Linear and Exponential Models

Introduction

This study guide explores the foundational concepts of linear and exponential models, which are essential for understanding patterns of change in calculus and applied mathematics. We will examine how to recognize, construct, and interpret these models using analytical, tabular, verbal, and graphical representations. Applications span finance, biology, demography, and technology.

Linear Models

Definition and General Form

• Linear Model: A function of the form or , where m is the slope (rate of change) and b is the y-intercept (initial value).

• Key Property: The output increases or decreases by a fixed amount for each unit increase in the input.

Example 1: Uber Fare Calculation

• Scenario: Flat rate of $5 plus $2 per mile.

• Table of Values:

• Equation:

• Interpretation: For each mile, the cost increases by $2 (constant rate).

• Example Calculation: For 20 miles:

• Graph: A straight line; slope = 2; independent variable = miles, dependent variable = cost.

• Function Definition: Each input (miles) yields exactly one output (cost).

Example 2: Business Revenue Prediction

• Scenario: Revenue as a function of employees (in tens).

• Table of Values:

• Equation:

• Projected Revenue (2024, x=70):

• Model Type: Linear (constant increase per 10 employees).

Example 3: Seeds per Row

• Scenario: Seeds as a function of row length.

• Table of Values:

• Equation:

• Interpretation: For each additional foot, 0.25 seeds can be planted (linear relationship).

• Example Calculation: For 200 feet: seeds.

Exponential Models

Definition and General Form

• Exponential Model: , where is the initial value and is the percent change (as a decimal).

• Key Property: The output increases or decreases by a fixed percentage, not a fixed amount.

• Growth vs. Decay: If , the function models growth; if , it models decay.

Linear vs. Exponential Growth

• Linear: Add or subtract a fixed amount each period.

• Exponential: Multiply by a fixed factor each period (e.g., doubling, tripling).

Example 1: Paper Tearing (Doubling)

• Table of Values:

• Equation:

• Model Type: Exponential (output doubles each time; 100% increase per tear).

• Example Calculation: For 50 tears: (a very large number, illustrating rapid exponential growth).

• Growth Description: Output increases slowly at first, then extremely rapidly.

Example 2: COVID Case Growth

• Table of Values:

• Equation: , where is the number of 10-day intervals since 11/20/2021.

• Model Type: Exponential (each value increases by 60%).

• Example Calculation: For (60 days): cases.

• Growth Description: Rapid, compounding increase characteristic of epidemics.

Example 3: U.S. Population Growth

• Initial Value: (2010)

• Growth Rate: (0.9% per year)

• Equation:

• Example Calculation: For (2060):

Example 4: Population Decay (China)

• Initial Value: (2009)

• Decay Rate: (0.5% decrease per year)

• Equation:

• Application: Predicting if the population will reach 700 million by 2050.

Identifying Model Types in Context

• Linear: Constant addition/subtraction (e.g., school enrollment increases by 50 students per year: ).

• Exponential: Constant percentage change (e.g., computer storage doubles every 2 years: ).

• Exponential Decay: Constant percentage decrease (e.g., TV price drops by 25% per year: ).

• Quadratic: Data forms a parabolic pattern (e.g., death rate vs. hours of sleep).

Common Applications of Exponential Models

• Population growth and decline

• Bacterial growth

• Inflation and compound interest

• Natural resource consumption

• Loudness of sound, radioactive decay

• Spread of information (social media, rumors)

Other Key Takeaways

• Exponential functions can model both rapid growth and rapid decay.

• Function definition: For each input, there is exactly one output.

• Fastest-growing functions: Exponential and factorial functions.

• Regression and probability: Linear regression and bell curves are important in statistics.