5.8 Sketching Functions & Derivatives

Reading between graphs

Slopes of f become y-values of f'. Where f has a local max/min, f' = 0. Where f is concave up, f' is increasing (and f'' > 0). Where f is steepest, f' has an extremum.

Both directions

Given f, sketch f' from its slopes; given f', deduce where f increases/decreases and its concavity. An inflection point of f lines up with an extremum of f'.

Key terms

f → f'

Slopes of f become values of f'.

Local extremum of f

Occurs where f' = 0.

Concave up

Corresponds to f' increasing.

Positive f'

f is increasing.

Inflection of f

Aligns with an extremum of f'.

Try an AP-style question

f increases up to x = 2, has a horizontal tangent there, and then decreases. What does f' do at x = 2?

1. f' has a relative maximum at x = 2
2. f' is undefined at x = 2
3. f' crosses zero, going from negative to positive
4. f' crosses zero, going from positive to negative