2.3 Estimating Derivatives at a Point

From a table

Estimate f'(a) with a difference quotient using a small h: [f(a+h) − f(a)]/h. The symmetric difference quotient [f(a+h) − f(a−h)]/(2h) uses both sides and is usually more accurate.

From a graph

Estimate f'(a) as the slope of the tangent line at x = a (or the slope between two very close points). Smaller h generally gives a better estimate.

Key terms

Estimating a derivative

Approximate f'(a) numerically or graphically.

Difference quotient

[f(a+h) − f(a)]/h with small h.

Symmetric difference quotient

[f(a+h) − f(a−h)]/(2h); usually more accurate.

Graphical estimate

Slope of the tangent line at a.

Accuracy

Smaller h gives a better estimate.

Try an AP-style question

What is the best estimate of f'(3.5)?

1. 8.5
2. 2
3. 3
4. 1