Short on time? This page is the recap. For the graph, the tables and the worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (h → 0) g(h) means “the limit as h approaches 0 of g(h)”. Topic 1.2 defines it properly.
Recap
- The average rate of change of f over [a, b] is (f(b) − f(a))/(b − a). Its units are output units per input unit.
- On a graph it is the slope of the secant line through (a, f(a)) and (b, f(b)).
- At a single instant the change in input is 0, so the formula gives 0/0, which is undefined. One division can never give the rate at an instant.
- Instead, use intervals that contain the instant and make them shorter. If the averages from both sides approach one value, that value is the instantaneous rate of change.
- On a graph, the secant slopes approach the slope of the tangent line.
- From a table you can only estimate: use the shortest interval containing the instant, ideally with data on both sides.
Key relationships
| Idea | Formula or rule | Meaning |
|---|---|---|
| Average rate over [a, b] | (f(b) − f(a))/(b − a) | Slope of a secant line |
| Average rate over [a, a + h] | (f(a + h) − f(a))/h, h ≠ 0 | h > 0: right of a; h < 0: left of a |
| Interval of length zero | (f(a) − f(a))/0 = 0/0 | Undefined, not 0 |
| Rate at the instant a | lim (h → 0) (f(a + h) − f(a))/h | Slope of the tangent line |
| Sign of a rate | positive / negative | quantity increasing / decreasing |
Assumptions
- The interval must contain the instant (inside it or at one end).
- h is never 0 in any quotient you calculate; you only look at what happens as h gets close to 0.
- A single rate exists only if the averages from the left and from the right approach the same value.
Mistakes to avoid
- Putting h = 0 into (f(a + h) − f(a))/h. That gives 0/0.
- Dividing an output by an input, such as s(3)/3. That is not the rate at t = 3.
- Using a long interval when a shorter one containing the instant is available.
- Averaging the outputs instead of dividing the change in output by the change in input.
- Dropping units or the sign when you interpret a rate.
Quick self-check
- Find the average rate of change of f(x) = x² over [1, 4]. (5: (16 − 1)/3 = 15/3)
- For f(x) = x², the average rate over [5, 5 + h] simplifies to 10 + h for h ≠ 0. What is the rate of change at x = 5? (10, because 10 + h approaches 10 as h approaches 0)
- A bakery has sold L(t) loaves t hours after opening, with L(1) = 40 and L(3) = 52. Find the average rate over [1, 3] and give its units. (6 loaves per hour)
Next: practice questions.