AP Calculus AB and BC glossary

Increasing/decreasing test

Also called: Monotonicity test

The increasing/decreasing test states that a function increases where its derivative is positive and decreases where it is negative. You apply it by checking the sign of the derivative on each interval between the critical numbers and any points where the function or its derivative is undefined.

Divide the number line at every critical number and at every point where ff or f′f' is undefined, then test one value per interval. One test value is enough because between consecutive dividers f′f' is never zero, and a derivative cannot skip values. It takes every value between any two it attains, so it must hold one sign across the whole interval.

Endpoints may be included wherever ff is continuous. Since f(x)=x2f(x) = x^2 is continuous at 00 and rising to the right of it, ff is increasing on [0,∞)[0, \infty) even though f′(0)=0f'(0) = 0. Open and closed intervals have both been accepted on the exam, so state the interval and justify it with the sign of f′f'.

The mistake

Merging intervals across a gap in the domain. For f(x)=1xf(x) = \frac{1}{x} the derivative f′(x)=−1x2f'(x) = -\frac{1}{x^2} is negative everywhere it exists, yet ff is not decreasing on the whole domain, because f(−1)=−1f(-1) = -1 is smaller than f(1)=1f(1) = 1. Report (−∞,0)(-\infty, 0) and (0,∞)(0, \infty) separately.

Appears in: Unit 5: Analytical Applications