Short on time? This page is the recap. For explanations, the tangent-line graph and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- A derivative is a rate in any setting, not only motion: litres per minute, dollars per item, people per year, grams per centimetre.
- The input need not be time. It can be altitude, distance, number of items and so on.
- f′(a) > 0: the quantity is increasing at a. f′(a) < 0: it is decreasing.
- f″(a) tells you whether the rate itself is rising or falling.
- From a table you can only estimate a derivative: use the difference quotient of the nearest values either side.
Key relationships
| Idea | Rule |
|---|---|
| Units of f′(x) | units of f ÷ units of x |
| Units of f″(x) | units of f ÷ (units of x)² |
| Estimate from a table | f′(c) ≈ (f(b) − f(a))/(b − a), with a < c < b as close as possible |
| Average rate on [a, b] | (f(b) − f(a))/(b − a): an interval, not an instant |
| f′ < 0 and f″ > 0 | Decreasing, but more slowly |
| f′ > 0 and f″ < 0 | Increasing, but more slowly |
The four-part sentence: At [input value with units], the [quantity] is [increasing/decreasing] at a rate of [size] [units].
Assumptions behind the method
- The function is differentiable at the input value you are interpreting.
- A table estimate assumes the function behaves smoothly between the table values.
- A model is only valid on the domain it is given for.
Mistakes to avoid
- Confusing f(a) with f′(a): an amount versus a rate. Check the units.
- Units upside down: litres per minute, not minutes per litre.
- Double negative: “decreasing at a rate of −60”. Drop the minus sign or drop the word “decreasing”.
- No instant: always say “at t = …” (or “at an altitude of …”).
- Calling a rate an exact change: f′(a) = −60 does not mean exactly 60 lost in the next unit.
- Reading f″ < 0 as “the quantity is falling”: it means the rate is falling.
Quick self-check
- N(t) is the number of visitors in a park, t in hours, and N(t) = 40 + 6t − t². Find N′(2) and its units. (2 visitors per hour: at t = 2 hours, the number of visitors is increasing at 2 visitors per hour.)
- In question 1, N″(t) = −2. What does this tell you at t = 2? (The number of visitors is still increasing, but the rate of increase is falling by 2 visitors per hour each hour.)
- R(q) is revenue in euros from selling q tickets. What are the units of R′(q)? (Euros per ticket.)
Next: practice questions.