**Set 17 Definite Integrals**

Students will be able to:- Evaluate definite integrals of functions where the antiderivatives can be found by the power rule, e^x rule, and ln(x) rule.
- Recognize properties of integrals
- Decide when/how to simplify functions so that the integral may be computed.

**Set 18 Applications of Integration**

Students will be able to:- Understand the relationship between some definite integrals and area on the plane.
- Evaluate a definite integral by sketching the function and using geometry area formulas to calculate area.
- Find the area under a given curve.
- Understand that the definite integral of a rate of change will give the total accumulated change over the interval.
- Compute total change in business applications.

**Set 19 Integration by Substitution**

Students will be able to:- Evaluate indefinite integrals using substitution method
- Evaluate definite integrals using substitution method

**Set 20 Consumer and Producer Surplus**

Students will be able to:- Compute consumer surplus and producer surplus given supply and demand functions
- Recall methods to find equilibrium points and demand functions as needed for above
- Interpret the connection between the integral formulas for consumer/producer surplus and areas between/under supply and demand curves