Linear Functions

Definition and Properties

Linear functions are foundational in business calculus, modeling relationships with constant rates of change. They are typically written as , where is the slope and is the y-intercept.

• Slope (): Measures the rate of change; calculated as .

• Point-Slope Form: , useful for finding the equation of a line given a point and slope.

• Applications: Used to model cost, revenue, and profit functions in business.

Cost, Revenue, and Profit Functions

• Cost Function (): Represents the total cost to produce units. Often linear: .

• Revenue Function (): Represents total income from selling units: , where is price per unit.

• Profit Function (): .

• Marginal Cost/Revenue/Profit: The rate of change of cost, revenue, or profit with respect to units produced or sold.

Nonlinear Functions

Exponential and Logarithmic Functions

Exponential and logarithmic functions model growth and decay, such as population growth or depreciation.

• Exponential Function:

• Logarithmic Function:

• Properties: Exponential growth/decay, logarithms as inverses of exponentials.

• Negative Exponents: reflects the graph across the y-axis.

Applications

• Compound interest, population models, and depreciation calculations.

Matrices and Systems of Equations

Matrix Properties

A matrix is a rectangular array of numbers. Used to represent systems of equations and organize data.

• Matrix Addition/Subtraction: Only possible for matrices of the same size.

• Scalar Multiplication: Multiply each entry by a constant.

• Matrix Multiplication: If is and is , their product is .

Solving Systems with Matrices

• Systems of equations can be written as .

• Inverse Matrix Method: if is invertible.

• Determinants: Used to determine if a matrix is invertible. For matrix , .

Market Equilibrium Example

• Supply and demand equations can be solved using matrices to find equilibrium price and quantity.

The Derivative

Definition and Interpretation

The derivative measures the instantaneous rate of change of a function. It is foundational for analyzing marginal cost, revenue, and profit in business.

• Limit Definition:

• Notation: ,

• Interpretation: Slope of the tangent line at a point on the function.

Rules for Finding Derivatives

• Power Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Examples

• Find for :

• Find marginal cost for :

Applications of the Derivative

Marginal Analysis

Marginal cost, revenue, and profit are derivatives of their respective functions and represent the rate of change with respect to quantity.

• Marginal Cost:

• Marginal Revenue:

• Marginal Profit:

Optimization

• Critical points occur where or is undefined.

• First Derivative Test: Determines intervals where a function is increasing or decreasing.

• Second Derivative Test: Determines concavity and identifies maxima/minima.

Integration

Basic Concepts

Integration is the reverse process of differentiation and is used to find areas under curves, total accumulated quantities, and more.

• Indefinite Integral:

• Definite Integral:

• Applications: Total cost, revenue, or profit over an interval.

Financial Mathematics

Simple and Compound Interest

• Simple Interest:

• Compound Interest:

• Effective Rate:

Present and Future Value

• Present Value:

• Future Value of Annuity:

• Present Value of Annuity:

Geometric Series

• Sum of First Terms:

Loan Amortization

• Monthly Payment:

Summary Table: Key Financial Formulas

Conclusion

These notes cover the essential topics in Business Calculus, including linear and nonlinear functions, derivatives and their applications, integration, matrices, and financial mathematics. Mastery of these concepts is crucial for analyzing and solving real-world business problems involving rates of change, optimization, and financial decision-making.