AP Calculus AB and BC glossary

Even function

An even function satisfies f of negative x equals f of x, making its graph symmetric about the y-axis. Over an interval symmetric about zero, its integral equals twice the integral over the right half, which often saves real work.

f(-x) = f(x) \implies \int_{-a}^{a} f(x)\,dx = 2\int_{0}^{a} f(x)\,dx

Cosine and every even power of $x$ are even. Differentiating an even function produces an odd one, which is a useful check on a derivative you are unsure of.

The mistake

Assuming a function must be even or odd. Most are neither, and the symmetry shortcuts only apply when you have verified which case you are in.

Appears in: Unit 6: Integration and Accumulation