3.3 Rates of Change

• Match the quantity with its rate of change:

  Volume ↔️ Flow Rate

  Distance ↔️ Speed

  Temperature ↔️ Temperature Change
• Match the concept with its definition:

  Average Rate of Change ↔️ Overall change over a time interval

  Instantaneous Rate of Change ↔️ Rate of change at a specific point
• Match the quantity with its rate of change formula:

  Volume ↔️ $\text{Volume} / \text{Time}$

  Distance ↔️ $\text{Distance} / \text{Time}$

  Temperature ↔️ $\text{Temperature} / \text{Time}$
• What is the formula for flow rate?

  $\text{Volume} / \text{Time}$
• The rate of change measures how one quantity changes over time.time
• The rate of change for distance is called speed, and its formula is distance divided by time.


  True
• To calculate the instantaneous rate of change, we need to take the derivative of the function.derivative
• If the temperature rises from 10°C to 30°C over 2 hours, the change in temperature is 20°C
• To estimate the instantaneous rate of change, draw a tangent line at the point of interest and calculate its slope.

  True
• The rate of change for distance is called speed
• Match the concept with its definition or formula:

  Average Rate of Change ↔️ Overall change over a time interval

  Instantaneous Rate of Change ↔️ Rate of change at a specific point
• The instantaneous rate of change is graphically represented by the slope of the tangent line.


  True
• To estimate the speed at t = 3 seconds, you draw a tangent
• The instantaneous rate of change describes the overall trend over a time interval

  False
• The formula for acceleration is change in velocity over time
• The relationship between distance and time in proportional rates is described by the formula for speed
• What are the two main types of rates of change?

  Average and Instantaneous
• The instantaneous rate of change is equivalent to the derivative
• What does the instantaneous rate of change represent graphically?

  Slope of the tangent line
• What is the key difference between the average and instantaneous rates of change?

  General trend vs. precise rate
• Steps to calculate the average rate of change:

  1️⃣ Identify the initial and final quantities

  2️⃣ Calculate the change in quantity

  3️⃣ Calculate the change in time

  4️⃣ Compute the average rate of change
• To calculate the slope of a tangent line, use the formula \frac{\Delta y}{\Delta x}
• The instantaneous rate of change gives the precise rate at a particular moment.

  True
• The instantaneous rate of change provides the precise rate at a particular moment, unlike the average rate of change


  True
• Steps to estimate the instantaneous rate of change:

  1️⃣ Draw a tangent line at the point of interest

  2️⃣ Calculate the slope of the tangent line

  3️⃣ The slope represents the instantaneous rate of change
• In proportional rates, the ratio between the quantities remains constant


  True
• What is the relationship between volume and time in proportional rates?

  Flow Rate
• What is the instantaneous speed at t = 3 seconds if the rise is 5 meters and the run is 1 second?

  5 meters/second
• The average rate of change is calculated as the change in quantity divided by the change in time
• The instantaneous rate of change represents the slope of the tangent line to the curve at a specific point in time.

  True
• Steps to calculate the average rate of change:

  1️⃣ Identify the initial and final quantities

  2️⃣ Calculate the change in quantity

  3️⃣ Calculate the change in time

  4️⃣ Compute the average rate of change
• In the example, the average rate of change in temperature is 10°C per hour.


  True
• What does the rate of change measure?

  How one quantity changes over time
• The rate of change for temperature is called temperature change.


  True
• The average rate of change is calculated as the overall change in quantity divided by the change in time
• To calculate the instantaneous rate of change, you need to take the derivative of the function.


  True
• The formula for average rate of change is (Change in Quantity) / (Change in Time)
• What is the average rate of change if the temperature rises from 10°C to 30°C over 2 hours?

  10°C/hour
• What is the instantaneous speed if the tangent line at t = 3 seconds has a rise of 5 meters and a run of 1 second?

  5 meters/second
• What does the term 'instantaneous rate of change' measure?

  Rate at a specific moment