Question

estimate the instantaneous rate of change at x = 3

Explanation:

Step1: Recall rate - of - change formula

The instantaneous rate of change of a linear function is its slope. For a linear function $y = mx + b$, the slope $m=\frac{\Delta y}{\Delta x}$.

Step2: Select two points on the line

Let's take two points $(x_1,y_1)$ and $(x_2,y_2)$ from the line. Suppose $(x_1,y_1)=(0,2)$ and $(x_2,y_2)=(2,6)$.

Step3: Calculate the slope

$m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{6 - 2}{2-0}=\frac{4}{2}=2$. Since the function is linear, the instantaneous rate of change at any point, including $x = 3$, is equal to the slope of the line.

Answer:

$2$