6.5 Interpreting Accumulation Functions

Behavior from f

For g(x) = ∫[a to x] f(t) dt, since g'(x) = f(x): g is increasing where f > 0 and decreasing where f < 0. Critical points of g occur where f(x) = 0.

Area signs

Area above the axis adds (positive), area below subtracts (negative). g has a local maximum where f changes from positive to negative.

Key terms

Accumulation behavior

g increases where f > 0, decreases where f < 0.

Critical points of g

Where f(x) = 0 (since g' = f).

Local max of g

Where f changes from + to −.

Signed area

Above axis adds; below axis subtracts.

Key link

g'(x) = f(x).

Try an AP-style question

Where does g have its relative maximum on (0, 6)?

1. x = 3
2. x = 1.5
3. x = 6
4. x = 0