Summary of Business Terms and Formulas

Revenue, Cost, and Profit Functions

Business calculus often uses functions to model revenue, cost, and profit. Understanding these functions is essential for analyzing business performance and making informed decisions.

• Total Revenue (R): , where x is the number of units sold and p is the price per unit (demand function).

• Average Cost: , where is the total cost for producing x units.

• Average Revenue:

• Average Profit: , where is the total profit.

Example:

• If , then average cost for 10 units is .

Compound Interest Formulas (Section 1.5)

Interest Compounded m-times per Year

Compound interest is a fundamental concept in finance, describing how money grows over time when interest is added periodically.

• Formula:

• Where:

  • A = final amount

  • P = principal (initial amount)

  • r = annual interest rate (decimal)

  • m = number of compounding periods per year

  • t = number of years

Interest Compounded Continuously

• Formula:

Example:

• If , , , , then .

Properties of Exponentials & Logarithms (Sections 1.5 & 1.6)

Exponentials

Exponential functions and logarithms are widely used in business calculus for modeling growth, decay, and solving equations involving rates.

• Basic Properties:

Logarithms

• Basic Properties:

Example:

• because .

Differentiation Rules (Sections 2.4, 2.5, 3.2, 3.3, 3.4)

Definition of the Derivative

The derivative measures the rate of change of a function. It is foundational for analyzing business models and optimization.

• Limit Definition:

Basic Differentiation Rules

• Constant Function Rule:

• Power Rule:

• Special Case:

• Constant Multiple Rule:

• Sum and Difference Rule:

Exponential and Logarithmic Derivatives

Product and Quotient Rules

• Product Rule:

• Quotient Rule:

General Power Rule

• If , then

Example:

Rate of Change and Marginal Analysis (Section 2.4)

Slope and Rate of Change

The derivative at a point gives the slope of the tangent line to the function at that point, representing the instantaneous rate of change.

• Slope at :

• Equation of Tangent Line: , where and

• Horizontal Tangent: Occurs when

Marginal Functions

• Marginal Profit:

• Marginal Revenue:

• Marginal Cost:

Example:

• If , then ; the marginal cost is $5$ per unit.

Study Guide Structure and Practice Recommendations

Sections and Exercises

The study guide references specific textbook sections, examples, matched problems, and exercises for practice. These cover:

• Compound Interest (Sections 1.5, 1.6)

• Finding Limits (Section 2.1)

• Derivatives via Limit Definition (Section 2.4)

• Finding Derivatives (Sections 2.5, 3.2, 3.3, 3.4)

• Equations of Tangent Lines (Sections 2.4, 2.5, 3.2, 3.3, 3.4)

• Horizontal Tangents (Sections 2.5, 3.3, 3.4)

• Rate of Change / Marginals (Sections 2.4, 2.5, 2.6, 2.7, 3.2, 3.3, 3.4)

Recommended Practice:

• Review examples and matched problems in each section.

• Complete exercises listed for each topic to reinforce understanding.

Table: Differentiation Rules Summary

Additional info:

• Sections referenced (1.5, 1.6, 2.1, 2.4, 2.5, 2.7, 3.2, 3.3, 3.4) correspond to standard business calculus topics: functions, limits, derivatives, and their applications.

• Practice exercises are essential for mastering calculation and application skills.