Business Calculus: Course Overview, Competencies, and Key Topics

Course Overview

Introduction to Business Calculus

This course provides an introduction to the fundamental concepts of differential and integral calculus, with a focus on applications relevant to business, economics, and the life sciences. Students will learn to analyze and solve problems involving limits, continuity, differentiation, and integration of various types of functions, including polynomials, exponential, and logarithmic functions.

• Prerequisite: Successful completion of MAC 1105 or MAC 1106 with a grade of C or higher, or departmental permission.

• Textbook: Calculus And Its Applications, 12th Edition by Bittinger, Ellenbogen, and Surgent.

• Technology: Scientific or graphing calculator (TI-84 recommended); access to MyMathLab for online assignments.

Course Schedule and Main Topics

Outline of Topics by Section

The following is a tentative schedule of topics covered in the course, organized by textbook sections:

• Limits and Continuity

  • Limits (1.1)

  • Algebraic Limits and Continuity (1.2)

  • Average Rates of Change (1.3)

• Differentiation

  • Differentiation Using Limits of Difference Quotients (1.4)

  • Differentiation Techniques: Power & Sum-Difference Rules (1.5)

  • Differentiation Techniques: Product & Quotient Rules (1.6)

  • The Chain Rule (1.7)

  • Higher-Order Derivatives (1.8)

• Exponential and Logarithmic Functions

  • Exponential Functions (2.1)

  • Logarithmic Functions (2.2)

  • Applications: Growth and Decay Models (2.3, 2.4)

  • Derivatives of Exponential and Logarithmic Functions (2.5)

  • Business Application: Amortization (2.6)

• Applications of Differentiation

  • Finding Maximum and Minimum Values (3.1, 3.2, 3.4)

  • Graph Sketching: Asymptotes and Rational Functions (3.3)

  • Maximum–Minimum Problems in Business and Economics (3.5)

  • Marginal Analysis and Differentials (3.6)

  • Implicit Differentiation and Related Rates (3.8, 3.9)

• Integration

  • Antidifferentiation (4.1)

  • Antiderivatives as Areas (4.2)

  • Area and Definite Integrals (4.3)

  • Properties of Definite Integrals (4.4)

  • Integration Techniques: Substitution (4.5)

• Functions of Several Variables

  • Functions of Several Variables (6.1)

  • Partial Derivatives (6.2)

Course Competencies

Key Learning Objectives

• Limits: Evaluate limits of algebraic, logarithmic, and exponential functions; determine continuity and discontinuity.

• Differentiation: Apply differentiation rules, including the chain rule and implicit differentiation; use derivatives to find slopes of tangent lines.

• Curve Sketching: Use first and second derivatives to analyze function behavior, including intervals of increase/decrease, concavity, and extrema.

• Applications of Derivatives: Solve rate of change, optimization, marginal analysis, and related rates problems; use differentials for approximation.

• Integration: Apply integration rules, use substitution, evaluate definite integrals, and find areas between curves.

Key Concepts and Definitions

Limits and Continuity

• Limit: The value that a function approaches as the input approaches a certain value.

• Continuity: A function is continuous at a point if the limit exists and equals the function's value at that point.

• Example: $\lim_{x \to a} f(x) = L$ means as $x$ approaches $a$, $f(x)$ approaches $L$.

Differentiation

• Derivative: Measures the rate at which a function changes; the slope of the tangent line at a point.

• Basic Rules: Power rule, sum/difference rule, product rule, quotient rule, chain rule.

• Example: $\frac{d}{dx} x^n = n x^{n-1}$ (Power Rule)

Exponential and Logarithmic Functions

• Exponential Function: $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Logarithmic Function: $f(x) = \log_a x$, the inverse of the exponential function.

• Derivative of Exponential: $\frac{d}{dx} e^x = e^x$

• Derivative of Logarithm: $\frac{d}{dx} \ln x = \frac{1}{x}$

Applications of Differentiation

• Optimization: Finding maximum or minimum values of functions, often in business contexts (e.g., maximizing profit).

• Marginal Analysis: Using derivatives to approximate changes in cost, revenue, or profit.

• Related Rates: Solving problems where two or more variables change with respect to time.

• Example: If $C(x)$ is the cost function, the marginal cost is $C'(x)$.

Integration

• Antiderivative: A function whose derivative is the given function.

• Definite Integral: Represents the area under a curve between two points.

• Fundamental Theorem of Calculus: Connects differentiation and integration.

• Example: $\int_a^b f(x) dx$ gives the net area under $f(x)$ from $x=a$ to $x=b$.

Functions of Several Variables and Partial Derivatives

• Function of Several Variables: A function with more than one input variable, e.g., $f(x, y)$.

• Partial Derivative: The derivative of a multivariable function with respect to one variable, holding others constant.

• Example: $\frac{\partial}{\partial x} f(x, y)$

Grading Policy

Assessment Components

Component	Weight
Online Homework	10%
Quizzes	10%
4 Exams (15% each)	60%
Final Exam (Cumulative)	20%

Grade Scale

Letter Grade	Percentage
A	90 - 100%
B	80 - 89%
C	70 - 79%
D	60 - 69%
F	0 - 59%

Additional Information

• Attendance: Regular attendance is required; missing more than 50% of a class may be considered an absence.

• Make-up Policy: No make-up tests; documented emergencies may allow the final exam to replace a missed test.

• Support Services: Academic support, disability services, and technical support are available through the college.

• AI Tools: Use of AI-powered instructional tools is permitted for homework, following ethical guidelines.