Average Rate of Change
Slope of a secant line.
Key idea
Slope of a secant line.
In depth
The average rate of change over [a, b] is (f(b) − f(a))/(b − a), the slope of the secant line. It is the average speed if f is position. Shrinking the interval leads to the derivative.
Worked example
f(x) = x² from 1 to 3: (9 − 1)/2 = 4.
Formula
Real-world uses
- Average speed over a journey
- Average growth of a company between two years
Common mistake
Mixing the order of subtraction in numerator and denominator.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Average Rate of Change?
Slope of a secant line. It is a topic at the advanced level in Calculus.
What is the formula for Average Rate of Change?
The key formula is (f(b)−f(a))/(b−a).
Where is Average Rate of Change used in real life?
Real-world examples: Average speed over a journey; Average growth of a company between two years.
What should I learn before Average Rate of Change?
Make sure you are comfortable with Intermediate Value Theorem, the previous topic in Calculus.
What is a common mistake with Average Rate of Change?
Mixing the order of subtraction in numerator and denominator.
How can I practise Average Rate of Change?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Average Rate of Change?
Move on to Definition of the Derivative next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.