Find the average rate of change of the function y = x³

Find the average rate of change of the function $y = x^3$ on the interval $1 \lt x \lt 2$.

Step 1: Identify the endpoints

The interval has endpoints $x_1 = 1$ and $x_2 = 2$, so the increment of the argument is

\[ \Delta x = x_2 - x_1 = 2 - 1 = 1. \]

Step 2: Compute the function values

\[ y_1 = x_1^3 = 1^3 = 1, \qquad y_2 = x_2^3 = 2^3 = 8. \]

The increment of the function is

\[ \Delta y = y_2 - y_1 = 8 - 1 = 7. \]

Step 3: Compute the average rate of change

The average rate of change (the mean velocity of change) is defined as

\[ \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}. \]

Substituting: $\frac{\Delta y}{\Delta x} = \frac{7}{1} = 7$.

Step 4: Verification

Using the finite-difference expansion of $y = x^3$ with $x_1 = 1$ and $\Delta x = 1$: \[\Delta y = (x_1 + \Delta x)^3 - x_1^3 = (1+1)^3 - 1^3 = 8 - 1 = 7 \;\;\checkmark\]