AP Calculus AB and BC glossary

Polynomial function

Also called: Polynomial

A polynomial function is a finite sum of terms, each a constant times x to a nonnegative integer power, such as 3x cubed minus 5x plus 2. Every polynomial is continuous and differentiable at every real number, which is why direct substitution always evaluates its limits.

p(x) = a_n x^{n} + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0

Two conditions do all the work: the exponents are whole numbers and the sum is finite. Each differentiation drops the degree by one, so the $n$ th derivative of a degree $n$ polynomial is the constant $n! \, a_n$ and every derivative past that one is 0. Nothing else in the course runs out of derivatives like this.

Continuity at every real number is what licenses $\lim_{x \to a} p(x) = p(a)$ , so a polynomial limit never needs factoring, a conjugate, or L'Hopital's rule. Far from the origin the leading term takes over, which is why a nonconstant polynomial has no horizontal asymptote: $\lim_{x \to \infty} p(x)$ is $\infty$ when $a_n > 0$ and $-\infty$ when $a_n < 0$ .

The mistake

Assuming the everywhere-continuous guarantee transfers to anything built out of powers. A negative or fractional exponent disqualifies an expression, so $x^{-1}$ and $x^{1/2}$ are power functions but not polynomials, and both break the guarantee: $\lim_{x \to 0} x^{-1}$ does not exist, and $x^{1/2}$ is not defined to the left of 0 at all.

Appears in: Unit 2: Defining the Derivative