AP Calculus AB key terms

The value that a function's outputs approach as the input approaches a specified number. The limit can exist even when the function is undefined or takes a different value at that input.

The value a function approaches as the input approaches a point from only one direction, either from below or from above. A two-sided limit exists only when both one-sided limits agree.

An expression such as 0/0 or infinity minus infinity that gives no direct information about a limit's value. Such forms require algebraic manipulation, factoring, or another technique before the limit can be evaluated.

If a function is trapped between two other functions that share the same limit at a point, then the trapped function also has that limit there. Useful for limits involving oscillation or piecewise behavior.

A function is continuous at a point when the function is defined there, its limit exists there, and the limit equals the function's value. All three conditions must hold simultaneously.

A hole in a graph where the limit exists but does not equal the function's value, or the function is undefined. The discontinuity can be repaired by redefining the function at that single point.

A break in a graph where the left-hand and right-hand limits both exist but are different. The function's values leap from one level to another and cannot be repaired by redefining a single point.

If a function is continuous on a closed interval, it takes every value between its endpoint values at least once. Commonly used to prove that an equation has a solution in a given interval.

A function continuous on a closed interval must attain both a maximum and a minimum value somewhere on that interval. Continuity on a closed interval is the essential hypothesis.

The expression giving the average rate of change of a function over an interval, formed as the change in output divided by the change in input. Its limit as the interval shrinks defines the derivative.

The limit of the difference quotient as the input change approaches zero, interpreted as the instantaneous rate of change or the slope of the tangent line at that point.

The line touching a curve at a point whose slope equals the derivative there. Its equation is often written using point-slope form with the derivative as the slope.

The property that a function's derivative exists at a point. Differentiability at a point implies continuity there, but continuity does not guarantee differentiability, as at a sharp corner.

A differentiation shortcut stating that the derivative of x raised to a constant power equals that power times x raised to one less. It applies to any constant real exponent.

A rule for differentiating a product of two functions: the derivative of the first times the second, plus the first times the derivative of the second. Order of the two terms does not matter.

A rule for differentiating a ratio of two functions: denominator times the derivative of the numerator minus numerator times the derivative of the denominator, all divided by the denominator squared.

A rule for differentiating a composition of functions: differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function.

A function formed by applying one function to the output of another, so the input passes through two stages. Its derivative is found with the chain rule.

A technique for finding derivatives when a relationship between variables is given by an equation rather than solved for one variable. Differentiate both sides with respect to the input and solve for the desired derivative.

A function that reverses another function's action, sending each output back to its original input. It exists only when the original function is one-to-one on its domain.

The derivative of an inverse function at a value equals the reciprocal of the original function's derivative at the corresponding input. The two slopes are reciprocals of each other.

A problem type where several quantities change with time and are linked by an equation. Differentiating that equation with respect to time connects their rates of change.

The total change in a function's output divided by the length of the input interval. Geometrically it is the slope of the secant line joining the interval's endpoints.

The rate at which a quantity changes at a single moment, given by the derivative. It corresponds to the slope of the tangent line rather than a secant line.

Estimating a function's value near a known point using its tangent line there. Accuracy is generally better when the point of interest is close to the known point.

A method for evaluating limits that produce indeterminate quotients by replacing the ratio of functions with the ratio of their derivatives, provided the resulting limit exists.

An interior input where the derivative is zero or fails to exist. These are the candidates where a function may have a local maximum, local minimum, or neither.

A method for classifying critical points by checking whether the derivative changes sign across them. Positive to negative indicates a local maximum; negative to positive indicates a local minimum.

A method for classifying critical points using concavity: negative second derivative at a stationary point indicates a local maximum, positive indicates a local minimum, and zero is inconclusive.

The direction a curve bends, determined by the sign of the second derivative. Positive second derivative means concave up; negative means concave down.

A point where a graph changes concavity, from up to down or down to up. At such a point the second derivative is typically zero or undefined.

For a function continuous on a closed interval and differentiable inside it, some interior point has derivative equal to the average rate of change over the interval. Guarantees a tangent parallel to the secant.

A function whose derivative is a given function. Antiderivatives of the same function differ only by an additive constant, so a family of them exists.

The family of all antiderivatives of a function, written with an integration constant. It represents the reverse of differentiation rather than an accumulated quantity.

A number representing the signed area between a curve and the horizontal axis over an interval. Area below the axis counts as negative.

An approximation of a definite integral formed by summing rectangle areas over subintervals. Using left endpoints, right endpoints, or midpoints gives different estimates.

The theorem linking differentiation and integration: the integral of a rate of change gives net change, and the derivative of an accumulation function recovers the integrand.

A function defined as the integral of a rate from a fixed starting point to a variable endpoint. Its derivative is the rate function itself.

An equation involving an unknown function and one or more of its derivatives. A solution is a function satisfying the equation on some interval.

A grid of short segments whose slopes equal a differential equation's prescribed derivative at each point. Solution curves follow the segment directions.

A solution technique that rearranges a differential equation so each variable appears on only one side, then integrates both sides. It applies when the equation can be factored into a product of single-variable pieces.

A differential equation stating that a quantity's rate of change is proportional to its current size. Its solutions are exponential functions with a constant multiplier in the exponent.

The integral of a function over an interval divided by the interval's length. It represents the constant height giving the same accumulated area.

A method for finding a solid's volume by integrating the area of slices taken perpendicular to an axis. The slice shape is specified and its area expressed in terms of position.

A volume technique for a solid of revolution that integrates circular slice areas, using the squared distance from the axis as the radius. A hole in the slice leads to the washer method.

The definite integral of a rate of change over an interval equals the total change in the quantity over that interval. Displacement versus distance traveled is a common application.