BGlossary

Terms and abbreviations used in this book, defined in plain terms. Each is also defined inline where it first appears.

Accumulation

The total a rate builds up when it acts over an interval — for example, the total distance built up by a speed over time. Accumulation is what the integral computes.

Continuity

The property of a function whose graph has no jumps, gaps, or holes — you could draw it without lifting your pen.

Derivative

The instantaneous rate of change of a function — the slope of its tangent line — obtained as the limit of secant slopes as the two points merge.

Discontinuity

A point where a function fails to be continuous — a place where you must lift your pen. The three kinds are a removable hole, a jump, and a blow-up.

Diverge to infinity

What a function does when its output grows without bound as the input nears a target, racing off toward positive or negative infinity. The limit does not exist as a finite number; 'infinity' names how it fails, not a value the output approaches.

Factoring

Rewriting a sum or difference as a product, such as x squared minus 1 = (x minus 1)(x plus 1); it reveals where an expression is zero and lets a shared factor cancel.

Function

A rule that turns each input number into exactly one output number; its graph is the picture of every input-output pair at once.

Function notation

The shorthand f(x) for 'the output of the function f when the input is x'; f(3) means the output at input 3, and the parentheses mark the input slot, not multiplication.

Fundamental Theorem of Calculus

The central fact that differentiation and integration are inverse operations: differentiating an accumulated total recovers its rate, and integrating a rate recovers the total.

Graph of a function

The picture formed by plotting every input-output pair as a point, with the input measured horizontally and the output vertically; the curve is all the pairs at once.

Instantaneous rate of change

How fast a function's output is changing at one exact instant, as opposed to averaged over an interval; another name for the derivative.

Integral

The exact running total of a quantity that is changing while it is being added up, obtained by slicing an interval into ever-thinner pieces and adding their contributions; geometrically, the area beneath a curve.

Intermediate Value Theorem

The guarantee that a continuous function on an interval takes every height between its starting and ending values at least once — so if it starts below a target and ends above it, it must cross that target somewhere in between.

Jump discontinuity

A break where the function approaches one height from the left and a different height from the right, so the two one-sided limits disagree and the graph shows a vertical step.

Limit

The single value a quantity heads toward as its input is nudged ever closer to some target, whether or not it ever exactly arrives.

Linear function

A function whose graph is a straight line, written f(x) = mx + b, where m is the slope and b is the y-intercept.

One-sided limit

The single value a function approaches as its input closes in on a target from just one direction — from the left (inputs below the target) or from the right (inputs above it). The two-sided limit exists only when both one-sided limits exist and agree.

Rate of change

How much a quantity shifts for each small step forward in its input — how steeply it is climbing or falling. Made exact at a single instant, it is the derivative.

Removable discontinuity

A break where the function heads toward a finite height (the limit exists) but the point there is missing or sits at the wrong height. It is fixable by placing or moving a single point; also called a removable hole.

Secant line

A straight line drawn through two points on a curve. Its slope is the average rate of change of the function between those two points.

Slope

How steep a line is: the amount the output rises for each step the input takes to the right (rise over run).

Tangent line

The straight line that just grazes a curve at a single point, matching its direction there. Its slope is the derivative at that point.

Vertical asymptote

A vertical line x = a that a graph races alongside, growing without bound toward positive or negative infinity as x nears a, without ever touching the line. It marks a blow-up (an infinite discontinuity).

y-intercept

The height at which a graph crosses the vertical axis — the output of the function when the input is 0.