Calculus AB glossary
Browse existing editorial definitions and examples, then follow deeper reading in the subject’s native course. Shared senses can belong to more than one subject. Historical source definitions and tables remain in the 242-entry Physics source glossary.
Calculus AB printables · Calculus AB interactive and source resources
106 of 106 editorial senses
Absolute extremum
Calculus AB · Calculus BC · Definition
An absolute extremum is the greatest or least function value on the domain or interval under discussion.
The comparison includes every allowed input, including included endpoints.
For example, x² on [−1,2] has absolute maximum 4 at x=2 and absolute minimum 0 at x=0.
An absolute extremum must be attained rather than merely approached.
Acceleration in calculus
Calculus AB · Calculus BC · Definition
Acceleration is the derivative of velocity with respect to time.
In one-dimensional motion, it is also the second derivative of position.
For example, s(t)=t² meters gives constant acceleration 2 meters per second squared.
A negative acceleration can increase speed when the velocity is also negative.
Accumulation function
Calculus AB · Calculus BC · Definition
An accumulation function uses a variable endpoint in a definite integral.
For example, A(x) defined as the integral of 2t from 0 to x equals x².
Where the integrand is continuous, the Fundamental Theorem of Calculus gives A′(x) equal to the integrand at x.
Variable endpoints that are themselves functions also require the chain rule.
Amplitude
Calculus AB · Calculus BC · Definition
The amplitude of a sinusoidal function is its maximum distance from the midline.
For y=A sin(Bx+C)+D, the amplitude is |A|.
For example, y=−3 cos x+2 has amplitude 3 and midline y=2.
Amplitude measures vertical variation rather than the interval required for one cycle.
Angle addition identity
Calculus AB · Calculus BC · Definition
Angle addition identities express a trigonometric function of a sum using the separate angles.
For example, sin(a+b)=sin a cos b+cos a sin b.
Using a=π/6 and b=π/3 gives sin(π/2)=1.
The addition and subtraction identities also help rewrite expressions for limits, derivatives, and integrals.
Antiderivative
Calculus AB · Calculus BC · Definition
An antiderivative of a function is another function whose derivative equals the original function on an interval.
Two antiderivatives on the same interval differ by a constant.
For example, x² is an antiderivative of 2x, and x²+7 is another.
An antiderivative is a function, whereas a definite integral is an accumulated value.
Antiderivative power rule
Calculus AB · Calculus BC · Definition
The antiderivative power rule integrates xⁿ to xⁿ⁺¹/(n+1)+C when n≠−1 on an interval where the powers are defined.
The exponent increases by one, and the result is divided by that new exponent.
For example, the indefinite integral of x² is x³/3+C.
The exceptional exponent −1 instead gives ln|x|+C on intervals excluding zero.
Average rate of change
Calculus AB · Calculus BC · Definition
The average rate of change is the change in output divided by the change in input over a nonzero interval.
For a function f, it is [f(b)−f(a)]/(b−a).
For example, a temperature rising from 10°C to 16°C in 3 hours has average rate 2°C per hour.
It describes the whole interval and need not equal the rate at either endpoint.
Average value of a function
Calculus AB · Calculus BC · Definition
The average value of an integrable function on [a,b], with a<b, is its integral divided by b−a.
It is the height of a constant function with the same signed accumulation over that interval.
For example, x² on [0,3] has average value 9/3=3.
This is an average of outputs, whereas an average rate of change compares endpoint outputs with input change.
Chain rule
Calculus AB · Calculus BC · Definition
The chain rule differentiates a composition by multiplying the outer derivative at the inner output by the inner derivative.
For differentiable functions, (f(g(x)))′=f′(g(x))g′(x).
For example, the derivative of sin(x²) is cos(x²)·2x.
Nested compositions require a factor for each successive layer.
Completing the square
Calculus AB · Calculus BC · Definition
Completing the square rewrites a quadratic using a squared linear expression plus a constant.
For a monic quadratic x²+bx+c, add and subtract (b/2)² to obtain (x+b/2)²+c−(b/2)².
For example, x²+4x+7 becomes (x+2)²+3.
The rewritten form can reveal a vertex or prepare an integral for substitution.
Composite function
Calculus AB · Calculus BC · Definition
A composite function applies one function to the output of another.
In f(g(x)), evaluate g first and then use its output as the input to f.
For example, if f(u)=u² and g(x)=x+1, then f(g(x))=(x+1)².
The composition is defined only where the inner output lies in the outer function’s domain.
Concavity
Calculus AB · Calculus BC · Definition
Concavity describes how a graph’s slope changes as the input increases.
A graph is concave up where its derivative increases and concave down where its derivative decreases.
For example, x² is concave up because its derivative 2x increases.
A positive second derivative establishes concave-up behavior on an interval, while a negative one establishes concave-down behavior.
Constant of integration
Calculus AB · Calculus BC · Definition
A constant of integration represents the arbitrary constant added when finding a family of antiderivatives.
It appears because differentiating a constant gives zero.
For example, every function x²+C has derivative 2x.
An initial condition may determine a particular value of C.
Constraint in optimization
Calculus AB · Calculus BC · Definition
A constraint restricts the quantities or choices permitted in a problem.
It can be an equation, an inequality, or a requirement from the physical context.
For example, a rectangle with perimeter 20 must satisfy 2l+2w=20 with positive side lengths.
Use constraints to determine the objective’s allowed domain before searching for extrema.
Continuity
Calculus AB · Calculus BC · Definition
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree.
At a domain endpoint, continuity uses the relevant one-sided approach.
For example, x² is continuous at 3 because both its value and its nearby limit are 9.
Continuity on an interval requires the appropriate condition at every input in that interval.
Critical point
Calculus AB · Calculus BC · Definition
A critical number is an input in the function’s domain where its derivative is zero or does not exist.
The corresponding point on the graph is called a critical point.
For example, x³ has a critical number at 0, but it has no local maximum or minimum there.
Critical numbers are candidates for extrema and need further analysis; inputs outside the domain are excluded.
Cross section
Calculus AB · Calculus BC · Definition
A cross section is the planar slice of a solid made by a specified cutting plane.
For volume by integration, its area must be expressed as a function of the position along the slicing direction.
For example, a square slice with side s(x) has area s(x)².
Integrating those areas over the allowed positions yields volume under the method’s assumptions.
Cylindrical shell method
Calculus AB · Definition
The cylindrical shell method computes volume by accumulating thin cylindrical layers parallel to a rotation axis.
The volume contribution is circumference times shell height times thickness.
For example, rotating a vertical strip at x>0 around the vertical axis gives a contribution 2πx·height·dx.
Choose the radius as a distance from the axis and check the shell height against the bounding region.
Definite integral
Calculus AB · Calculus BC · Definition
A definite integral accumulates a function over a specified interval as a limit of Riemann sums when that limit exists.
Its sign includes the function’s sign and the orientation of the limits.
For example, the integral of the constant −2 from 0 to 3 is −6, although the geometric area is 6.
In a rate model, the result has rate units multiplied by input units.
Derivative
Calculus AB · Calculus BC · Definition
A derivative gives the instantaneous rate of change of a function with respect to its input.
At a point, it is the finite limit of the difference quotient when that limit exists.
For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6.
A derivative function collects those rates across the inputs where the function is differentiable.
Difference quotient
Calculus AB · Calculus BC · Definition
A difference quotient divides a change in a function’s output by the associated nonzero input change.
The form [f(a+h)−f(a)]/h is an average rate over an interval of length h.
For example, for f(x)=x² it simplifies to 2a+h when h≠0.
Taking its limit as h approaches 0 defines the derivative at a if the finite limit exists.
Differentiability
Calculus AB · Calculus BC · Definition
Differentiability means that a function has a finite derivative at the point under discussion.
Differentiability at a point implies continuity there, but continuity alone does not imply differentiability.
For example, |x| is continuous at 0 but its left and right tangent slopes differ, so it is not differentiable there.
Corners, cusps, vertical tangents, and discontinuities can prevent a finite derivative.
Differential
Calculus AB · Calculus BC · Definition
For a differentiable function y=f(x), the differential dy is defined as f′(x) dx.
It represents the tangent-line estimate of the output change associated with an input change dx.
For example, y=x² at x=3 and dx=0.01 gives dy=6·0.01=0.06.
The actual change in y can differ from this linear estimate.
Differential equation
Calculus AB · Calculus BC · Definition
A differential equation relates an unknown function to one or more of its derivatives.
A solution is a function that makes the equation true throughout an appropriate interval.
For example, y′=2x has solutions y=x²+C.
An initial condition can select a particular member of a family of solutions.
Disc method
Calculus AB · Calculus BC · Definition
The disc method finds a solid’s volume using circular slices perpendicular to the rotation axis.
A slice with radius R has area πR².
For example, rotating the region under y=x on [0,1] around the horizontal axis gives volume equal to the integral of πx².
If the region leaves a hole around the axis, a washer rather than a full disc is needed.
Discontinuity
Calculus AB · Calculus BC · Definition
A discontinuity is a failure of continuity at an input being considered.
The obstruction can be a missing or mismatched value, unequal one-sided limits, oscillation, or unbounded behavior.
For example, assigning f(0)=5 while f(x)=x elsewhere creates a discontinuity at 0.
Identifying the obstruction determines whether changing one value can repair it.
Domain
Calculus AB · Calculus BC · Definition
The domain of a function is the set of inputs for which the function is defined.
An expression and its context can both restrict the allowed inputs.
For example, the real function f(x)=1/(x−2) has domain all real numbers except 2.
For a model of elapsed time, a sensible domain may also require t≥0.
Double angle identity
Calculus AB · Calculus BC · Definition
A double angle identity rewrites a trigonometric expression with twice an angle.
For example, cos(2θ)=2cos²θ−1 and sin(2θ)=2sin θ cos θ.
The cosine identity gives cos²θ=[1+cos(2θ)]/2, which can simplify a trigonometric integral.
These identities follow from the angle addition formulas.
Exponential growth and decay
Calculus AB · Calculus BC · Definition
An exponential growth or decay model makes a quantity’s instantaneous rate proportional to the quantity itself.
The differential equation y′=ky gives y=Ceᵏˣ, with positive k describing growth and negative k describing decay for positive y.
For example, a population with a constant relative growth rate doubles in equal time intervals.
Its relative rate stays constant, while its absolute rate changes with the amount present.
Extreme Value Theorem
Calculus AB · Calculus BC · Definition
The Extreme Value Theorem guarantees an absolute maximum and absolute minimum for a function continuous on a closed bounded interval.
The extrema can occur at endpoints or interior points.
For example, x² on [−1,2] attains minimum 0 at 0 and maximum 4 at 2.
An open interval or a missing continuity hypothesis does not carry the same guarantee.
First derivative test
Calculus AB · Calculus BC · Definition
The first derivative test applies to a function continuous at a critical input and differentiable on nearby intervals on both sides.
A positive-to-negative sign change gives a local maximum, and a negative-to-positive change gives a local minimum.
For example, the derivative 2x of x² changes from negative to positive at 0, identifying a local minimum.
If the sign does not change, this test does not identify a local extremum there.
Function
Calculus AB · Calculus BC · Definition
A mathematical function assigns exactly one output to each input in its domain.
Inputs can share an output, but one input cannot have two different outputs.
For example, f(x)=x² assigns the output 9 to both 3 and −3.
Function notation names the rule and its input; f(3) means the output when the input is 3.
Fundamental Theorem of Calculus
Calculus AB · Calculus BC · Definition
The Fundamental Theorem of Calculus connects accumulation with differentiation.
For a continuous integrand, differentiating its accumulation function returns the integrand.
It also allows a definite integral of a continuous function to be evaluated by the endpoint change of an antiderivative.
For example, the integral of 2x from 1 to 3 equals 3²−1²=8.
General solution
Calculus AB · Calculus BC · Definition
A general solution describes a family of functions satisfying a differential equation.
Arbitrary constants account for the freedom left by the equation.
For example, y=x²+C is a general solution of y′=2x.
Separation or other manipulations can miss special solutions, so the family and any excluded cases need checking.
Graph reflection
Calculus AB · Calculus BC · Definition
A graph reflection flips points across a specified line.
Replacing f(x) by −f(x) reflects across the horizontal axis, while f(−x) reflects across the vertical axis.
For example, −x² is the reflection of y=x² across the horizontal axis.
Reflecting across y=x interchanges input and output coordinates and can produce an inverse relation.
Graph stretching and compression
Calculus AB · Calculus BC · Definition
Graph scaling multiplies output values or changes how quickly the input is traversed.
For y=A f(x), a factor |A|>1 stretches vertically, while 0<|A|<1 compresses vertically.
For example, f(2x) compresses the graph horizontally by a factor of 1/2 relative to f(x).
Negative factors also introduce reflections, and a zero factor is a separate degenerate case.
Graph translation
Calculus AB · Calculus BC · Definition
A graph translation shifts every point by the same horizontal and vertical amounts.
The graph y=f(x−h)+k moves the original graph h units right and k units up.
For example, y=(x−2)²+3 shifts y=x² right 2 and up 3.
Negative shifts move in the opposite directions.
Half angle identity
Calculus AB · Calculus BC · Definition
A half angle identity expresses a trigonometric function of half an angle using the original angle.
For example, sin²(θ/2)=(1−cos θ)/2.
Taking a square root gives a magnitude, and the sign of sin(θ/2) depends on the angle’s quadrant.
For θ=π, the formula yields sin(π/2)=1 using the positive sign.
Half-life
Calculus AB · Calculus BC · Definition
A half-life is the time required for a positive exponentially decaying quantity to fall to half its current amount.
For y(t)=y₀eᵏᵗ with k<0, it equals ln 2 divided by −k.
For example, a quantity with half-life 3 hours falls to one quarter of its initial amount after 6 hours.
The same half-life applies from any starting time within the exponential model.
Higher-order derivative
Calculus AB · Calculus BC · Definition
A higher-order derivative is obtained by differentiating a function repeatedly.
The nth derivative means that the derivative operation has been applied n times.
For example, f(x)=x³ has third derivative 6 and fourth derivative 0.
Each differentiation requires the preceding derivative to exist on the region being considered.
Horizontal asymptote
Calculus AB · Calculus BC · Definition
A horizontal asymptote y=L describes a finite limiting output as the input tends to positive or negative infinity.
The two ends of a graph may have different horizontal asymptotes.
For example, 1/x has horizontal asymptote y=0 at both ends.
A graph may cross a horizontal asymptote at finite inputs.
Implicit differentiation
Calculus AB · Calculus BC · Definition
Implicit differentiation finds a derivative from an equation relating variables without first solving explicitly for one variable.
When y depends on x, differentiating a term containing y requires the chain rule and a factor of dy/dx.
For example, x²+y²=1 gives 2x+2y(dy/dx)=0.
Solving that relation for dy/dx gives −x/y only where y≠0 and the relation supports the local function being considered.
Improper integral
Calculus AB · Calculus BC · Definition
An improper integral has an infinite integration interval or an integrand that becomes unbounded within the interval or at an endpoint.
It is defined using limits of proper integrals rather than substituting infinity as a number.
For example, the integral of 1/x² from 1 to infinity converges to 1.
If an interior singularity splits an interval, the separate improper integrals must both converge.
Indefinite integral
Calculus AB · Calculus BC · Definition
An indefinite integral denotes the family of antiderivatives of its integrand on an interval.
The constant of integration records the possible constant differences among those functions.
For example, the indefinite integral of cos x is sin x+C.
It has no pair of numerical bounds and does not by itself give a single accumulated amount.
Indeterminate form
Calculus AB · Calculus BC · Definition
An indeterminate form is a pattern in a proposed limit that does not by itself determine the limit’s value.
The forms 0/0 and infinity divided by infinity can lead to several different outcomes.
For example, x/x approaches 1 as x approaches 0, while x²/x approaches 0 despite the same 0/0 substitution pattern.
The form signals that a justified further analysis is needed.
Inflection point
Calculus AB · Calculus BC · Definition
An inflection point is a point on a continuous graph where its concavity changes.
A zero or undefined second derivative may identify a candidate but does not establish that change.
For example, x³ changes concavity at 0, while x⁴ has second derivative zero there without changing concavity.
Check the behavior on both sides of the candidate.
Initial condition
Calculus AB · Calculus BC · Definition
An initial condition specifies a solution’s value at a particular input.
It can determine the constant in a family of solutions to a differential equation.
For example, y′=2x with y(0)=3 selects y=x²+3.
The solution must satisfy both the equation and the stated condition on a suitable interval.
Instantaneous rate of change
Calculus AB · Calculus BC · Definition
An instantaneous rate of change is the limiting average rate around one input when that limit exists.
For a differentiable function, it is its derivative at that input.
For example, position s(t)=t² meters has instantaneous rate 4 meters per second at t=2 seconds.
The rate’s units are output units per input unit.
Integral
Calculus AB · Calculus BC · Definition
Integration can mean accumulating a function over an interval or finding an antiderivative.
A definite integral gives a number, while an indefinite integral represents a family of antiderivatives.
For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4.
The Fundamental Theorem of Calculus connects these two operations under its hypotheses.
Integrand
Calculus AB · Calculus BC · Definition
The integrand is the function or expression being integrated.
The differential identifies the variable of integration.
For example, in the integral of x² with respect to x, the integrand is x².
Bounds belong to the integral’s interval rather than to the integrand itself.
Intermediate Value Theorem
Calculus AB · Calculus BC · Definition
The Intermediate Value Theorem states that a function continuous on a closed interval takes every value between its endpoint values.
It establishes the existence of an input rather than a formula for that input or a guarantee of uniqueness.
For example, a continuous function with f(0)=−1 and f(2)=3 must equal 0 somewhere between 0 and 2.
The continuity hypothesis must be checked before using the theorem.
Interval
Calculus AB · Calculus BC · Definition
An interval is a set of real numbers containing every number between any two of its members.
Parentheses exclude an endpoint, while square brackets include it.
For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1.
An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
Inverse function
Calculus AB · Calculus BC · Definition
An inverse function reverses another function’s input-output assignment.
A function needs to be one-to-one on its chosen domain for an inverse function to exist.
For example, x³ has inverse the cube-root function, while x² needs a domain restriction such as x≥0.
The notation f⁻¹ denotes an inverse function rather than the reciprocal 1/f.
Inverse trigonometric function
Calculus AB · Calculus BC · Definition
An inverse trigonometric function returns a principal angle from a trigonometric output.
The original trigonometric function is restricted to a one-to-one interval to define that inverse.
For example, arcsin(1/2)=π/6 under the usual range [−π/2,π/2].
The notation sin⁻¹x means arcsin x, whereas the reciprocal of sine is cosecant.
Jump discontinuity
Calculus AB · Calculus BC · Definition
A jump discontinuity occurs when finite left-hand and right-hand limits exist but differ.
The graph approaches different heights from the two sides.
For example, a function equal to 0 for x<1 and 2 for x≥1 has a jump at 1.
Changing its value at 1 does not make the two side limits equal.
L’Hospital’s rule
Calculus AB · Calculus BC · Definition
L’Hospital’s rule can evaluate certain quotient limits with the indeterminate forms 0/0 or infinity divided by infinity.
Its hypotheses require appropriate differentiability near the target, a nonzero denominator derivative there, and a suitable limit of the derivative quotient.
For example, lim(x→0) sin x/x equals lim(x→0) cos x/1=1.
Differentiating numerator and denominator separately here is a limit theorem, not the quotient rule for derivatives.
Limit
Calculus AB · Calculus BC · Definition
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound.
For an input approaching a finite target, its value at that exact input does not determine the limit.
For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1.
A finite two-sided limit requires the left and right approaches to agree.
Limit laws
Calculus AB · Calculus BC · Definition
Limit laws allow limits of sums, products, and suitable quotients to be assembled from the limits of their parts.
The component limits must exist as required, and a quotient law needs a nonzero denominator limit.
For example, if f approaches 2 and g approaches 3, then f+g approaches 5 and fg approaches 6.
Direct substitution into a quotient with denominator limit 0 is not justified by the quotient limit law.
Linearization
Calculus AB · Calculus BC · Definition
A linearization replaces a differentiable function near an input with its tangent line.
At x=a, the approximation is L(x)=f(a)+f′(a)(x−a).
For example, √x near 4 is approximated by 2+(x−4)/4.
The approximation is local, and its error can grow as the input moves away from a.
Local extremum
Calculus AB · Calculus BC · Definition
A local extremum is a value at least as large as, or at least as small as, all values at nearby allowed inputs.
A local maximum compares nearby heights above, while a local minimum compares them below.
For example, x² has a local minimum at 0.
A local extremum need not be the greatest or least value over the entire domain.
Logarithm
Calculus AB · Calculus BC · Definition
A logarithm gives the exponent required to obtain a number from a specified base.
For real logarithms, the argument is positive and the base is positive and different from 1.
For example, log₂8=3 because 2³=8, while ln uses the base e.
Logarithm rules convert products into sums and powers into constant multiples, subject to the domain restrictions.
Mean Value Theorem
Calculus AB · Calculus BC · Definition
The Mean Value Theorem connects an average rate on a closed interval to an instantaneous rate at an interior point.
It requires continuity on the closed interval and differentiability on its interior.
For example, f(x)=x² on [1,3] has average slope 4 and derivative 4 at x=2.
The theorem guarantees at least one such interior point but need not identify a unique one.
Midpoint rule
Calculus AB · Calculus BC · Definition
The midpoint rule approximates a definite integral by sampling at the center of each subinterval.
Each sampled height is multiplied by its own subinterval width.
For example, one midpoint rectangle for x² on [0,2] has height 1 and width 2, giving 2.
For a concave-up function the midpoint sum underestimates the integral, while for a concave-down function it overestimates it.
Net change
Calculus AB · Calculus BC · Definition
Net change is the final amount minus the initial amount.
Integrating a rate gives this signed change when the model and integral are valid.
For example, moving 5 meters right and then 2 meters left gives net displacement 3 meters.
Opposite contributions cancel, so net change can differ from the total amount of movement or activity.
Newton’s method
Calculus AB · Definition
Newton’s method estimates a root by repeatedly replacing the current input with the horizontal intercept of its tangent line.
The update is xₙ₊₁=xₙ−f(xₙ)/f′(xₙ) where the denominator is nonzero.
For example, starting at 2 for f(x)=x²−2 produces the next estimate 1.5.
A poor starting value or problematic derivative can cause failure, so convergence is not guaranteed.
One-sided limit
Calculus AB · Calculus BC · Definition
A one-sided limit uses inputs approaching a target from just one side.
A left-hand limit uses smaller inputs, and a right-hand limit uses larger inputs.
For example, a function that equals 0 left of 2 and 1 right of 2 has different one-sided limits at 2.
The two-sided limit exists as a finite real number only when both one-sided limits exist and agree.
One-to-one function
Calculus AB · Calculus BC · Definition
A function is one-to-one when different inputs always have different outputs.
Equivalently, each horizontal line meets its graph at most once.
For example, x³ is one-to-one on the real numbers, whereas x² is not because 2 and −2 have the same output.
This property allows a function’s inverse relation to be a function.
Optimization
Calculus AB · Calculus BC · Definition
Optimization finds the greatest or least value of an objective while satisfying the problem’s constraints.
Express the objective in suitable variables, determine the allowed domain, and compare all relevant candidates.
For example, among rectangles with a fixed perimeter, writing area in terms of one side reduces the search to a one-variable problem.
Endpoint and feasibility checks are part of the argument, not optional final details.
Particular solution
Calculus AB · Calculus BC · Definition
A particular solution is one specific function satisfying a differential equation.
Initial conditions often select it from a general family.
For example, y=x²+3 is the solution of y′=2x with y(0)=3.
Verify the selected function in both the differential equation and the initial condition.
Period of a function
Calculus AB · Calculus BC · Definition
A positive period of a function is an input shift that leaves all its outputs unchanged.
The fundamental period is the smallest such positive shift when one exists.
For example, sin x has fundamental period 2π, while sin(2x) has fundamental period π.
The period describes repetition along the input axis rather than the height of an oscillation.
Phase shift
Calculus AB · Calculus BC · Definition
A phase shift moves a periodic graph horizontally relative to a chosen reference graph.
Writing y=A sin(B(x−h))+D makes the shift h visible when B≠0.
For example, sin(x−π/2) is the sine graph shifted π/2 units to the right.
For a form such as sin(Bx+C), factor B before reading the shift as −C/B.
Polynomial
Calculus AB · Calculus BC · Definition
A polynomial in one variable is a finite sum of constant multiples of nonnegative integer powers of that variable.
The degree is the highest exponent with a nonzero coefficient.
For example, 2x³−x+5 is a polynomial of degree 3.
Expressions containing 1/x or √x are not polynomials in x.
Power rule
Calculus AB · Calculus BC · Definition
The derivative power rule gives d(xⁿ)/dx=n xⁿ⁻¹ wherever the real power function is differentiable.
Constant multiples remain as constant factors.
For example, the derivative of 3x⁴ is 12x³.
A composite power such as (x²+1)³ also requires the chain rule.
Product rule
Calculus AB · Calculus BC · Definition
The product rule differentiates a product of two differentiable functions.
It gives (fg)′=f′g+fg′, changing one factor at a time and adding the results.
For example, the derivative of x sin x is sin x+x cos x.
The product of the two derivatives is generally not the derivative of the product.
Quadrant
Calculus AB · Calculus BC · Definition
The coordinate axes divide the plane into four quadrants.
Quadrant I has two positive coordinates, and the numbering continues counterclockwise.
For example, (−2,3) lies in Quadrant II, where cosine is negative and sine is positive.
Points on an axis are not in any quadrant.
Quotient rule
Calculus AB · Calculus BC · Definition
The quotient rule differentiates a quotient where the denominator is nonzero and both functions are differentiable.
It gives (f/g)′=(f′g−fg′)/g².
For example, differentiating x/(x+1) produces 1/(x+1)² for x≠−1.
The order of the subtraction and the denominator square both matter.
Radian
Calculus AB · Calculus BC · Definition
A radian measures an angle by dividing the intercepted arc length by the circle’s radius.
One full turn is 2π radians, so π radians equals 180 degrees.
For example, an arc with length equal to its radius subtends an angle of one radian.
The familiar derivative formulas for sine and cosine use radian inputs.
Range of a function
Calculus AB · Calculus BC · Definition
The range of a function is the set of outputs it actually produces over its domain.
It can differ from the set of outputs that were merely allowed when defining the function.
For example, f(x)=x² on all real inputs has range [0,∞).
Restricting the domain to [1,2] changes that range to [1,4].
Rational function
Calculus AB · Calculus BC · Definition
A rational function is a quotient of two polynomials.
Its domain excludes inputs that make the original denominator zero.
For example, (x²−1)/(x−1) agrees with x+1 away from x=1, but remains undefined there.
Canceling a common factor does not automatically restore an excluded input.
Reciprocal derivative rule
Calculus AB · Calculus BC · Definition
The reciprocal derivative rule differentiates 1/f(x) wherever f is differentiable and nonzero.
It gives −f′(x)/[f(x)]².
For example, the derivative of 1/(x²+1) is −2x/(x²+1)².
This is a special case of the quotient rule and can also be derived with the chain rule.
Removable discontinuity
Calculus AB · Calculus BC · Definition
A removable discontinuity occurs when a finite two-sided limit exists but the function is missing or has a different value at the target.
Changing or defining that single value to equal the limit repairs continuity.
For example, (x²−1)/(x−1) has a removable discontinuity at 1 and can be repaired by setting its value there to 2.
A jump with unequal one-sided limits cannot be repaired this way.
Riemann sum
Calculus AB · Calculus BC · Definition
A Riemann sum approximates accumulation by adding sample heights times subinterval widths.
Left, right, and midpoint sums choose different sample positions in each subinterval.
For example, a left sum for x² on [0,2] with two equal widths uses heights 0 and 1, giving 1.
A definite integral is the limiting value as the largest subinterval width approaches zero when that limit exists.
Rolle’s theorem
Calculus AB · Calculus BC · Definition
Rolle’s theorem is the equal-endpoint case of the Mean Value Theorem.
A function continuous on [a,b], differentiable on (a,b), and satisfying f(a)=f(b) has an interior point with derivative 0.
For example, x² on [−1,1] has equal endpoint values and derivative 0 at x=0.
A sharp corner can break the differentiability hypothesis even when the endpoints agree.
Secant line
Calculus AB · Calculus BC · Definition
A secant line passes through two distinct points on a curve.
Its slope is the average rate of change between their inputs.
For example, the secant to y=x² through inputs 1 and 3 has slope (9−1)/(3−1)=4.
Tangent slopes arise from limits of these secant slopes when the second point approaches the first.
Second derivative
Calculus AB · Calculus BC · Definition
A second derivative is the derivative of the first derivative.
It measures how the first rate of change changes with the input; further repeated differentiation gives higher-order derivatives.
For example, f(x)=x³ has f′(x)=3x² and f″(x)=6x.
For position measured against time, the second derivative is acceleration.
Second derivative test
Calculus AB · Calculus BC · Definition
The second derivative test classifies a stationary point when its second derivative is nonzero.
If f′(c)=0, a positive f″(c) gives a local minimum and a negative f″(c) gives a local maximum under the theorem’s smoothness conditions.
For example, f(x)=x² has f′(0)=0 and f″(0)=2, so 0 is a local minimum.
If f″(c)=0, the test is inconclusive and another argument is required.
Separation of variables
Calculus AB · Calculus BC · Definition
Separation of variables solves certain differential equations by placing the dependent and independent variables on opposite sides.
For y′=g(x)h(y), rewrite dy/h(y)=g(x) dx where h(y)≠0 and integrate.
For example, y′=xy leads to ln|y|=x²/2+C for nonzero solutions.
Check any solutions lost by division, such as y=0 in this example.
Sinusoidal function
Calculus AB · Calculus BC · Definition
A sinusoidal function has the shape of a sine or cosine graph after shifts, scaling, and possible reflection.
A common form is A sin(B(x−h))+D with nonzero A and B.
For example, 2 sin(3x)+1 has amplitude 2, midline 1, and period 2π/3.
The parameters control vertical variation, repetition, and position of the wave.
Slope field
Calculus AB · Calculus BC · Definition
A slope field displays short line segments with slopes specified by a first-order differential equation.
For y′=F(x,y), the segment at (x,y) has slope F(x,y).
For example, y′=y gives horizontal segments along y=0 and positive slopes above that axis.
Solution curves follow the local slopes, and an initial point helps choose the relevant curve.
Solid of revolution
Calculus AB · Calculus BC · Definition
A solid of revolution is formed by rotating a plane region around a specified line.
The axis and region determine the resulting radii and any central hole.
For example, rotating the region under y=x on [0,1] around the horizontal axis produces a cone.
Disc, washer, and shell methods describe different ways to slice or assemble such a solid.
Squeeze theorem
Calculus AB · Calculus BC · Definition
The squeeze theorem determines a limit by trapping a function between two others with the same limit.
The bounding inequalities must hold sufficiently near the target where the functions are being compared.
For example, −x²≤x² sin(1/x)≤x² for x≠0 shows that the middle expression approaches 0 as x approaches 0.
A picture suggesting a trap does not replace the required inequality.
Stationary point
Calculus AB · Calculus BC · Definition
A stationary point is a point where a function’s derivative is zero.
Its tangent is horizontal when the derivative exists as a finite number.
For example, x³ has a stationary point at 0 but continues increasing through it.
A stationary point can be a local extremum or neither, so classification requires additional evidence.
Substitution for integration
Calculus AB · Calculus BC · Definition
Substitution rewrites an integral using a new variable whose differential accounts for a derivative factor.
It reverses the chain rule for suitable compositions.
For example, with u=x²+1 and du=2x dx, the integral of 2x(x²+1)³ becomes the integral of u³.
For a definite integral, transform the bounds consistently or return to the original variable before evaluating them.
Tangent line
Calculus AB · Calculus BC · Definition
A tangent line at a differentiable point passes through the point and has slope equal to the derivative there.
It describes the function’s local linear behavior.
For example, y=x² has tangent line y=4+4(x−2) at x=2.
A tangent can cross the curve, so touching at exactly one point is not its general definition.
Total distance traveled
Calculus AB · Calculus BC · Definition
Total distance traveled adds the lengths of every part of a motion path without canceling opposite directions.
In one dimension it is the integral of speed, |v(t)|, over the time interval.
For example, moving 5 meters right and then 2 meters left gives total distance 7 meters.
When velocity changes sign, split the integral at those sign changes or integrate its absolute value correctly.
Trapezoidal rule
Calculus AB · Calculus BC · Definition
The trapezoidal rule approximates a definite integral using a straight segment between each pair of adjacent endpoint values.
Each piece contributes its width times the average of its two endpoint heights.
For example, one trapezoid for x² on [0,2] has width 2 and heights 0 and 4, giving 4.
For a concave-up function the trapezoidal sum overestimates the integral, while for a concave-down function it underestimates it.
Trigonometric identity
Calculus AB · Calculus BC · Definition
A trigonometric identity is an equality that holds for every input where both sides are defined.
Identities let you rewrite expressions without changing their values on that shared domain.
For example, sin²θ+cos²θ=1 follows from the unit circle.
Any division used while rewriting an identity can introduce domain restrictions.
Trigonometric substitution
Calculus AB · Calculus BC · Definition
Trigonometric substitution replaces the variable of an integral with a trigonometric expression to simplify a radical.
For example, x=a sin θ can turn a²−x² into a² cos²θ when a>0.
The differential, bounds, and chosen angle interval must change consistently.
The angle interval also determines signs when taking square roots, so √(cos²θ) cannot always be replaced by cos θ.
Unit circle
Calculus AB · Calculus BC · Definition
The unit circle is the circle of radius 1 centered at the origin.
At angle θ measured from the positive horizontal axis, its point has coordinates (cos θ, sin θ).
For example, θ=π/2 gives the point (0,1), so cosine is 0 and sine is 1.
The circle also helps track signs and periodic behavior of trigonometric functions.
Velocity in calculus
Calculus AB · Calculus BC · Definition
In one-dimensional motion, velocity is the derivative of position with respect to time.
Its sign records direction relative to the chosen positive axis.
For example, s(t)=t²−4t meters gives velocity 2t−4 meters per second.
Speed is the magnitude of velocity and therefore cannot be negative.
Vertical asymptote
Calculus AB · Calculus BC · Definition
A vertical asymptote x=c occurs when function values tend to positive or negative infinity as x approaches c from at least one side.
Being unbounded near c is insufficient if the values oscillate without tending to either signed infinity.
For example, 1/x has vertical asymptote x=0, tending to positive infinity from the right and negative infinity from the left.
A zero denominator alone does not prove an asymptote because cancellation can instead leave a removable hole.
Vertical shift
Calculus AB · Calculus BC · Definition
A vertical shift adds the same constant to every output of a function.
The graph of f(x)+k is shifted upward by k when k is positive and downward when k is negative.
For example, sin x+3 has midline y=3.
The shift changes the midline without changing a sinusoid’s amplitude or period.
Vertical tangent
Calculus AB · Calculus BC · Definition
A vertical tangent is a tangent line parallel to the vertical axis, with no finite slope.
A graph can have a vertical tangent even though its derivative does not exist as a finite real number at that input.
For example, y=∛x has the vertical tangent x=0 at the origin as its nearby slopes tend to positive infinity.
This case is separate from the finite-slope tangent approximation available at a differentiable point.
Washer method
Calculus AB · Calculus BC · Definition
The washer method uses circular slices with a central hole to compute volume.
A slice with outer radius R and inner radius r has area π(R²−r²).
For example, rotating the region 1≤y≤2 on a horizontal interval around the horizontal axis gives washer area 3π.
Both radii are distances from the rotation axis and must be expressed using the same slice variable.