AP Calculus BC Vocabulary

Absolute convergence

A series converges absolutely if the series of the absolute values of its terms also converges. Absolute convergence implies convergence.

Example: "The series was shown to have          because the sum of the absolute values of its terms was finite."

Absolute extrema

The largest (absolute maximum) or smallest (absolute minimum) value of a function over its entire domain or a closed interval.

Similar definitions: global extrema, global maximum, global minimum

Example: "To find the          on a closed interval, evaluate the function at all critical points and endpoints."

Absolute maximum

The greatest value of a function on a given interval or its entire domain. On a closed interval, it must occur at a critical point or an endpoint.

Example: "The          of f(x) = -x^2 + 4x on [0, 5] is found by comparing f at critical points and endpoints."

Absolute minimum

The smallest value of a function on a given interval or its entire domain. On a closed interval, it must occur at a critical point or an endpoint.

Example: "The          of f(x) = x^2 on [-3, 2] is f(0) = 0."

Absolute value function

The function f(x) = |x|, which returns the non-negative magnitude of x. It is continuous everywhere but not differentiable at x = 0.

Example: "The          has a sharp corner at x = 0, making it a classic example of a continuous but non-differentiable point."

Absolute value inequality

An inequality involving |f(x)|, often used to express convergence conditions like |x - a| < R for power series.

Example: "The power series converges when |x - 3| < 5, which is an          equivalent to -2 < x < 8."

Acceleration

The rate of change of velocity with respect to time; the second derivative of the position function.

Example: "If position is s(t) = t^3 - 6t, then          is a(t) = 6t."

Acceleration function

The function a(t) = v'(t) = s''(t) that describes how velocity changes over time.

Example: "If v(t) = 3t^2 - 12, then the          is a(t) = 6t."

Acceleration vector

For a particle with position r(t) =

Example: "The          for r(t) =

Accumulation function

A function defined as F(x) = integral from a to x of f(t) dt, representing the accumulated area under f from a to x.

Example: "The          F(x) = integral from 0 to x of cos(t) dt gives the net area under cos(t) from 0 to x."

Accumulation problem

A problem involving the integral of a rate function to find total accumulated quantity, such as total water in a tank or total distance traveled.

Example: "An          might ask for the total amount of water that flows into a tank given a rate function r(t)."

Additive property of integrals

The integral from a to c equals the integral from a to b plus the integral from b to c, for any b between a and c.

Example: "By the         , integral from 0 to 5 of f = integral from 0 to 2 of f + integral from 2 to 5 of f."

Algebraic manipulation of series

Adding, subtracting, or multiplying convergent series term by term, and multiplying by constants, to create new series.

Example: "Using         , if sum an = 3 and sum bn = 5, then sum (2an + bn) = 11."

Alternating harmonic series

The series 1 - 1/2 + 1/3 - 1/4 + ..., which converges conditionally to ln(2) by the Alternating Series Test.

Example: "The          is the classic example of conditional convergence: it converges, but its absolute value series (the harmonic series) diverges."

Alternating series

A series whose terms alternate in sign, typically written as the sum of (-1)^n * a_n where a_n > 0.

Example: "The series 1 - 1/2 + 1/3 - 1/4 + … is an          that converges by the Alternating Series Test."

Alternating Series Error Bound

For a convergent alternating series, the absolute error from using a partial sum is at most the absolute value of the first omitted term.

Example: "Using the         , the error from approximating the sum with the first 5 terms is less than the 6th term."

Alternating series remainder

For a convergent alternating series satisfying the Alternating Series Test, the error |S - S_n| is bounded by |a_{n+1}|, the absolute value of the first omitted term.

Example: "The          for the series approximation using 10 terms is at most |a_{11}|."

Alternating Series Test

An alternating series converges if the absolute values of its terms decrease monotonically to zero.

Similar definitions: Leibniz's test

Example: "By the         , the series converges because its terms decrease in absolute value and approach zero."

Alternating sign pattern

A pattern of alternating positive and negative terms, typically expressed using (-1)^n or (-1)^(n+1) in series.

Example: "The          in the Maclaurin series for cos(x) gives 1 - x^2/2! + x^4/4! - x^6/6! + …"

Angle of inclination

The angle that the tangent line to a curve makes with the positive x-axis, related to the slope by tan(alpha) = dy/dx.

Example: "If the slope of the tangent is 1, the          is pi/4 or 45 degrees."

Antiderivative

A function F(x) whose derivative equals f(x). If F'(x) = f(x), then F is an antiderivative of f.

Similar definitions: indefinite integral, primitive function

Example: "Since the derivative of sin(x) is cos(x), sin(x) is an          of cos(x)."

Antiderivative rules

The collection of basic integration formulas: power rule, exponential rule, logarithmic rule, and trigonometric antiderivatives.

Example: "         include integral of sec^2(x) dx = tan(x) + C and integral of e^x dx = e^x + C."

Antidifferentiation

The process of finding an antiderivative; the reverse of differentiation. Also called integration.

Example: "         of cos(x) yields sin(x) + C."

Approximation by differentials

Using dy = f'(x)dx to estimate small changes in f: f(x + dx) is approximately f(x) + f'(x)dx.

Example: "        : estimate cube root of 8.1 as 2 + (1/12)(0.1) = 2.00833…"

Approximation by Taylor polynomial

Using a Taylor polynomial P_n(x) to estimate f(x) near the center a, with accuracy improving as n increases.

Example: "        : e^(0.1) is approximately 1 + 0.1 + 0.005 + 0.000167 = 1.10517 using a 3rd degree polynomial."

Arc length

The distance along a curve between two points, calculated by integrating sqrt(1 + (dy/dx)^2) dx for Cartesian curves, or sqrt((dx/dt)^2 + (dy/dt)^2) dt for parametric curves.

Example: "The          of the curve y = x^2 from x = 0 to x = 1 is found by integrating sqrt(1 + 4x^2)."

Arc length in parametric form

The length of a parametric curve (x(t), y(t)) from t = a to t = b, given by the integral of sqrt((dx/dt)^2 + (dy/dt)^2) dt.

Example: "The          for x = cos(t), y = sin(t) from 0 to 2pi equals 2pi, the circumference of a unit circle."

Arc length in polar form

The length of a polar curve r = f(theta) from theta = a to theta = b, given by the integral of sqrt(r^2 + (dr/dtheta)^2) d(theta).

Example: "To find the          of r = 1 + cos(theta), integrate sqrt((1+cos(theta))^2 + sin^2(theta)) d(theta)."

Area between curves

The area enclosed between two curves, found by integrating the absolute difference of the functions over the interval where one is above the other.

Example: "The          y = x and y = x^2 from x = 0 to x = 1 is the integral of (x - x^2) dx."

Area between polar curves

The area between two polar curves r1 = f(theta) and r2 = g(theta) is (1/2) integral of (r1^2 - r2^2) d(theta) over the appropriate interval.

Example: "The          r = 2 and r = 2(1+cos(theta)) requires finding the angles of intersection first."

Area in polar coordinates

The area enclosed by a polar curve r = f(theta), computed as (1/2) times the integral of r^2 d(theta) over the desired interval.

Example: "The          enclosed by r = 2cos(theta) is found using (1/2) integral of (2cos(theta))^2 d(theta)."

Area of a region

The total positive area of a region in the plane, found using definite integrals, possibly requiring multiple integrals for complex regions.

Example: "To find the          bounded by y = x^2 and y = x, set up the integral of (x - x^2) from 0 to 1."

Area under a curve

The region between a function's graph and the x-axis over a given interval, computed using a definite integral.

Example: "The          f(x) = x^2 from 0 to 3 equals the definite integral of x^2 dx from 0 to 3."

Area using parametric equations

The area under a parametric curve y = g(t), x = f(t) from t = a to t = b is the integral of g(t)*f'(t) dt.

Example: "The          enclosed by x = cos(t), y = sin(t) from 0 to 2pi is the integral of sin(t)*(-sin(t)) dt = pi."

Area with respect to y

Computing area by integrating with respect to y, using horizontal rectangles: A = integral from c to d of [right(y) - left(y)] dy.

Example: "Finding the          between x = y^2 and x = y + 2 is easier by integrating with respect to y."

Asymptote

A line that a graph approaches but never reaches. Types include horizontal, vertical, and oblique (slant) asymptotes.

Example: "The function f(x) = (x+1)/(x-2) has a vertical          at x = 2 and a horizontal one at y = 1."

Average acceleration

The change in velocity divided by the change in time: [v(b) - v(a)]/(b - a).

Example: "The          from t = 1 to t = 5 is (v(5) - v(1))/4."

Average rate of change

The change in a function's output divided by the change in input over an interval: [f(b) - f(a)] / (b - a). Geometrically, it is the slope of the secant line.

Example: "The          of f(x) = x^2 from x = 1 to x = 3 is (9 - 1)/(3 - 1) = 4."

Average value of a function

The mean value of a continuous function on [a, b], calculated as (1/(b-a)) times the integral of f(x) from a to b.

Example: "The          f(x) = x^2 on [0, 3] is (1/3) times the integral of x^2 from 0 to 3."

Average velocity

The displacement divided by the elapsed time: [s(b) - s(a)]/(b - a), which equals the average value of v(t) on [a, b].

Example: "The          from t = 0 to t = 4 is (s(4) - s(0))/4."

Binomial series

The Taylor series expansion of (1 + x)^k for any real number k, given by the sum of (k choose n) * x^n for n = 0 to infinity, converging for |x| < 1.

Example: "Using the         , (1 + x)^(1/2) can be expanded as 1 + (1/2)x - (1/8)x^2 + … for |x| < 1."

Bounded above

A sequence or function that has an upper bound M such that a_n

Bounded below

A sequence or function that has a lower bound m such that a_n >= m for all n.

Example: "The sequence 1/n is          by 0 since 1/n > 0 for all positive n."

Bounded monotonic sequence theorem

Every bounded monotonic sequence converges. This theorem is essential for proving convergence of many sequences.

Example: "By the         , a_n = (n+1)/n is increasing and bounded above by 2, so it converges."

Bounded sequence

A sequence {a_n} for which there exist real numbers M and m such that m

Calculating Taylor coefficients

The coefficient of (x-a)^n in a Taylor series is c_n = f^(n)(a)/n!, found by evaluating the nth derivative at the center.

Example: "         for e^x at a = 0: c_n = e^0/n! = 1/n! for all n."

Candidates Test

A method for finding absolute extrema on a closed interval by evaluating the function at all critical points and endpoints, then comparing values.

Similar definitions: Closed Interval Method

Example: "Using the         , evaluate f at x = 0, x = 2 (critical point), and x = 5 to find the absolute max and min on [0, 5]."

Cardioid

A heart-shaped polar curve of the form r = a(1 + cos(theta)) or r = a(1 + sin(theta)).

Example: "The          r = 1 + cos(theta) passes through the origin when theta = pi and has maximum r = 2 when theta = 0."

Carrying capacity

In logistic growth, the maximum population size L that an environment can sustain, where the growth rate equals zero.

Example: "In the logistic model dP/dt = 0.5P(1 - P/1000), the          is 1000."

Cartesian coordinates

The standard (x, y) coordinate system using perpendicular axes, as opposed to polar coordinates.

Example: "Converting from polar to         : the point (r, theta) becomes (rcos(theta), rsin(theta))."

Center of a power series

The value a in a power series sum of c_n(x - a)^n, around which the series is expanded.

Example: "The          for the Taylor series of ln(x) centered at 1 is a = 1."

Centroid

The geometric center of a plane region, with coordinates (x-bar, y-bar) found using integrals that average the x and y positions over the region.

Example: "The          of the region under y = x^2 from 0 to 1 is found by computing the moment integrals divided by the total area."

Chain rule

A differentiation rule stating that the derivative of a composite function f(g(x)) equals f'(g(x)) * g'(x).

Example: "Using the         , the derivative of sin(3x) is cos(3x) * 3."

Chain rule for integrals

When the FTC Part 1 involves a composite upper limit, d/dx[integral from a to g(x) of f(t) dt] = f(g(x)) * g'(x).

Example: "Using the         , d/dx[integral from 0 to x^2 of sin(t) dt] = sin(x^2) * 2x."

Chain rule with multiple variables

When differentiating a composition involving multiple intermediate functions, the chain rule extends by summing partial contributions from each variable.

Example: "Using the         , if z = f(x, y) and both x and y depend on t, then dz/dt = (partial f/partial x)(dx/dt) + (partial f/partial y)(dy/dt)."

Closed interval

An interval [a, b] that includes both endpoints. Important for the Extreme Value Theorem and definite integrals.

Example: "The Extreme Value Theorem requires f to be continuous on a         ."

Coefficient

In a power series sum of c_n(x-a)^n, c_n is the coefficient of the nth term. For a Taylor series, c_n = f^(n)(a)/n!.

Example: "The          of x^3 in the Maclaurin series for sin(x) is -1/6."

Common ratio

In a geometric series a + ar + ar^2 + ..., the constant ratio r between consecutive terms. The series converges if and only if |r| < 1.

Example: "The geometric series 3 + 3/4 + 3/16 + … has a          of 1/4."

Comparison of growth rates

The relative rates at which functions grow as x approaches infinity: logarithmic < polynomial < exponential < factorial. Used for limits and L'Hopital's Rule.

Example: "By         , lim(x->infinity) x^100/e^x = 0 because exponential growth dominates polynomial growth."

Comparison Test

A convergence test that compares a series to a known convergent or divergent series. If 0

Completing the square

An algebraic technique used to rewrite quadratic expressions in a form suitable for integration, especially with inverse trig antiderivatives.

Example: "To integrate 1/(x^2 + 4x + 8), first use          to rewrite the denominator as (x+2)^2 + 4."

Composite function

A function formed by applying one function to the output of another: (f o g)(x) = f(g(x)). Differentiated using the chain rule.

Example: "sin(x^2) is a          where f(x) = sin(x) and g(x) = x^2."

Concave down

A curve is concave down on an interval where f''(x) < 0, meaning the curve lies below its tangent lines and the slope is decreasing.

Example: "f(x) = -x^2 is          everywhere because f''(x) = -2 < 0."

Concave up

A curve is concave up on an interval where f''(x) > 0, meaning the curve lies above its tangent lines and the slope is increasing.

Example: "f(x) = x^2 is          everywhere because f''(x) = 2 > 0."

Concavity

A description of the curvature of a graph. A function is concave up where its second derivative is positive and concave down where its second derivative is negative.

Example: "The          of f(x) = x^3 changes from down to up at x = 0 because f''(x) = 6x changes sign there."

Concavity test

Using the second derivative to determine concavity: f''(x) > 0 means concave up, f''(x) < 0 means concave down.

Example: "The          shows f(x) = x^4 is concave up everywhere except at x = 0 where f'' = 0."

Conditional convergence

A series that converges but does not converge absolutely; the series of absolute values diverges while the original series converges.

Example: "The alternating harmonic series exhibits          because it converges, but the harmonic series diverges."

Connected region

A region in the plane that cannot be divided into separate pieces, important for path-based area calculations.

Example: "When finding area between curves, identify each          separately if the curves intersect multiple times."

Constant function

A function f(x) = c for all x, whose derivative is always 0 and whose integral from a to b is c(b - a).

Example: "The derivative of a          is zero, which is why antiderivatives differ only by a constant."

Constant multiple rule

The derivative of a constant times a function equals the constant times the derivative: d/dx[c*f(x)] = c*f'(x).

Example: "By the         , the derivative of 5x^3 is 5 * 3x^2 = 15x^2."

Constant of integration

The arbitrary constant C added to an indefinite integral, representing the family of all antiderivatives differing by a constant.

Example: "The integral of 2x dx is x^2 + C, where C is the         ."

Constant term of Taylor series

The first term of a Taylor series, equal to f(a), the value of the function at the center of expansion.

Example: "The          for the Maclaurin series of e^x is e^0 = 1."

Continuity

A function is continuous at a point if the limit as x approaches that point equals the function's value there. No breaks, jumps, or holes exist at the point.

Example: "To verify          at x = 2, check that the limit of f(x) as x approaches 2 equals f(2)."

Continuity on an interval

A function is continuous on an interval if it is continuous at every point in the interval. On a closed interval [a,b], it must also be right-continuous at a and left-continuous at b.

Example: "The function f(x) = sqrt(x) has          [0, infinity) because it is defined and continuous at every point in that interval."

Continuous function properties

Continuous functions on closed intervals satisfy the EVT, IVT, and MVT. They can be integrated and their integrals define accumulation functions.

Example: "The          guarantee that a function attains all values between its max and min on a closed interval."

Continuous on a closed interval

A function that is continuous on (a, b) and is right-continuous at a and left-continuous at b.

Example: "For the EVT to apply, f must be          [a, b]."

Converge conditionally

A series converges conditionally if it converges but does not converge absolutely.

Example: "The alternating harmonic series         s because it converges, but the harmonic series (its absolute version) diverges."

Convergence

The property of a sequence, series, or improper integral having a finite limiting value.

Example: "The          of an infinite series means its partial sums approach a finite number."

Convergence at endpoints

After finding the radius of convergence of a power series, each endpoint must be tested separately by substitution to determine if the series converges there.

Example: "         of sum x^n/n: at x = 1 it becomes the harmonic series (diverges), at x = -1 it becomes the alternating harmonic (converges)."

Convergence of a sequence

A sequence {a_n} converges to L if for every epsilon > 0 there exists N such that |a_n - L| < epsilon for all n > N.

Example: "The          {(2n+1)/(3n-1)} is to L = 2/3 as n approaches infinity."

Convergence of geometric series

The geometric series a + ar + ar^2 + ... converges if and only if |r| < 1, with sum a/(1-r).

Example: "        : 10 + 10(0.9) + 10(0.81) + … converges to 10/(1-0.9) = 100."

Convergence of p-series

The p-series sum of 1/n^p converges when p > 1 and diverges when p

Convergence tests

Methods for determining whether a series converges or diverges, including the Ratio, Root, Comparison, Limit Comparison, Integral, Alternating Series, and Divergence Tests.

Example: "Choosing the right          depends on the structure of the series terms."

Convergent improper integral

An improper integral whose limit exists and is finite.

Example: "The integral of 1/x^2 from 1 to infinity is a          equal to 1."

Convergent sequence

A sequence whose terms approach a finite limit L as n approaches infinity.

Example: "The sequence a_n = 1/n is a          because its terms approach 0 as n increases."

Convergent series

An infinite series whose partial sums approach a finite limit.

Example: "The geometric series 1 + 1/2 + 1/4 + 1/8 + … is a          with a sum of 2."

Converging to a function

A power series converges to the function f(x) on an interval if the Taylor remainder R_n(x) approaches 0 as n approaches infinity.

Example: "The Maclaurin series for sin(x)         s sin(x) for all real x."

Coordinate conversion

Converting between polar and rectangular coordinates using x = r*cos(theta), y = r*sin(theta) and r = sqrt(x^2+y^2), theta = arctan(y/x).

Example: "         allows us to graph polar equations in Cartesian form and vice versa."

Critical number

A value x = c in the domain of f where f'(c) = 0 or f'(c) does not exist.

Similar definitions: critical point, critical value

Example: "For f(x) = |x|, x = 0 is a          because f'(0) does not exist."

Critical point

A point in the domain of a function where the derivative is zero or undefined.

Similar definitions: critical number, critical value

Example: "For f(x) = x^3 - 3x, the         s occur at x = -1 and x = 1 where f'(x) = 0."

Cross sections

Slices of a solid perpendicular to an axis, used to compute volumes by integrating the area of each cross-sectional shape.

Example: "The volume of the solid with square          perpendicular to the x-axis is found by integrating the square of the side length."

Cross-sectional area

The area A(x) of a slice of a solid perpendicular to an axis at position x, used in the formula V = integral of A(x) dx.

Example: "If the          at position x is a square with side length (1-x), then A(x) = (1-x)^2."

Curvature

A measure of how sharply a curve bends at a given point. For y = f(x), curvature = |f''(x)| / (1 + (f'(x))^2)^(3/2).

Example: "The          of a circle of radius R is constant and equal to 1/R."

Curve defined by vector-valued function

A vector-valued function r(t) =

Example: "The          r(t) =

Curve sketching

The process of analyzing a function's domain, intercepts, symmetry, asymptotes, critical points, intervals of increase/decrease, concavity, and inflection points to draw its graph.

Example: "When         , use the first and second derivatives to determine the shape and key features of the graph."

Cusp

A sharp point on a curve where the function is continuous but not differentiable, and the tangent line is vertical or changes direction abruptly.

Example: "The function f(x) = x^(2/3) has a          at x = 0 because the derivative approaches infinity from both sides."

Decay constant

The constant k in exponential decay y = Ce^(kt) where k < 0, determining the rate at which the quantity decreases.

Example: "If a substance has a          of k = -0.05, it decays 5% per unit time approximately."

Decreasing function

A function where f(x1) > f(x2) whenever x1 < x2 on an interval. Equivalently, f'(x) < 0 on that interval.

Example: "f(x) = -x^2 is a          on (0, infinity) because f'(x) = -2x < 0 for x > 0."

Definite integral

The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis.

Example: "The          of f(x) = 2x from 0 to 3 equals 9."

Definite integral as net area

The definite integral gives the net (signed) area: regions above the x-axis contribute positively and regions below contribute negatively.

Example: "The          of sin(x) from 0 to 2pi is 0 because the positive and negative areas cancel."