Calculus BC glossary
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157 of 157 editorial senses
Absolute convergence
Calculus BC · Definition
A series converges absolutely when the series of its absolute term values converges.
Absolute convergence also guarantees convergence of the original signed series.
For example, an alternating series with magnitudes 1/n² converges absolutely.
An absolutely convergent series retains its sum under any rearrangement of its terms.
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.
Alternating series
Calculus BC · Definition
An alternating series has successive terms that switch between positive and negative signs.
The alternating series test establishes convergence when the term magnitudes eventually decrease and approach 0.
For example, 1−1/2+1/3−1/4+⋯ satisfies those conditions.
Alternating signs alone do not guarantee convergence.
Alternating series error bound
Calculus BC · Definition
For an alternating series satisfying the decreasing-to-zero magnitude conditions, the error after n terms is at most the next term’s magnitude.
The actual sum lies between two consecutive partial sums.
For example, stopping 1−1/2+1/3−⋯ after its tenth term gives error at most 1/11.
The bound requires the alternating series hypotheses and does not apply automatically to every signed series.
Alternating series test
Calculus BC · Definition
The alternating series test applies to alternating terms whose positive magnitudes eventually decrease to 0.
Those two conditions together ensure the partial sums approach a finite limit.
For example, 1−1/2+1/3−1/4+⋯ converges because 1/n decreases to 0.
If one condition fails, this test cannot by itself determine the result.
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.
Arc length
Calculus BC · Definition
Arc length measures the distance along a curve over a specified parameter interval.
For a smooth graph y=f(x), its length is the integral of √(1+[f′(x)]²) with respect to x.
For example, the graph y=x from x=0 to x=1 has length √2.
Parametric curves instead integrate the magnitude of their velocity, accounting for how the curve is traced.
Area in polar coordinates
Calculus BC · Definition
Polar area accumulates thin sectors over a specified angle interval.
For a single boundary r(θ), the basic contribution is one half of r(θ)² dθ.
For example, r=2 over 0≤θ≤π/2 encloses area π when that interval traces the quarter-circle region once.
Check intersections, radial order, and repeated tracing before choosing the bounds or subtracting two squared radii.
Arithmetic sequence
Calculus BC · Definition
An arithmetic sequence has a constant difference between consecutive terms.
With first term a₁ and common difference d, its nth term is a₁+(n−1)d.
For example, 3,7,11,15 has common difference 4.
An infinite arithmetic sequence with nonzero common difference does not approach a finite limit.
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.
Bounded sequence
Calculus BC · Definition
A sequence is bounded when all its terms lie between two finite fixed numbers.
Being bounded alone does not guarantee that its terms approach one limit.
For example, the sequence (−1)ⁿ is bounded between −1 and 1 but does not converge.
A bounded monotone sequence does converge to a finite limit.
Carrying capacity
Calculus BC · Definition
Carrying capacity is the positive equilibrium level built into a logistic growth model.
In y′=ky(1−y/K) with k>0, it is K.
For example, K=1000 models a population approaching 1000 when it starts between 0 and 1000.
It is a modeling parameter whose interpretation depends on the situation and assumptions.
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.
Common ratio
Calculus BC · Definition
A common ratio is the fixed multiplier between consecutive terms of a geometric sequence or series.
When the earlier term is nonzero, it equals the next term divided by that term.
For example, 2,6,18,54 has common ratio 3.
For a nonzero geometric series, its magnitude must be less than 1 for the series to converge.
Comparison tests
Calculus BC · Definition
Comparison tests determine convergence by relating nonnegative terms to a known benchmark series.
Direct comparison uses a correctly directed inequality, while limit comparison uses a positive finite limit of the term ratio.
For example, 0≤1/(n²+1)≤1/n² shows convergence because the upper p-series converges.
Being smaller than a divergent series or larger than a convergent one does not settle convergence.
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.
Conditional convergence
Calculus BC · Definition
A series converges conditionally when it converges but the series of its absolute term values diverges.
Cancellation between positive and negative terms is essential to this convergence.
For example, 1−1/2+1/3−1/4+⋯ converges while the harmonic series of its magnitudes diverges.
Rearranging a conditionally convergent series can change its sum or destroy convergence.
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.
Convergence of a sequence
Calculus BC · Definition
An infinite sequence converges when its terms approach one finite value as the index tends to infinity.
It diverges when no finite limit exists.
For example, 1/n converges to 0, while (−1)ⁿ diverges by oscillating between two values.
A sequence’s convergence concerns its individual terms; a series uses the limits of their partial sums instead.
Convergence of a series
Calculus BC · Definition
A series converges when the sequence of its partial sums approaches a finite limit.
It diverges when there is no such finite limit.
For example, 1/2+1/4+1/8+⋯ converges to 1, while the harmonic series diverges.
Convergence of the individual terms to 0 is necessary but does not by itself establish convergence of their sum.
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.
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.
Error bound
Calculus BC · Definition
An error bound gives a justified upper limit for the magnitude of an approximation’s error.
It states how far the approximate value can be from the true value.
For example, an error bound of 0.01 places the true value within 0.01 of the approximation.
A bound requires its stated hypotheses and is not a claim that the error actually equals the bound.
Euler’s method
Calculus BC · Definition
Euler’s method approximates a differential-equation solution by stepping along successive tangent lines.
For y′=F(x,y) with step size h, the update is yₙ₊₁=yₙ+hF(xₙ,yₙ).
For example, starting at (0,1) for y′=y with h=0.1 gives the next estimate y=1.1 at x=0.1.
Errors accumulate, and a smaller step often improves accuracy but does not make an approximation exact.
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.
Factorial
Calculus BC · Definition
The factorial n! multiplies the positive integers from 1 through n for a positive integer n.
By definition, 0!=1.
For example, 4!=4·3·2·1=24.
Factorials occur in Taylor coefficients and their rapid growth often makes the ratio test useful.
Fibonacci sequence
Calculus BC · Definition
A Fibonacci sequence begins with specified starting terms and obtains each later term by adding the preceding two.
A common starting convention uses 0 and 1.
For example, that convention gives 0,1,1,2,3,5,8, and so on.
Because the indexing convention can vary, check the stated first terms before identifying a particular indexed term.
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.
Fourier series
Calculus BC · Definition
A Fourier series represents periodic behavior using constant, sine, and cosine components with related frequencies.
The coefficients measure those components through integrals over a period.
For example, a square wave can be approximated by adding suitable odd sine harmonics.
Convergence depends on the function; under common regularity conditions, the series at a jump approaches the average of the one-sided values.
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.
Geometric sequence
Calculus BC · Definition
A geometric sequence multiplies each term by the same fixed ratio to obtain the next.
With first term a₁ and ratio r, its nth term is a₁rⁿ⁻¹.
For example, 8,4,2,1 has common ratio 1/2 and approaches 0.
This list of terms is distinct from the geometric series obtained by adding them.
Geometric series
Calculus BC · Definition
A geometric series adds terms that have a fixed common ratio.
With first term a and ratio r, it converges to a/(1−r) when |r|<1.
For example, 1+1/2+1/4+⋯ sums to 2.
For a nonzero first term and |r|≥1, the infinite series diverges.
Gibbs phenomenon
Calculus BC · Definition
The Gibbs phenomenon is persistent overshoot near a jump when a function is approximated by finite Fourier sums.
Adding terms narrows the region of the overshoot but does not make its peak relative to the jump disappear.
For example, Fourier approximations to a square wave can rise above its high level near a transition.
This local behavior does not mean that every Fourier series diverges everywhere.
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.
Harmonic series
Calculus BC · Definition
The harmonic series is 1+1/2+1/3+1/4+⋯.
It diverges even though its individual terms approach 0.
For example, grouping terms from 1/2 through 1/3+1/4 and then 1/5+⋯+1/8 shows repeatedly added blocks of at least 1/2.
It is a standard comparison that separates a necessary term condition from actual convergence.
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.
Index of a sequence or sum
Calculus BC · Definition
An index identifies the position of a term in a sequence or finite or infinite sum.
The starting index determines which term is included first.
For example, a sum indexed by n=0 starts with the term obtained by substituting 0 into its term formula.
Reindexing is valid only when the formula and bounds change together so that the same terms are retained.
Infinite series
Calculus BC · Definition
An infinite series is the sum of the terms of an infinite sequence, defined through its partial sums.
The series converges when those partial sums approach a finite value.
For example, 1/2+1/4+1/8+⋯ converges to 1.
Terms approaching 0 are necessary for convergence, but that condition alone is insufficient.
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.
Integral test
Calculus BC · Definition
The integral test compares a positive-term series with an improper integral of a matching function.
The function must be continuous, positive, and decreasing for all sufficiently large inputs, with f(n) equal to the series terms.
For example, applying it to 1/n² uses the convergent integral of 1/x² over [1,∞).
The test determines convergence together, but the integral’s value is generally not the series sum.
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.
Integration by parts
Calculus BC · Definition
Integration by parts rewrites an integral using the derivative product rule in reverse.
It takes the form ∫u dv=uv−∫v du with suitable choices of u and dv.
For example, choosing u=x and dv=eˣ dx turns ∫x eˣ dx into x eˣ−∫eˣ dx.
The choices should simplify the remaining integral, and definite bounds must be carried through consistently.
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.
Interval of convergence
Calculus BC · Definition
The interval of convergence is the full set of real inputs where a power series converges.
It consists of the interior determined by its radius plus any convergent endpoints.
For example, the series x−x²/2+x³/3−⋯ has interval (−1,1].
The radius alone cannot determine whether either endpoint is included.
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.
Lagrange error bound
Calculus BC · Definition
The Lagrange error bound controls the difference between a sufficiently differentiable function and its degree-n Taylor polynomial.
If the magnitude of the next derivative is at most M between the center c and the input x, the error is at most M|x−c|ⁿ⁺¹/(n+1)!.
For example, approximating eˣ near 0 can use a bound on eᵗ over the interval between 0 and x.
The derivative bound must hold across that whole interval, not merely at the center.
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.
Linear differential equation
Calculus BC · Definition
A differential equation is linear when the unknown function and its derivatives appear only to the first power and are not multiplied together.
Its coefficients may depend on the independent variable.
For example, y′+xy=x² is linear, while y′+y²=0 is nonlinear.
The sum of solutions is guaranteed to remain a solution for a homogeneous linear equation, but generally not for a nonhomogeneous one.
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.
Logistic model
Calculus BC · Definition
A logistic model limits positive growth as a quantity approaches a carrying capacity.
A standard form is y′=ky(1−y/K), with k>0 and capacity K>0.
For example, with K=1000, a population rising from 100 to 900 has its relative growth rate y′/y decrease from 0.9k to 0.1k.
The equilibrium solutions y=0 and y=K must be considered separately when solving by separation.
Maclaurin series
Calculus BC · Definition
A Maclaurin series is a Taylor series centered at 0.
Its coefficients are determined by the function’s successive derivatives evaluated at 0.
For example, sin x has Maclaurin series x−x³/3!+x⁵/5!−⋯.
As with any Taylor series, equality to the function requires the appropriate convergence and remainder justification.
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.
Normal distribution in integration
Calculus BC · Definition
A normal distribution is a continuous bell-shaped probability model determined by a mean and a positive standard deviation.
The standard normal model has mean 0 and standard deviation 1.
For example, the probability that a standard normal variable lies between 0 and 1 is the integral of its density over that interval.
The full density has total area 1, and interval areas rather than single-point heights give probabilities.
nth-term test for divergence
Calculus BC · Definition
The nth-term test shows that a series diverges if its terms fail to approach 0.
If the term limit is nonzero or does not exist, a finite sum is impossible.
For example, the series with terms n/(n+1) diverges because its terms approach 1.
When the term limit is 0, the test is inconclusive.
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.
Ordinary differential equation
Calculus BC · Definition
An ordinary differential equation uses derivatives of unknown functions with respect to one independent variable.
Its order is the highest derivative appearing in the equation.
For example, y″+y=0 is an ordinary differential equation of order 2.
Equations involving partial derivatives with respect to several independent variables belong to a different class.
p-series
Calculus BC · Definition
A p-series has terms 1/nᵖ for positive integers n.
It converges exactly when p>1.
For example, the p-series with p=2 converges, while p=1 gives the divergent harmonic series.
The boundary p=1 belongs to the divergent cases.
Parametric equations
Calculus BC · Definition
Parametric equations describe coordinates as functions of a shared parameter.
For x=x(t) and y=y(t), the point (x(t),y(t)) traces a curve as t varies.
For example, x=cos t and y=sin t trace the unit circle.
The slope dy/dx equals y′(t)/x′(t) where x′(t)≠0, and the parameter also records orientation and repeated tracing.
Partial derivative
Calculus BC · Definition
A partial derivative measures how a multivariable function changes with one input while the other inputs are held constant.
The chosen variable must be specified.
For example, for f(x,y)=x²y, the partial derivative with respect to x is 2xy.
This is different from differentiating y as an unknown function of x in implicit differentiation.
Partial fractions
Calculus BC · Definition
Partial fraction decomposition writes a rational expression as a sum of simpler fractions.
If the numerator’s degree is at least the denominator’s, polynomial division comes first.
For example, 1/[x(x+1)] equals 1/x−1/(x+1) wherever the original expression is defined.
The factor types and any repeated factors determine the form of the decomposition.
Partial sum
Calculus BC · Definition
A partial sum adds the first finitely many terms of a series.
The nth partial sum is Sₙ=a₁+a₂+⋯+aₙ when indexing begins at 1.
For example, the first three terms of 1/2+1/4+1/8+⋯ give partial sum 7/8.
Convergence of a series concerns the limit of these sums, not just the limit of the individual terms.
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.
Polar coordinates
Calculus BC · Definition
Polar coordinates describe a plane point by a directed radial distance r and angle θ.
The corresponding rectangular coordinates are x=r cos θ and y=r sin θ.
For example, (r,θ)=(2,π/2) describes the point (0,2).
Angles differing by a full turn and suitable negative-radius descriptions can name the same point, so tracing a polar curve needs care.
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.
Power series
Calculus BC · Definition
A power series is an infinite series of constant multiples of powers of x−c about a center c.
It can converge for some inputs and diverge for others.
For example, 1+x+x²+⋯ represents 1/(1−x) when |x|<1.
Determining the radius and checking the endpoints establishes its interval of convergence.
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.
Radius of convergence
Calculus BC · Definition
The radius of convergence R describes how far a power series converges from its center.
It converges absolutely for |x−c|<R and diverges for |x−c|>R.
For example, the series 1+x+x²+⋯ has center 0 and radius 1.
When R is finite and positive, convergence at each endpoint must be checked separately.
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].
Ratio test
Calculus BC · Definition
The ratio test examines the limiting absolute ratio of consecutive series terms.
A limit below 1 gives absolute convergence, and a limit above 1 or infinite gives divergence.
For example, terms 1/n! give ratio 1/(n+1), whose limit is 0, so the series converges absolutely.
A limit equal to 1 is inconclusive.
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.
Recursive sequence
Calculus BC · Definition
A recursive sequence specifies later terms using earlier terms and suitable starting values.
The recursion alone is incomplete until those starting values are given.
For example, the Fibonacci sequence can start with 0 and 1 and add the previous two terms to obtain each next term.
To find a later term from the recursion, compute or otherwise determine the earlier terms it depends on.
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 rearrangement theorem
Calculus BC · Definition
The Riemann rearrangement theorem concerns changing the order of terms in a conditionally convergent real series.
Such a series can be rearranged to converge to any chosen real value or to diverge.
For example, the alternating harmonic series has rearrangements with a sum different from its original one.
An absolutely convergent series instead retains its sum under every rearrangement.
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.
Root test
Calculus BC · Definition
The root test examines the limiting nth root of the absolute value of the nth series term.
A limit below 1 gives absolute convergence, and a limit above 1 or infinite gives divergence.
For example, terms (1/3)ⁿ have nth-root limit 1/3 and their series converges absolutely.
A limit equal to 1 is inconclusive.
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.
Sequence
Calculus BC · Definition
A sequence is an ordered list of terms indexed by integers.
A rule can specify each term directly or recursively from earlier terms.
For example, aₙ=1/n produces 1, 1/2, 1/3, and so on.
An infinite sequence converges when its terms approach a finite limit as the index grows without bound.
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.
Taylor polynomial
Calculus BC · Definition
A Taylor polynomial approximates a function near a center using its value and successive derivatives at that center.
The degree-n polynomial uses the terms through the nth power of x−c, with derivative coefficients divided by factorials.
For example, a degree-2 Taylor polynomial for eˣ at 0 is 1+x+x²/2.
An error argument is needed to judge the approximation away from the center.
Taylor series
Calculus BC · Definition
A Taylor series is the infinite power series built from a function’s derivatives at a chosen center.
Its coefficients are the derivatives at the center divided by the corresponding factorials.
For example, eˣ has Taylor series 1+x+x²/2!+x³/3!+⋯ centered at 0.
A Taylor series represents the function only where convergence and the vanishing remainder justify that equality.
Telescoping series
Calculus BC · Definition
A telescoping series has partial sums in which cancellation reduces the result to a fixed number of boundary terms.
Write out a partial sum before taking its limit so that the cancellation is justified.
For example, the first n terms of 1/[k(k+1)]=1/k−1/(k+1) sum to 1−1/(n+1).
Taking that limit shows the series starting at k=1 sums to 1.
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.
Vector in calculus
Calculus BC · Definition
A vector records a magnitude and direction and can be represented by ordered components.
In planar motion, the components describe changes or rates along the horizontal and vertical axes.
For example, velocity ⟨3,4⟩ has speed 5 because its magnitude is √(3²+4²).
A vector-valued function lets those components vary with a parameter.
Vector-valued function
Calculus BC · Definition
A vector-valued function assigns a vector to each allowed input.
Its component functions can represent a moving point’s coordinates.
For example, r(t)=⟨cos t,sin t⟩ traces a circle, and its derivative is ⟨−sin t,cos t⟩.
Derivatives and integrals are computed component by component, while speed is the magnitude of the velocity vector.
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.