AP Calculus BC key terms

The value a function's outputs approach as the input approaches a specified point, whether or not the function is defined there. Written lim x→a f(x)=L, it means outputs can be made arbitrarily close to L by taking inputs close enough to a.

The value a function approaches as the input approaches a point from only one direction, either from below (left-hand limit) or from above (right-hand limit). A two-sided limit exists exactly when both one-sided limits exist and agree.

A function is continuous at x=a when the two-sided limit at a exists, the function is defined at a, and the limit equals the function value. Informally, the graph has no hole, jump, or break at that point.

A point where a function is undefined or misdefined but the two-sided limit exists, producing a hole in the graph. Redefining the function at that single point to equal the limit would make it continuous there.

A point where the left-hand and right-hand limits both exist but are different, so the graph breaks with a vertical gap. No redefinition at that point can repair it, because the two-sided limit does not exist.

If f is continuous on a closed interval and k lies between the values of f at the endpoints, then f attains k somewhere inside the interval. It guarantees a solution to f(x)=k and justifies root existence.

A line a graph approaches arbitrarily closely as the input or output grows without bound. Horizontal asymptotes come from limits at infinity; vertical asymptotes arise where a function grows without bound near a finite input.

The limit of the difference quotient as the input change shrinks to zero, equal to the slope of the tangent line and the instantaneous rate of change. Its existence requires matching left and right difference-quotient limits.

The expression [f(x+h)−f(x)]/h, giving the average rate of change of f over an interval of length h. Its limit as h approaches zero defines the derivative.

The property of having a derivative at a point, which requires continuity there but is stronger than continuity. Corners, cusps, vertical tangents, and jumps are places where a function fails to be differentiable.

The differentiation rule stating that the derivative of x^n is n·x^(n−1) for any constant exponent n. Combined with linearity, it differentiates all polynomials term by term.

The rule that the derivative of a product f·g equals f′·g + f·g′. It expresses how the rate of a product combines the rates and values of its two factors.

The rule that the derivative of f/g equals (f′g − fg′)/g², valid where g is nonzero. The squared denominator and the subtraction order are the parts students most often misremember.

The rule for differentiating a composition: the derivative of f(g(x)) is f′(g(x))·g′(x). It says the outer rate is multiplied by the inner rate, and it underlies implicit and related-rates work.

A technique for finding dy/dx when a curve is given by an equation relating x and y rather than as y=f(x). Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.

The relationship that if g is the inverse of f, then g′(b)=1/f′(a) where f(a)=b and f′(a) is nonzero. Slopes of inverse graphs are reciprocals at mirrored points.

A method for relating the rates at which two or more changing quantities vary, by differentiating an equation linking them with respect to time. Used for expanding balloons, sliding ladders, and draining tanks.

Approximating a function near a point by its tangent line, f(a)+f′(a)(x−a). It converts a small input change into an estimated output change using the local rate of change.

The change in a function's value divided by the length of the input interval, equal to the slope of the secant line. It contrasts with instantaneous rate of change, which is the derivative.

An input in the domain where the derivative is zero or fails to exist. Critical points are the candidates for local extremes and must be tested or compared.

A procedure classifying a critical point by the sign change of f′ around it: positive to negative means a local maximum, negative to positive a local minimum, and no change means neither.

Describes how a curve bends: concave up where the derivative increases and the graph lies above its tangent lines, concave down where the derivative decreases. The sign of the second derivative determines it.

A point on a curve where concavity changes sign, which requires the second derivative to switch between positive and negative. A zero second derivative alone does not guarantee one.

If f is continuous on a closed interval and differentiable inside it, then some interior point has derivative equal to the interval's average rate of change. It links secant slopes to tangent slopes.

A function continuous on a closed interval must attain both a maximum and a minimum value somewhere on that interval. It guarantees extrema exist and justifies checking endpoints and critical points.

An approximation of a definite integral by summing f evaluated at sample points times subinterval widths. Left, right, and midpoint versions differ in sample placement; trapezoidal sums average adjacent endpoint values.

The limit of Riemann sums over an interval, interpreted as signed area between a graph and the axis. It accumulates net change and is evaluated using antiderivatives via the Fundamental Theorem.

Connects differentiation and integration: the integral of a rate gives net change, and the derivative of an accumulation function recovers the integrand. It provides the main tool for evaluating definite integrals.

A function whose derivative is the given function. Antiderivatives of a single function differ only by an additive constant, which is why indefinite integrals include +C.

A technique reversing the chain rule by renaming an inner expression u and rewriting the integral and differential in terms of u. For definite integrals, the limits must also be converted.

The rule ∫u dv = uv − ∫v du, derived from the product rule, used to integrate products. Choosing u and dv well, often via the LIATE guideline, simplifies the remaining integral.

An equation involving an unknown function and its derivatives. Its solutions are families of functions; an initial condition selects one particular solution.

A solution method for differential equations that rearranges all terms in one variable to one side and all terms in the other variable to the other side, then integrates both sides.

A grid of short segments whose slopes equal the differential equation's prescribed value at each point. It visualizes solution curves without solving the equation.

A numerical technique approximating a solution curve by repeatedly stepping along tangent-line directions from the slope field. Smaller step sizes generally improve accuracy.

A model where a quantity's rate of change is proportional to its current size, giving y = y₀e^(kt). Positive k yields growth, negative k decay, and the constant k sets the rate.

The region's area computed as the integral of the upper curve minus the lower curve over the interval where they bound it. Integrating with respect to y handles regions better described by left and right functions.

A volume found by integrating cross-sectional areas of washers, π(R² − r²), when a region is revolved about an axis. The radii are distances from the axis to the outer and inner curves.

The integral of a function over an interval divided by the interval's length, giving the constant height with the same accumulated area. It equals some function value for continuous functions.

A pair of equations expressing x and y separately in terms of a third variable, the parameter, tracing a curve as it varies. Slope is dy/dx = (dy/dt)/(dx/dt).

A system locating points by distance r from the origin and angle θ from a reference direction. Curves like circles and roses have simple polar equations, and area uses ½∫r² dθ.

A function whose output is a vector, typically written with component functions of a parameter. Its derivative is taken componentwise and gives a velocity vector tangent to the path.

The length of a curve, computed by integrating the square root of one plus the squared slope, or the analogous parametric and polar formulas. It sums infinitely many tiny segment lengths.

An ordered list of numbers indexed by positive integers. A sequence converges when its terms approach a single finite value as the index grows without bound.

The sum of the terms of a sequence, understood as the limit of its partial sums. The series converges when that sequence of partial sums approaches a finite value.

A series whose consecutive terms share a constant ratio r. It converges to a/(1−r) exactly when |r| is less than 1, and diverges otherwise.

A convergence test examining the limit of the absolute ratio of consecutive terms. A limit below 1 gives absolute convergence, above 1 divergence, and exactly 1 is inconclusive.

A power series representing a function using its derivatives at a single center point. Truncating it gives Taylor polynomials that approximate the function near that center.

A Taylor series centered at zero, expressing a function as a power series in x. Common examples include the series for e^x, sin x, and 1/(1−x).

The distance from a power series' center within which the series converges absolutely. Behavior at the two endpoints must be checked separately because convergence there varies.

A convergence test for series whose terms alternate in sign, decrease steadily in size, and approach zero. Under these conditions the series converges, and the truncation error is bounded by the next term.