CLEP Calculus key terms

The value a function's output approaches as the input approaches a specified number, whether or not the function actually equals that value at that input.

The value a function approaches as the input approaches a point from only the left side or only the right side, written with a minus or plus superscript.

A limit that exists only when both the left-hand and right-hand limits at a point exist and are equal to the same number.

A limit in which the function's values grow without bound, positive or negative, as the input approaches a point; it describes a vertical asymptote rather than a finite number.

The value a function approaches as the input grows without bound in the positive or negative direction; it describes the function's end behavior.

A property holding when a function's limit at a point exists, the function is defined there, and the limit equals the function's value at that point.

A gap in a graph where the limit exists but does not equal the function's value, or the function is undefined; the hole can be patched by redefining a single point.

A break in a graph where the left-hand and right-hand limits both exist but differ, so the function's values leap from one level to another.

A vertical line that a graph approaches ever more closely as the input nears a particular value, typically where the function grows without bound.

A horizontal line that a graph approaches as the input grows without bound in one or both directions, determined by limits at infinity.

An expression such as zero over zero or infinity over infinity arising from direct substitution that gives no immediate answer and requires algebraic manipulation or another technique.

A principle stating that if a function is trapped between two others that share the same limit at a point, then the trapped function also has that limit there.

A principle stating that a function continuous on a closed interval takes every value between its endpoint values somewhere inside that interval.

A principle stating that a function continuous on a closed interval attains both a maximum and a minimum value somewhere on that interval.

The instantaneous rate of change of a function with respect to its input, defined as the limit of a difference quotient and equal to the slope of the tangent line.

The fraction formed by dividing the change in a function's output by the change in its input over an interval; its limit as the interval shrinks gives the derivative.

The slope of the secant line between two points on a graph, computed as the change in output divided by the change in input over that interval.

A straight line touching a curve at a point and having the same slope as the curve there, given by the derivative at that point.

The line perpendicular to the tangent line at a point on a curve, whose slope is the negative reciprocal of the derivative there.

The property of having a derivative at a point; it requires continuity there, and it fails where a graph has a corner, cusp, or vertical tangent.

A differentiation formula stating that the derivative of a product of two functions equals the first times the derivative of the second plus the second times the derivative of the first.

A differentiation formula for a ratio of two functions: the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared.

A differentiation formula for composed functions: the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

A technique for finding a derivative when a relationship between variables is given by an equation rather than solved for one variable, differentiating both sides and solving for the derivative.

A method for finding how quickly one changing quantity changes by writing an equation linking it to another quantity and differentiating both sides with respect to time.

An input value in a function's domain where the derivative is zero or undefined, marking a candidate location for a local maximum, local minimum, or neither.

A point where a function's value is the largest or smallest among nearby points, called a local maximum or local minimum respectively.

A procedure that classifies a critical point by checking whether the derivative changes sign there: positive to negative means a local maximum, negative to positive a local minimum.

A procedure that classifies a critical point using the second derivative: positive indicates a local minimum, negative a local maximum, and zero is inconclusive.

The way a curve bends, described as concave up when the slope increases and concave down when the slope decreases, as indicated by the sign of the second derivative.

A point on a curve where the direction of concavity changes, often where the second derivative is zero or undefined and changes sign.

A principle stating that for a function continuous on a closed interval and differentiable inside it, some interior point has a tangent slope equal to the interval's average rate of change.

A special case of the Mean Value Theorem stating that a function continuous on a closed interval, differentiable inside, and equal at both endpoints has a horizontal tangent somewhere inside.

A technique for evaluating indeterminate limits by replacing the ratio of two functions with the ratio of their derivatives, provided the resulting limit exists.

The process of finding the largest or smallest possible value of a quantity subject to constraints, typically by building a function and locating its extreme values.

A function whose derivative equals a given function; antiderivatives of the same function differ only by an additive constant.

The family of all antiderivatives of a function, written with an integral sign and a constant of integration, representing a collection of functions rather than a number.

An arbitrary additive constant included when writing an indefinite integral, accounting for the fact that antiderivatives are determined only up to a constant.

A number computed as the limit of Riemann sums over an interval, interpreted as the signed area between a graph and the horizontal axis.

An approximation of a definite integral formed by adding up areas of rectangles whose heights come from function values sampled within subintervals.

The link between differentiation and integration: the definite integral of a continuous function is found using any antiderivative evaluated at the endpoints, and the integral of a function with variable upper limit differentiates back to the integrand.

A function defined by an integral with a variable upper limit, giving the signed area accumulated from a fixed starting point up to that input.

The region enclosed by two graphs, computed as the definite integral of the upper function minus the lower function across the interval where they bound the region.

The volume of a solid formed by rotating a region about an axis, computed by integrating cross-sectional areas such as disks, washers, or cylindrical shells.

An integration technique that reverses the Chain Rule by replacing an inner expression with a new variable, rewriting the integral in simpler form before evaluating.

An integration technique derived from the Product Rule that rewrites an integral of a product as a boundary term minus another integral, useful when the new integral is simpler.

The total change in a quantity over an interval, equal to the definite integral of its rate of change over that interval.