Short on time? This page is the recap. For explanations, the figures and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: f′(a) = lim (h → 0) (f(a + h) − f(a))/h; the symbol ≈ means “approximately”.
Recap
- f′(a) is a limit, so a table or graph only lets you estimate it.
- Table: use the closest data points on an interval that contains a. Divide by the change in x, not the number of rows.
- Graph: f′(a) is the gradient of the tangent at (a, f(a)). Read two grid points on the tangent line, not on the curve.
- Sign check: rising graph, f′ > 0; falling graph, f′ < 0; smooth peak or trough, f′ = 0.
- Calculator: write the setup, such as f′(3), then the value to three decimal places. Use radian mode.
- Units: units of f per unit of x.
Key relationships
| Estimate | Formula (h > 0) | Notes |
|---|---|---|
| Forward | (f(a + h) − f(a))/h | Uses a and a point to the right |
| Backward | (f(a) − f(a − h))/h | Uses a point to the left and a |
| Symmetric | (f(a + h) − f(a − h))/(2h) | Usually the most accurate for the same h |
| Table, a between entries p and q | (f(q) − f(p))/(q − p) | Use the closest entries either side of a |
| Tangent through (x₁, y₁) and (x₂, y₂) | (y₂ − y₁)/(x₂ − x₁) | Points must lie on the tangent |
With even spacing, symmetric = (forward + backward)/2.
Assumptions behind the method
- The function is differentiable at a; otherwise there is nothing to estimate (Topic 2.4).
- The function changes smoothly between table values, so a short secant is close to the tangent.
- The tangent line on a graph is drawn accurately, and the grid points you read lie on it.
Mistakes to avoid
- Computing f(a)/a instead of a change in f over a change in x.
- Using an interval that does not contain a.
- Giving only a number with no quotient shown.
- Reading points on the curve when a tangent is drawn.
- Missing or wrong units, or describing a total change instead of a rate.
- Writing = instead of ≈ for an estimate.
- Trusting a calculator at a corner. It can give a number where no derivative exists.
Quick self-check
- f(1.0) = 3, f(1.2) = 3.8 and f(1.4) = 5.1. Estimate f′(1.2). (f′(1.2) ≈ (5.1 − 3)/(1.4 − 1.0) = 5.25)
- The tangent to y = g(x) at x = 2 passes through (0, 5) and (4, −1). Estimate g′(2). (−6/4 = −1.5)
- h(t) is the height of a balloon in metres, t seconds after release, and h′(7) ≈ 2.4. What does this mean? (At t = 7 seconds the balloon is rising at about 2.4 metres per second.)
Next: practice questions.