2.3 Estimating Derivatives of a Function at a Point

Estimating slope from a table

When given a table of values, estimate a derivative using a difference quotient: (f(b) - f(a)) / (b - a). For the best estimate at a point, use the two table values that straddle it (a symmetric difference), since this typically gives a closer approximation than a one-sided quotient.

Estimating slope from a graph

On a graph, estimate f'(x) by drawing a secant line through two nearby points and computing its slope, or by sketching the tangent line at the point and estimating rise over run. The closer the two points are to the point of interest, the better the estimate of the instantaneous rate of change.

Average rate vs. instantaneous rate

A difference quotient computed from a table or graph gives the average rate of change over an interval, which serves as an approximation for the instantaneous derivative at a point within that interval. Smaller intervals around the point generally produce more accurate estimates.

Key terms

Difference quotient

(f(b) - f(a)) / (b - a); represents the average rate of change of f over [a,b], used to estimate a derivative.

Symmetric difference quotient

Using two points that straddle the point of interest to estimate the derivative there; usually more accurate than a one-sided estimate.

Estimating derivative from a table

Choose the closest surrounding x-values and compute the difference quotient using those points.

Estimating derivative from a graph

Draw or estimate the tangent line at the point and calculate its slope (rise/run).

Average rate of change vs. instantaneous rate of change

Average rate of change is calculated over an interval; instantaneous rate of change (the derivative) occurs at a single point, and can be approximated by the average rate over a small interval.

Try an AP-style question

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

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