3.3 Rates of Change

• In a rate of change, the quantity being measured or observed is the dependent variable.
• Rates of change are interpreted in real-world contexts.
• Match the representation with the example:

  Ratio ↔️ 60 miles per hour

  Fraction ↔️ 60 miles / 1 hour
• The independent variable is manipulated to observe changes in the dependent variable.

  True
• What is the general formula for calculating a rate of change?

  $\frac{\Delta y}{\Delta x}$
• Rates of change are essential for measuring growth rates in populations.


  True
• What is a rate of change usually expressed as?

  Ratio or fraction
• When studying plant growth, the amount of sunlight received is the independent variable.


  True
• Calculating rates of change requires specific formulas.

  True
• How is a rate of change defined in terms of two quantities?

  Change in one quantity relative to another
• Why are rates of change important in real-world applications?

  Measure various changes
• What does the independent variable cause in a rate of change?

  Changes in dependent variable
• When calculating the rate of change between points (1, 3) and (4, 9), Δy is equal to 6.
• What is the role of the dependent variable in a rate of change?

  Changes in relation to another variable
• In a rate of change, the quantity that changes in relation to another variable is called the dependent variable.
• What does the rate of change measure in the formula Δy/Δx?

  Amount y changes per unit change in x
• The formula for the instantaneous rate of change involves a limit as Δx approaches zero.
• What does Δy represent in the rate of change formula?

  Change in y-values
• What is the key difference between average and instantaneous rates of change?

  Finite vs. specific interval
• If a car travels at 60 miles per hour, it means it moves 60 miles in every hour.


  True
• If a runner travels 10 miles in 2 hours, what is the runner's speed in miles per hour?

  5 miles per hour
• What is a rate of change typically expressed as?

  Ratio or fraction
• What is the difference between average and instantaneous rates of change?

  Timeframe for calculation
• Speed is an example of the rate of change of distance with respect to time.

  True
• The dependent variable is the quantity that changes in relation to another variable.
• Arrange the following steps to calculate a rate of change using the formula: Δy / Δx

  1️⃣ Identify the y-values

  2️⃣ Identify the x-values

  3️⃣ Calculate Δy

  4️⃣ Calculate Δx

  5️⃣ Divide Δy by Δx
• What is the rate of change between the points (1, 3) and (4, 9)?

  2
• The independent variable is the quantity being manipulated or controlled.
• What is the independent variable in a rate of change?

  Quantity being varied
• The formula for calculating the rate of change is Δy/Δx.
• The rate of change between points (1, 3) and (4, 9) is 2.


  True
• What does the rate of change "miles per hour" describe in the context of a vehicle's speed?

  Miles covered each hour
• The rate of change is represented by the formula Δy/Δx.
• Steps to solve word problems involving rates of change

  1️⃣ Identify the knowns and unknowns

  2️⃣ Set up an equation using the rate of change formula

  3️⃣ Solve the equation

  4️⃣ State the answer with appropriate units