AP Calculus glossary

A series converges absolutely when the series formed by taking the absolute value of every term still converges. Absolute convergence is stronger than ordinary convergence and always implies it.

Acceleration is the derivative of velocity with respect to time, and therefore the second derivative of position. Comparing its sign with the sign of velocity tells you whether the object is speeding up or slowing down.

An accumulation function is a definite integral whose upper bound is the variable, so it defines a new function measuring how much has accumulated from a fixed starting point. Its derivative is the integrand evaluated at the upper bound.

For an alternating series that satisfies the alternating series test, the error in using a partial sum is less than the absolute value of the first omitted term. It is the simplest error bound in the course.

The alternating series test says that a series whose terms alternate in sign converges if the absolute values of its terms decrease and approach zero. Both conditions are needed, and together they are enough.

An antiderivative of a function is any function whose derivative equals it. Because the derivative of a constant is zero, a function has infinitely many antiderivatives, all differing from one another by a constant.

Arc length is the distance measured along a curve rather than straight across. It comes from adding up infinitesimal hypotenuses, which produces an integral of the square root of one plus the square of the derivative.

The average rate of change of a function over an interval is the change in output divided by the change in input, which is the slope of the secant line joining the endpoints. It needs no calculus, unlike the instantaneous rate of change.

The average value of a function on an interval is the definite integral over that interval divided by the length of the interval. It is the constant height a rectangle would need to have the same area as the region under the curve.

The candidates test finds absolute extrema on a closed interval by evaluating the function at every critical point and at both endpoints, then comparing the resulting values. The largest is the absolute maximum and the smallest is the absolute minimum.

The direct comparison test settles a series of positive terms by bounding it against a series whose behaviour is known. If it is term by term smaller than a convergent series it converges, and if it is larger than a divergent series it diverges.

A composite function is one function evaluated inside another, so the output of the inner function becomes the input of the outer one. Spotting a composition is what tells you to use the chain rule.

Concavity describes which way a graph bends. A graph is concave up where the second derivative is positive and the slope is increasing, and concave down where the second derivative is negative and the slope is decreasing.

A series converges conditionally when it converges as written but the series of its absolute values diverges. The cancellation between positive and negative terms is what makes it work, so the convergence depends on the order of the terms.

The constant multiple rule says a constant factor passes straight through a derivative, and the sum rule says the derivative of a sum is the sum of the derivatives. Together they make differentiation linear.

The constant of integration is the arbitrary constant added to an indefinite integral. It is required because differentiating any constant gives zero, so a function has infinitely many antiderivatives that differ only by a constant.

A function is continuous at a point when three things hold: the function is defined there, the limit exists there, and the limit equals the function value. Informally, you can draw the graph through that point without lifting your pencil.

A series converges when its sequence of partial sums approaches a finite limit, and diverges when it does not. Divergence covers both partial sums that grow without bound and ones that oscillate without settling.

A corner is a point where a graph changes direction abruptly, with left and right derivatives that are different finite numbers. The function is continuous there but not differentiable, because no single tangent slope exists.

A critical point of a function is an interior point of its domain where the derivative is either zero or undefined. Every local maximum and minimum occurs at a critical point, though not every critical point is an extremum.

A cross-section is the two-dimensional shape you get by slicing a solid perpendicular to an axis. If you know the area of each cross-section as a function of position, integrating that area function along the axis gives the volume.

A cusp is a sharp point on a graph where the one-sided slopes grow without bound in opposite directions. The function stays continuous there but is not differentiable, because the tangent slope is not a finite number.

A definite integral is a number defined as the limit of Riemann sums as the subintervals shrink to zero width. It represents the net signed accumulation of a quantity between two bounds, with area below the axis counting as negative.

The derivative of a function at a point is its instantaneous rate of change there, defined as the limit of the difference quotient as the gap shrinks to zero. Geometrically it is the slope of the tangent line at that point.

The difference quotient is the slope of the secant line joining two points on a curve, written as the change in output over the change in input. Taking its limit as the gap shrinks to zero produces the derivative.

A function is differentiable at a point when the limit defining the derivative exists there, which requires the graph to be locally smooth with a single well-defined tangent slope. Differentiability always implies continuity, but continuity does not imply differentiability.

The differential dy is the change in output along the tangent line corresponding to a small change dx in the input. It approximates the true change in the function and is the notation that makes substitution and separable equations work.

A differential equation is an equation that relates an unknown function to its own derivatives. Its solution is a function rather than a number, and there is usually a whole family of them until an initial condition picks one out.

Direct substitution evaluates a limit by putting the value straight into the function. It is valid exactly when the function is continuous at that point, which makes it the first thing to try on any limit.

The disk method computes the volume of a solid of revolution by slicing it into circular disks perpendicular to the axis of rotation and integrating the area of each. It applies when the region being rotated touches the axis, leaving no hole.

Displacement is the net change in position over a time interval, computed by integrating velocity. Total distance travelled integrates the absolute value of velocity instead, so it counts backtracking as additional distance.

The domain of a function is the set of inputs for which it produces a value. Domain restrictions determine where a function can be continuous or differentiable, and they are what make some answers to calculus problems invalid.

The epsilon-delta definition states that a limit equals L if, for every tolerance around L, there is a distance around the input point that keeps all outputs within that tolerance. It is the precise version of the informal idea of approaching a value.

Euler's method approximates the solution of a differential equation by starting at a known point and repeatedly stepping along the tangent line for a small increment. Each step uses the differential equation to compute the slope at the current point.

Exponential growth describes a quantity whose rate of change is proportional to its current amount. The differential equation says the derivative equals a constant times the quantity, and its solution is a constant multiple of e raised to that constant times time.

The Extreme Value Theorem says that a function continuous on a closed, bounded interval must attain both an absolute maximum and an absolute minimum somewhere on that interval. It guarantees the extrema exist without saying where they are.

An extremum is a maximum or minimum value of a function. A local extremum is largest or smallest compared only to nearby points, while an absolute extremum is largest or smallest across the entire domain or interval under consideration.

The first derivative test classifies a critical point by examining the sign of the derivative on both sides. A change from positive to negative gives a local maximum, negative to positive gives a local minimum, and no sign change means neither.

Part 1 of the Fundamental Theorem of Calculus says that differentiating an integral with a variable upper bound returns the integrand evaluated at that bound. It is the statement that differentiation and integration are inverse operations.

Part 2 of the Fundamental Theorem of Calculus says a definite integral equals any antiderivative evaluated at the upper bound minus the same antiderivative at the lower bound. It is what makes definite integrals computable without Riemann sums.

The general solution of a differential equation is the entire family of functions that satisfy it, expressed with an arbitrary constant. Every specific solution is obtained by choosing a value for that constant.

A geometric series is one where each term is a fixed multiple of the previous term. It converges exactly when the absolute value of that common ratio is less than one, and then its sum is the first term divided by one minus the ratio.

The harmonic series is the sum of the reciprocals of the positive integers. Its terms shrink to zero, yet the series diverges, which makes it the standard proof that shrinking terms are not enough for convergence.

A higher-order derivative is what you get by differentiating repeatedly. The second derivative is the derivative of the derivative and measures how the rate of change is itself changing, which is what determines concavity and acceleration.

A horizontal asymptote is a horizontal line that a graph approaches as the input grows without bound in either direction. A function has one exactly when its limit at infinity or at negative infinity is a finite number.

An implicit function is one defined by an equation relating the variables without the output isolated, such as the equation of a circle. Its derivative is found by differentiating both sides and solving for the derivative rather than by rearranging first.

An improper integral is one with an infinite bound of integration or an integrand that grows without bound somewhere on the interval. It is defined as a limit of ordinary definite integrals, and it converges if that limit is finite and diverges otherwise.

A function is increasing on an interval where its derivative is positive and decreasing where its derivative is negative. Increasing behaviour is a property of an interval, not of a single point.

An indefinite integral is the collection of all antiderivatives of a function, written with an integral sign and no limits. Its result is a family of functions plus a constant of integration, which is what distinguishes it from a definite integral.

An indeterminate form is a limit expression whose value cannot be determined from its form alone, such as zero over zero or infinity minus infinity. It is not an answer but a signal that you need algebra or L'Hopital's rule to resolve the limit.

An infinite discontinuity occurs where at least one one-sided limit is infinite, so the function grows without bound near the point. It always corresponds to a vertical asymptote and can never be removed by redefining a value.

An infinite limit means the function grows without bound as the input approaches a point, written as equal to positive or negative infinity. Because infinity is not a number, an infinite limit is a description of how the limit fails to exist, not a value.

An inflection point is a point on a graph where the concavity changes from up to down or from down to up. Finding one requires the second derivative to change sign there, which is a stronger condition than simply equalling zero.

The integral test says that for a positive, decreasing, continuous function, the series of its values and the improper integral of the function either both converge or both diverge. It is what proves the p-series rule.

The integrand is the function being integrated, written between the integral sign and the differential. Identifying it clearly is what tells you which integration technique applies.

The Intermediate Value Theorem says that if a function is continuous on a closed interval, it attains every value between the two endpoint values somewhere on that interval. It is the standard tool for proving a solution exists without finding it.

The interval of convergence is the complete set of inputs for which a power series converges. You find the radius with the ratio test, then test each endpoint separately, because the ratio test is inconclusive exactly there.

The derivative of an inverse function at a point equals the reciprocal of the original function's derivative, evaluated at the matching point. You never need a formula for the inverse itself, only the point correspondence.

A jump discontinuity occurs at a point where both one-sided limits exist but disagree, so the graph steps abruptly from one value to another. Because the one-sided limits differ, the two-sided limit does not exist and no redefinition can repair the break.

The Lagrange error bound limits how far a Taylor polynomial can be from the function it approximates. It uses the maximum size of the next derivative on the interval between the centre and the point of interest.

Leibniz notation writes a derivative as dy over dx, which names both the changing quantity and the variable it changes with respect to. That explicitness makes it the standard notation for related rates, implicit differentiation, and any problem with several variables.

A limit is the single value a function approaches as its input approaches a given number. The limit describes where the function is heading, not where it lands, so a limit can exist at a point where the function is undefined.

A limit at infinity is the value a function approaches as its input grows without bound in the positive or negative direction. It describes the end behavior of the graph, and a finite limit at infinity is exactly what produces a horizontal asymptote.

The limit comparison test compares two series of positive terms by taking the limit of their ratio. If that limit is finite and positive, both series converge or both diverge together.

The limit laws let you distribute a limit across sums, differences, products, quotients, and powers. Each law requires the individual limits to exist, and the quotient law additionally requires the denominator's limit to be nonzero.

Linearization approximates a function near a point by using its tangent line there. Because a smooth curve and its tangent line are nearly identical close to the point of tangency, the line gives a good estimate for inputs near that point.

Logarithmic differentiation is a technique where you take the natural logarithm of both sides of an equation before differentiating implicitly. It converts products into sums and exponents into coefficients, which is what makes a variable base raised to a variable power differentiable.

Logistic growth models a quantity that grows nearly exponentially when small but levels off as it approaches a maximum called the carrying capacity. Its growth rate is fastest when the quantity is exactly half the carrying capacity.

A Maclaurin series is a Taylor series centred at zero. It is not a different object, just the most common special case, and the standard ones for the exponential, sine, and cosine functions are worth knowing by heart.

The Mean Value Theorem says that if a function is continuous on a closed interval and differentiable on the open interval, then somewhere inside there is a point where the instantaneous rate of change equals the average rate of change across the whole interval.

The net change theorem says that integrating a rate of change over an interval gives the net change in the underlying quantity. It is the interpretation of the Fundamental Theorem of Calculus that applied problems actually use.

The normal line at a point on a curve is the line through that point perpendicular to the tangent line. Its slope is the negative reciprocal of the derivative there, provided the derivative is not zero.

The nth term test says that if the terms of a series do not approach zero, the series diverges. It can only ever prove divergence, because terms approaching zero says nothing about whether the series converges.

The objective function in an optimization problem is the quantity being maximized or minimized. The constraint is a separate equation relating the variables, used to rewrite the objective in terms of a single variable before differentiating.

A one-sided limit is the value a function approaches as the input comes in from one direction only. The left-hand limit uses inputs below the point and the right-hand limit uses inputs above it. The two-sided limit exists exactly when both agree.

A p-series is the sum of the reciprocals of the positive integers raised to a fixed power p. It converges when p is greater than one and diverges when p is less than or equal to one.

Parametric equations define the coordinates of a point separately as functions of a third variable called the parameter, usually time. They describe not just the shape of a curve but the direction and speed with which it is traced.

Partial fraction decomposition rewrites a rational function as a sum of simpler fractions with linear denominators. Each piece then integrates to a natural logarithm, which is what makes otherwise impossible rational integrals routine.

The nth partial sum of a series is the total of its first n terms. The series converges precisely when this sequence of partial sums approaches a finite limit, which is what makes an infinite sum meaningful.

A particular solution is the one member of a differential equation's solution family that passes through a specified point. The initial condition is what determines the arbitrary constant and selects that single curve.

Polar coordinates locate a point by its distance from the origin and the angle it makes with the positive horizontal axis. A polar curve expresses that distance as a function of the angle, which makes circles and roses far simpler to describe than in rectangular form.

The power rule says that to differentiate a power of the variable, multiply by the exponent and then reduce the exponent by one. It holds for every real exponent, including negative and fractional ones.

A power series is an infinite sum of constant multiples of powers of the variable minus a fixed centre. It converges for the centre always, and generally on an interval around it whose half-width is the radius of convergence.

The product rule differentiates a product of two functions: multiply the first by the derivative of the second, then add the second times the derivative of the first. The derivative of a product is never the product of the derivatives.

Definite integrals obey a small set of rules: swapping the bounds reverses the sign, equal bounds give zero, constants factor out, sums split apart, and an integral can be broken at any interior point.

The quotient rule differentiates a fraction: the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the denominator squared. The order matters because of the subtraction.

The radius of convergence is the distance from the centre of a power series to the edge of the region where it converges. It is found by applying the ratio test and solving the resulting inequality for the variable.

A related rate is the rate of change of one quantity expressed in terms of the rate of change of another that it depends on. These problems link the quantities with an equation, then differentiate both sides with respect to time.

A removable discontinuity is a point where the limit exists but the function either is not defined there or is defined to be a different value. It appears as a hole in the graph, and redefining the single point would make the function continuous.

A Riemann sum approximates a definite integral by slicing the interval into subintervals and adding up the area of a rectangle on each one. As the number of subintervals grows without bound, the sum approaches the exact value of the integral.

Rolle's Theorem says that if a function is continuous on a closed interval, differentiable inside it, and takes the same value at both endpoints, then there is an interior point where the derivative is zero. It is the Mean Value Theorem with equal endpoint values.

The root test takes the limit of the nth root of the absolute value of the terms. A limit less than one means the series converges absolutely, greater than one means it diverges, and exactly one means the test tells you nothing.

A secant line is a straight line through two points on a curve. Its slope equals the average rate of change between those points, and as the two points slide together the secant line approaches the tangent line.

The second derivative test classifies a critical point by evaluating the second derivative there. A negative value means a local maximum, a positive value means a local minimum, and a value of zero makes the test inconclusive.

A sequence is an ordered list of numbers indexed by the positive integers. A series is the sum of the terms of a sequence, so a sequence is a list while a series is a total.

A series is the sum of the terms of a sequence, often infinitely many. It converges when the sequence of its partial sums approaches a finite limit, and that limit is defined to be the sum of the series.

The shell method computes a volume of revolution by slicing the region parallel to the axis of rotation and treating each slice as a thin cylindrical shell. Its integrand is two pi times the radius times the height.

A slope field is a grid of short line segments, each drawn with the slope that a differential equation assigns to that point. It shows the shape of every solution curve at once without solving the equation.

A solid of revolution is the three-dimensional shape swept out when a plane region is rotated about a line. Its volume is computed by integrating cross-sectional areas, using the disk method, the washer method, or the shell method.

Speed is the absolute value of velocity, so it is never negative and carries no direction information. Speed increases exactly when velocity and acceleration have the same sign.

The squeeze theorem says that if a function is trapped between two others near a point, and those two share the same limit there, the trapped function must have that limit too. It is the standard way to evaluate limits that resist algebra, such as x squared times sine of one over x.

The tangent line to a curve at a point is the straight line through that point whose slope equals the derivative there. It is the best linear approximation of the curve near that point.

A Taylor polynomial is the finite piece of a Taylor series obtained by stopping after a chosen degree. It approximates the function near the centre, and higher degree generally means a better approximation over a wider range.

A Taylor series represents a function as an infinite polynomial whose coefficients come from the function's derivatives at a single centre point. Each coefficient is a derivative at the centre divided by the factorial of its order.

The trapezoidal rule approximates a definite integral by joining consecutive points with straight segments and summing the resulting trapezoids. It equals the average of the left and right Riemann sums and is usually more accurate than either.

A vector-valued function assigns a vector to each value of a parameter, most often describing the position of a particle over time. Differentiating each component separately gives the velocity vector, and differentiating again gives acceleration.

Velocity is the derivative of position with respect to time. It is a signed quantity, so its sign tells you the direction of motion, and its absolute value is the speed.

A vertical asymptote is a vertical line that a graph approaches without ever reaching, occurring wherever at least one one-sided limit is infinite. For a rational function, these sit at the zeros of the denominator that remain after all common factors cancel.

A vertical tangent occurs at a point where the derivative grows without bound with the same sign from both sides, so the tangent line is vertical. The function remains continuous there but is not differentiable, since infinite slope is not a number.

The washer method computes the volume of a solid of revolution with a hole through it by subtracting the inner circle's area from the outer circle's area on every slice. It is used whenever the rotated region does not touch the axis of rotation.