Why is the secant slope formula considered the average rate of change over a finite interval?

Conditions

• Two distinct points on the curve are selected: $(x_0, f(x_0))$ and $(x_0 + \Delta x, f(x_0 + \Delta x))$.
• $\Delta x \neq 0$.

Reasoning, step by step

1. Identify the vertical change: $f(x_0 + \Delta x) - f(x_0)$.
2. Identify the horizontal change: $\Delta x$.
3. Divide the vertical change by the horizontal change to get the slope.
4. Recognize that this ratio averages the local variations of the function over the entire span of $\Delta x$.

Example

The script introduces the green secant line drawn between $(x_0, f(x_0))$ and $(x_0 + \Delta x, f(x_0 + \Delta x))$. Its slope is written algebraically as $[f(x_0 + \Delta x) − f(x_0)] / \Delta x$, representing average rate of change over an interval.

Common misconceptions

• Believing the secant slope gives the exact velocity at the starting point $x_0$.
• Thinking that average rate of change is meaningless for non-linear functions.

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Explore next

• How does shrinking the interval transform the average rate into an instantaneous rate?
• What is the geometric interpretation of the secant line?