2.1 Rate of Change

Car Rental Company Example

• Charges Overview

  • Flat Fee: $50

  • Daily Rate: $20 per day

Cost Calculation Example

• Cost Formula: Total Cost = Flat Fee + (Daily Rate × Number of Days)

• Example Calculation: For 3 days

  • Total Cost = $50 + (20 × 3)

  • Total Cost = $50 + $60 = $110

Rate of Change Definition

• What does the $20 represent?:

  • It is the cost per day for renting the car.

  • Definition: Cost per day (CPD) is the amount charged each day for the rental.

• Rate of Change Interpretation:

  • Rate of change is the cost incurred based on the number of days rented.

  • Expressed as:

    • For every 1 day rented, cost increases by $20.

• General Definition of Rate of Change:

  • Compares how one quantity changes in relation to another quantity.

Mathematical Concepts

• Slope and Rate of Change:

  • Rate of change is equivalent to the slope of a line in math.

• Formula for Slope (Rate of Change):

  • m = \frac{y2 - y1}{x2 - x1}

  • Another representation: Change in $y$ over Change in $x$ ($\Delta$y/$\Delta$x)

Slope Interpretation in Graphs

• Graphical Representation:

  • On a graph, the slope is referred to as rise over run.

• Equation of a Line:

  • Common forms:

    • y = mx + b

    • y = b + mx

  • Where $m$ represents the slope (rate of change) and $b$ is the y-intercept.

Finding Slope Through Examples

• Example 1:

  • Find slope between the points (2, 5) and (6, 13):

    • Using m = \frac{y2 - y1}{x2 - x1} :

    • Assign points:

      • (x1, y1) = (2, 5)

      • (x2, y2) = (6, 13)

    • Calculation:

      • m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2

    • Result: Slope is 2.

Practice Problems

• Example Task:

  • Find slope between the points (1, 2) and (3, 10):

    • Assign points:

    • (x1, y1) = (1, 2)

    • (x2, y2) = (3, 10)

    • Calculation:

    • m = \frac{10 - 2}{3 - 1} = \frac{8}{2} = 4

    • Result: Slope is 4.

Additional Example Problem

• Example 2:

  • A graph shows a line passing through (0, 2) and (4, 10):

    • Assign points:

    • (x1, y1) = (0, 2)

    • (x2, y2) = (4, 10)

    • Calculation:

    • m = \frac{10 - 2}{4 - 0} = \frac{8}{4} = 2

    • Result: Slope is 2.

Real-World Application of Slope

• Rate of Change Application:

  • Problem: A plumber charges $75 for 2 hours of work and $150 for 5 hours of work.

    • Identify changing quantities involves picking data points based on service fees and time.

    • Assign:

    • Let (2, 75) and (5, 150) be the points.

    • Using slope formula:

    • m = \frac{150 - 75}{5 - 2} = \frac{75}{3} = 25

    • Interpretation: The plumber's rate of change is $25 per hour.