MAT 1400 Business Calculus
Course Objectives: MAT 1400 is a one-semester, 4-credit Calculus course that incorporates both differential and integral Calculus. The course is intended to be presented from a “mainstream” point of view and will de-emphasize certain theoretical aspects of Calculus, focusing instead on the understanding and interpretation of the primary concepts. To the extent possible, applications will be from business, economics, and finance.
- Some review of functions and pre-calculus material, including the exponential and natural logarithm functions;
- The concept of derivative and its meaning both as slope and as rate of change; marginal cost and revenue;
- Basic computational rules for derivatives; Power, Product, Quotient and Chain Rules; derivatives of exponential (and the relation to constant relative growth rate/APR/nominal interest rate) and logarithmic functions;
- Implicit differentiation and partial derivatives [optional]
- Higher order derivatives; the Second Derivative Test; Concavity/Convexity
- Applications of the derivative: Graphs, Optimization (including max/min problems, finding critical points, checking endpoints), Elasticity of Demand;
- The concept of the definite integral and its meaning both as area and as accumulated change;
- Basic techniques for evaluating an integral; substitution; integration by parts [optional]
- Applications of the integral: Probability, average values, Consumer and Producer Surplus, the Gini Index.
Text: Applied Calculus, 4th edition; Hughes-Hallett, Gleason, et al; 2010; Wiley.
- The Villanova University bookshop will carry new, paperbound copies of the text, bundled with a one-year subscription to the WileyPlus online homework software and the electronic version of the book, at a cost of $160.95. (ISBN#: 978-047-0578773)
- The electronic version of the text (no hard copy), including a one-year subscription to the WileyPlus online homework software, is available directly from the publisher. In 2011, the student price for this is $61.95. (ISBN#: 978-047-0446447) (publisher's website: https://www.wileyplus.com/WileyCDA/)
- Used copies of the text may also be available at various prices at the Villanova University bookshop or through online vendors.
Technology. Use of Excel is required. (Templates will be available for numerical evaluation of limits, derivatives, definite integrals, and other operations relevant to the course.) At their discretion, instructors may incorporate use of a graphing calculator, such as a TI-83, or of Maple as well.
Online homework. At their discretion, instructors may use the WileyPlus online homework package for this textbook. (Students may find this package beneficial even if their instructor does not require it.) Each section of the course will be registered with WileyPlus and each instructor will have access to the online homework and assessment tools that WileyPlus offers.
Textbook coverage: Chapters 1-9.
The following core sections are recommended; optional sections are indicated with square brackets [...]; omitted sections are indicated with red double square brackets [[...]]; The Focus on Theory section of Chapter 2 must be included.
1. Functions and change.
1.1: What is a function?
1.2: Linear functions.
1.3: Average rate of change and relative change.
1.4: Applications of functions to economics.
1.5: Exponential functions.
1.6: The natural logarithm.
1.7: Exponential growth and decay.
1.8: New functions from old.
1.9: [Proportionality and power functions.]
1.10: [[Periodic functions.]]
2. Rate of change: The derivative.
2.1: Instantaneous rate of change.
Focus on Theory: Limits, continuity, and the definition of the derivative.
2.2: The derivative function
2.3: Interpretations of the derivative.
2.4: The second derivative.
2.5: Marginal cost and revenue.
3. Shortcuts to differentiation.
3.1: Derivative formulas for powers and polynomials.
3.2: Exponential and logarithmic functions.
3.3: The chain rule.
3.4: The product and quotient rules.
3.5: [[Derivatives of periodic functions.]]
4. Using the derivative.
4.1: Local maxima and minima.
4.2: Inflection points.
4.3: Global maxima and minima.
4.4: Profit, cost, and revenue.
4.5: Average cost
4.6: Elasticity of demand.
4.7: [Logistic growth.]
4.8: [[The surge function and drug concentration.]]
5. Accumulated change: The definite integral.
5.1: Distance and accumulated change.
5.2: The definite integral.
5.3: The definite integral as area.
5.4: Interpretations of the definite integral.
5.5: The Fundamental Theorem of Calculus.
6. Using the definite integral.
6.1: Average value.
6.2: Consumer and producer surplus.
6.3: Present and future value.
6.4: [Integrating relative growth rates.]
7.1: Constructing antiderivatives analytically.
7.2: Integration by substitution.
7.3: Using the Fundamental Theorem to find definite integrals.
7.4: [Integration by parts.]
7.5: [Analyzing antiderivatives graphically and numerically.]
8.1: Density functions.
8.2: Cumulative distribution functions and probability.
8.3: [The median and the mean.]
9. Functions of several variables.
9.1: [[Understanding functions of two variables.]]
9.2: [[Contour diagrams.]]
9.3: [Partial derivatives.]
- The course syllabus includes 37 required sections of the text, including the Focus on Theory section from Chapter 2. Thus, it will be necessary to discuss about three sections per week over a fourteen-week semester. Some sections can be combined to create more time for discussion of optional topics and/or more class sessions for review and tests. For example, sections 4.2/4.3 can be combined, as can sections 6.2/6.3, or sections 8.1/8.2.
- For classes offered in the format of two 75-minute periods and one 50-minute period per week, there are about 42 class sessions. Thus if one section per session, on average, is discussed, there will remain about five sessions for tests and review. If more test and review sessions or more sessions for optional sections are desired, some sections of the text can be combined, as suggested in the previous paragraph.
- For classes offered in the format of four 50-minute periods per week, there are about 56 class sessions. Discussing about three sections of the text per week, on average, would require about 49 or 50 sessions, leaving five or six sessions for tests and review. Combining some sections of the text, as suggested above, would create time for more discussion of optional topics or allow for more test and review sessions.