1.4.2. Example

✳️ Definition: The average rate of change of a function 𝑓(𝑥) over an interval [𝑎,𝑏] is the change in the function's value divided by the change in 𝑥:

Average Rate of Change = $\frac{𝑓(𝑏)−𝑓(𝑎)}{𝑏 −𝑎}$

📊 Example: Let 𝑓(𝑥) = 𝑥². Find the average rate of change from 𝑥 = 1 to 𝑥 = 3.

$f(3) = 9,\quad f(1) = 1 \\ \text{AROC} = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4$

This means that on average, the function increases by 4 units for every 1 unit increase in 𝑥 over this interval.

The average rate of change (AROC) measures how much a function changes on average over a specific interval. To find it, we use the formula that calculates the difference in the function's values at two points, divided by the interval length. In our example, with the function 𝑓(𝑥) = 𝑥², we first find the function values at 𝑥 = 1 and 𝑥 = 3. The difference in the function values is 9 (at 𝑥 = 3) minus 1 (at 𝑥 = 1), giving us 8. The difference in x-values is 3 - 1, which is 2. Dividing these values, we get 4, indicating that, on average, for every 1 unit increase in x, the function increases by 4 units.

Imagine driving a car on a straight road. If you travel 80 kilometers in 2 hours, your average speed is 40 kilometers per hour. Just like how the average rate of change in a function measures overall change in y-values for a given change in x-values, average speed measures distance traveled over time.

⚡ Instantaneous Rate of Change (IROC) ✳️ Definition: The instantaneous rate of change is the rate at which a function is changing at a single point. This is given by the derivative of the function at that point. Mathematically:

$IROC at 𝑥 = 𝑎 = \lim_{ℎ→0} \frac{𝑓(𝑎+ℎ)−𝑓(𝑎)}{ℎ}$

🧮 Example: Let 𝑓(𝑥) = 𝑥². Find the IROC at 𝑥 = 2.

$f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = 2x \\ ext{At } x = 2: f'(2) = 4$

The function is changing at a rate of 4 units per 1 unit change in 𝑥 at 𝑥 = 2.

The instantaneous rate of change (IROC) tells us how a function is changing right at a specific point. It is found using the derivative, which calculates the limit as a very small change approaches zero. For our example with 𝑓(𝑥) = 𝑥², we derive the formula for the derivative and then substitute 𝑥 = 2. After calculation, we find that the rate of change at this point is 4, indicating that at 𝑥 = 2, for every 1 unit increase in 𝑥, the value of the function increases by 4 units.

Think of riding a bike. If you are pedaling, your speed (how fast you are going at that moment) is like the IROC—it tells you how quickly you're moving at a specific instant, much like how the speedometer on a bike measures your speed at that exact time.

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