AP Calculus BC Vocabulary

Absolute Convergence

A series Σaₙ converges absolutely if the series of absolute values Σ|aₙ| also converges. Absolute convergence implies ordinary convergence.

Similar definitions: unconditional convergence

Example: "Because the series of absolute values converged, the series satisfied         , guaranteeing it also converged in the ordinary sense."

Absolute Maximum

The largest value of a function on a given interval or its entire domain; f(c) ≥ f(x) for all x in the interval.

Similar definitions: global maximum

Example: "By evaluating the function at each critical point and both endpoints, the student identified the          on the closed interval."

Absolute Minimum

The smallest value of a function on a given interval or its entire domain; f(c) ≤ f(x) for all x in the interval.

Similar definitions: global minimum

Example: "The          of the cost function occurred at the boundary of the feasible region."

Acceleration

The rate of change of velocity with respect to time; the second derivative of the position function: a(t) = v'(t) = s''(t).

Example: "The particle's          was positive, indicating the velocity was increasing even though the particle was moving in the negative direction."

Alternating Series

A series whose terms alternate in sign, written as Σ(-1)ⁿbₙ or Σ(-1)ⁿ⁺¹bₙ where bₙ > 0.

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

Alternating Series Error Bound

The absolute error in approximating an alternating series by its nth partial sum is at most |aₙ₊₁|, the absolute value of the first omitted term.

Example: "Using the         , the student determined that truncating after three terms introduced an error of less than 1/4."

Alternating Series Test

An alternating series Σ(-1)ⁿbₙ converges if the terms bₙ are positive, eventually decreasing, and approach zero as n → ∞.

Example: "The          confirmed convergence because the absolute value of each term decreased monotonically to zero."

Antiderivative

A function F(x) such that F'(x) = f(x); the general antiderivative includes the constant of integration C.

Similar definitions: indefinite integral, primitive function

Example: "Since the derivative of x³/3 is x², the expression x³/3 + C is the          of x²."

Antidifferentiation

The process of finding an antiderivative; reversing differentiation to find a function F(x) such that F'(x) = f(x).

Similar definitions: integration, reverse differentiation

Example: "By performing          on f(x) = 3x², the student found F(x) = x³ + C."

Arc Length

The length of a curve over an interval; for y = f(x) on [a,b]: L = ∫ₐᵇ √(1 + [f'(x)]²) dx; for a parametric curve: L = ∫ √((dx/dt)² + (dy/dt)²) dt.

Example: "The student computed the          of the curve y = x^(3/2) from x = 0 to x = 4 using the arc length integral formula."

Area Between Curves

The area of the region between two functions f(x) and g(x) on [a, b], given by ∫ₐᵇ [f(x) - g(x)] dx when f(x) ≥ g(x).

Example: "To compute the         , the student first found the intersection points and then integrated the difference of the two functions."

Area in Polar Coordinates

The area enclosed by a polar curve r = f(θ) from θ = α to θ = β is A = (1/2) ∫ₐᵝ [r(θ)]² dθ.

Example: "Finding the          of the rose curve required setting up an integral from 0 to π/4 using the polar area formula."

Average Rate of Change

The change in output divided by the change in input over an interval [a, b]: (f(b) - f(a)) / (b - a); equals the slope of the secant line.

Example: "The          of the function on [1, 3] was 4, meaning the function increased by an average of 4 units per unit of input."

Average Value of a Function

The mean output value of a continuous function f on [a, b], given by (1/(b-a)) ∫ₐᵇ f(x) dx.

Example: "The          of the temperature function over the 12-hour period was found by dividing the definite integral by 12."

Bounded Sequence

A sequence {aₙ} that is bounded above if aₙ ≤ M for all n and bounded below if aₙ ≥ m for all n; a sequence that is both monotonic and bounded must converge.

Example: "The student showed the sequence {1/n} was a          because all terms satisfied 0 < aₙ ≤ 1."

Candidates Test

A method to find absolute extrema on a closed interval [a, b] by evaluating f at all critical points and both endpoints, then comparing values.

Similar definitions: closed interval method

Example: "Applying the         , the student evaluated the function at x = 0, x = 2, and x = 5 to find the absolute maximum."

Carrying Capacity

The maximum population or quantity L that a logistic growth model can sustain; the horizontal asymptote of the logistic function.

Example: "In the logistic model, the population approached its          of 10,000 as time increased without bound."

Chain Rule

The differentiation rule for composite functions: d/dx[f(g(x))] = f'(g(x)) · g'(x); the derivative of the outside times the derivative of the inside.

Example: "To differentiate sin(x²), the student applied the          to get 2x·cos(x²)."

Comparison Test

A convergence test: if 0 ≤ aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges; if Σaₙ diverges, then Σbₙ diverges.

Similar definitions: direct comparison test

Example: "Using the         , the student bounded the series above by a convergent p-series to establish convergence."

Concave Down

A function is concave down on an interval where f''(x) < 0; the graph bends downward like a frown, and tangent lines lie above the curve.

Example: "Because f''(x) was negative on (2, 5), the graph was          on that interval."

Concave Up

A function is concave up on an interval where f''(x) > 0; the graph bends upward like a smile, and tangent lines lie below the curve.

Example: "The function was          near the local minimum, confirming the second derivative test result."

Concavity Test

The second derivative is used to determine concavity: if f''(x) > 0 on an interval, f is concave up there; if f''(x) < 0, f is concave down; used to classify inflection points and apply the second derivative test.

Example: "The student applied the          to confirm the graph was concave down on (-1, 3) where f''(x) < 0."

Conditional Convergence

A series converges conditionally if it converges but the series of absolute values diverges; the alternating harmonic series Σ(-1)ⁿ⁺¹/n is the classic example.

Example: "Because the alternating harmonic series converged but the harmonic series diverged, the series exhibited          rather than absolute convergence."

Constant of Integration

The arbitrary constant C added to every indefinite integral, representing the family of all antiderivatives of a given function.

Example: "Forgetting the          when evaluating an indefinite integral means missing an infinite family of valid antiderivatives."

Continuity

A function f is continuous at x = c if three conditions hold: f(c) is defined, lim_{x→c} f(x) exists, and the limit equals f(c).

Example: "The function failed the test for          at x = 3 because the limit from the left did not equal the limit from the right."

Continuous Function

A function that is continuous at every point in its domain; no holes, jumps, or vertical asymptotes in the interior; formally, lim_{x→c} f(x) = f(c) for all c in the domain.

Similar definitions: everywhere continuous

Example: "Because f had no breaks on [0, 5], the student confirmed it was a          and applied the Extreme Value Theorem."

Convergence

A sequence or series converges if its terms (or partial sums) approach a finite limit L as n → ∞.

Example: "The ratio test confirmed          because the limiting ratio of consecutive terms was less than 1."

Convergence Tests

A collection of methods used to determine whether an infinite series converges or diverges, including the Ratio Test, Integral Test, Comparison Test, Limit Comparison Test, Alternating Series Test, and nth-Term Divergence Test.

Example: "After reviewing all available         , the student chose the Ratio Test because the series contained factorials."

Critical Point

A point x = c in the domain of f where f'(c) = 0 or f'(c) is undefined; potential location of a local extremum or inflection point.

Similar definitions: critical number, stationary point

Example: "The          at x = 2 was identified by setting the derivative equal to zero and solving."

Decreasing/Increasing Function Test

A function is increasing on an interval where f'(x) > 0 and decreasing where f'(x) < 0; used in the first derivative test to classify critical points as local maxima or minima.

Example: "Using the         , the student found the function was increasing on (-∞, 2) and decreasing on (2, ∞), confirming a local maximum at x = 2."

Definite Integral

An integral evaluated between two bounds a and b, written ∫ₐᵇ f(x) dx; represents the signed area between f and the x-axis on [a, b].

Example: "The          from 0 to 3 of the velocity function gave the displacement of the particle."

Definite Integral as Accumulation

The definite integral ∫ₐˣ f(t) dt represents the accumulated total of f from a to x; when f is a rate of change, the integral gives the total accumulated quantity (net change).

Example: "The student interpreted the integral of the velocity function as a         , finding the total displacement over the time interval."

Derivative

The instantaneous rate of change of a function at a point; equals the slope of the tangent line and is defined as f'(x) = lim_{h→0} [(f(x+h) - f(x))/h].

Similar definitions: rate of change, slope of tangent

Example: "The          of the position function is velocity, showing how quickly the particle moves."

Derivative of Inverse Function

If f and g are inverse functions, then g'(x) = 1 / f'(g(x)); the derivative of the inverse at a point is the reciprocal of the derivative of the original function at the corresponding point.

Example: "Using the          formula, the student found g'(3) by computing 1 / f'(g(3))."

Differentiable

A function is differentiable at x = c if its derivative exists there; differentiability requires the function to be continuous and have no sharp corners or vertical tangents.

Example: "The absolute value function is continuous but not          at x = 0 because it has a corner there."

Differential Equation

An equation that relates a function with one or more of its derivatives; solutions are functions rather than numbers.

Example: "The          dy/dx = ky models exponential growth and has the general solution y = Ce^(kx)."

Disc Method

A technique for finding the volume of a solid of revolution about an axis with no hole: V = π ∫ₐᵇ [f(x)]² dx, where each cross-section perpendicular to the axis is a full disc.

Similar definitions: disk method

Example: "To find the volume formed by rotating y = √x around the x-axis from x = 0 to x = 4, the student applied the         ."

Displacement

The net change in position of a particle over a time interval; computed as ∫ₐᵇ v(t) dt, which accounts for direction of motion.

Example: "The          was -3 meters, meaning the particle ended up 3 meters to the left of its starting position."

Divergence

A sequence or series diverges if its terms or partial sums do not approach a finite limit; the series has no sum.

Example: "Because the terms of the series failed to approach zero,          was confirmed by the nth-term test."

End Behavior

The behavior of a function f(x) as x → +∞ or x → -∞; determined by evaluating limits at infinity and used to identify horizontal asymptotes or confirm unbounded growth or oscillation.

Example: "By analyzing the          of the rational function, the student identified y = 2 as a horizontal asymptote."

Error Bound

An upper bound on the difference between an approximation and the exact value; AP BC uses the alternating series error bound and the Lagrange (Taylor) error bound for polynomial approximations.

Similar definitions: remainder bound, truncation error

Example: "The student used the          to guarantee that the third-degree Taylor polynomial approximated f(0.1) to within 0.001."

Euler's Method

A numerical technique for approximating solutions to differential equations by stepping along tangent lines: yₙ₊₁ = yₙ + f(xₙ, yₙ) · Δx.

Example: "The student used          with step size 0.1 to estimate the solution to the initial value problem at x = 0.3."

Exponential Function

A function of the form f(x) = aˣ or f(x) = eˣ; the natural exponential function eˣ is its own derivative and antiderivative, making it fundamental to differential equations and integration.

Example: "The student recognized that the          eˣ satisfies dy/dx = y, making it the solution to the simplest separable differential equation."

Exponential Growth and Decay

A model where a quantity grows or decays at a rate proportional to itself: dy/dt = ky, with solution y = Ce^(kt); k > 0 is growth, k < 0 is decay.

Example: "Radioactive decay follows the          model, where the amount present decreases exponentially over time."

Extreme Value Theorem

If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and an absolute minimum on that interval.

Example: "The          guaranteed that the continuous function had both a highest and lowest value on [0, 5]."

First Derivative Test

A method for classifying critical points: if f' changes from positive to negative at c, then c is a local maximum; if from negative to positive, c is a local minimum.

Example: "Applying the         , the student confirmed that x = 3 was a local minimum because f' changed from negative to positive there."

Fundamental Theorem of Calculus, Part 1

If F(x) = ∫ₐˣ f(t) dt where f is continuous, then F'(x) = f(x); differentiation and integration are inverse operations.

Example: "Using the         , the derivative of ∫₀ˣ sin(t) dt was immediately found to be sin(x)."

Fundamental Theorem of Calculus, Part 2

If F is any antiderivative of f on [a, b], then ∫ₐᵇ f(x) dx = F(b) - F(a); used to evaluate definite integrals exactly.

Example: "By applying the         , the student evaluated ∫₁³ 2x dx as F(3) - F(1) = 9 - 1 = 8."

General Solution

The complete family of solutions to a differential equation, expressed with an arbitrary constant C; a particular solution results from applying an initial condition.

Example: "The          to dy/dx = 2x was y = x² + C, which represents infinitely many possible curves."

Geometric Series

A series of the form Σarⁿ with constant ratio r; converges to a/(1-r) when |r| < 1, and diverges when |r| ≥ 1.

Example: "The          with first term 1 and ratio 1/2 converges to 2."

Harmonic Series

The series Σ(1/n) = 1 + 1/2 + 1/3 + 1/4 + ..., which diverges despite its terms approaching zero; a key counterexample in series convergence.

Example: "Although individual terms decrease to zero, the          diverges, as proven by the integral test."

Horizontal Asymptote

A horizontal line y = L that the graph of f approaches as x → +∞ or x → -∞; determined by evaluating limits at infinity.

Example: "The function had a          at y = 3 because the limit of f(x) as x → ∞ equaled 3."

Horizontal Tangent Line

A tangent line with slope zero; occurs where f'(x) = 0; for parametric curves, occurs where dy/dt = 0 and dx/dt ≠ 0.

Example: "The student found the points of          by solving f'(x) = 0, then verified each was a local extremum using the second derivative test."

Implicit Differentiation

A technique for finding dy/dx when y is not isolated as a function of x; differentiate both sides with respect to x, treating y as a function of x and applying the chain rule.

Example: "To find the slope of the circle x² + y² = 25, the student used          and solved for dy/dx."

Improper Integral

An integral with an infinite limit of integration or an unbounded integrand; evaluated using limits, e.g., ∫₁^∞ f(x) dx = lim_{b→∞} ∫₁ᵇ f(x) dx.

Example: "The          ∫₁^∞ (1/x²) dx converges to 1 because the limit of the antiderivative exists as the upper bound approaches infinity."

Indefinite Integral

The general antiderivative of a function, written ∫f(x) dx = F(x) + C; represents the entire family of antiderivatives.

Similar definitions: antiderivative, primitive

Example: "Evaluating the          of 3x² yielded x³ + C, not a single function but a family of functions."

Indeterminate Form

A limit expression that cannot be evaluated by direct substitution because it yields ambiguous forms such as 0/0, ∞/∞, 0·∞, or ∞ - ∞; L'Hôpital's Rule is often applied.

Example: "Direct substitution produced the          0/0, prompting the student to apply L'Hôpital's Rule."

Infinite Limit

A limit in which the function value grows without bound; lim_{x→c} f(x) = ∞ (or -∞); indicates a vertical asymptote at x = c.

Example: "The          as x → 0⁺ of 1/x equals +∞, confirming a vertical asymptote at the origin."

Inflection Point

A point on the graph where concavity changes from up to down or down to up; requires f''(c) = 0 or undefined AND a sign change in f''.

Similar definitions: point of inflection

Example: "The graph had an          at x = 2 where the second derivative changed from negative to positive."

Initial Condition

A known value of the dependent variable at a specific point, given as y(x₀) = y₀; used to determine the constant of integration and find a particular solution.

Example: "The          y(0) = 5 allowed the student to solve for C and write the particular solution y = 3e^(2x) + 2."

Initial Value Problem

A differential equation paired with an initial condition, used to determine a unique particular solution from the family of general solutions.

Example: "The          dy/dx = 2x with y(0) = 1 was solved to get the particular solution y = x² + 1."

Instantaneous Rate of Change

The derivative of a function at a specific point; the slope of the tangent line at that exact location, found as the limit of the average rate of change.

Example: "The          of the population model at t = 5 was 200 individuals per year."

Integral Test

For a positive, continuous, decreasing function f where aₙ = f(n), the series Σaₙ and the integral ∫₁^∞ f(x) dx either both converge or both diverge.

Example: "The          was applied to Σ(1/n²) by evaluating ∫₁^∞ (1/x²) dx and finding it convergent."

Integration by Parts

A technique for integrating products of functions based on the product rule: ∫u dv = uv - ∫v du; typically used when the integrand is a product of polynomial and transcendental functions.

Example: "To evaluate ∫x·eˣ dx, the student used          with u = x and dv = eˣ dx."

Integration by Substitution

A technique using the substitution u = g(x) to transform a complex integral into a simpler one; the reverse of the chain rule.

Similar definitions: u-substitution, change of variables

Example: "Setting u = x² + 1, the student used          to evaluate ∫2x(x²+1)⁵ dx."

Integration of Trigonometric Functions

Standard antiderivative formulas for trig functions: ∫sin x dx = -cos x + C, ∫cos x dx = sin x + C, ∫sec²x dx = tan x + C, and related formulas; essential for AP BC integration problems.

Example: "Using         , the student found that ∫(cos x - sin x) dx = sin x + cos x + C."

Intermediate Value Theorem

If f is continuous on [a, b] and N is any value between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N.

Example: "The          guaranteed that the continuous function must equal zero somewhere on [1, 3] because f(1) < 0 and f(3) > 0."

Interval of Convergence

The set of all x-values for which a power series converges; always an interval centered at a, with endpoints requiring separate testing.

Example: "The          for the power series was found using the ratio test, then the endpoints were checked individually."

Inverse Trigonometric Derivative

The derivatives of inverse trig functions; for example, d/dx[arcsin x] = 1/√(1-x²) and d/dx[arctan x] = 1/(1+x²).

Example: "Recognizing the integrand as a          form, the student integrated 1/(1+x²) to get arctan x + C."

Jump Discontinuity

A type of discontinuity where both one-sided limits exist but are not equal; the function "jumps" from one finite value to another.

Example: "The piecewise function had a          at x = 2 because the left-hand limit was 4 and the right-hand limit was 7."

L'Hôpital's Rule

If lim f(x)/g(x) yields 0/0 or ±∞/±∞, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the new limit exists; may be applied repeatedly.

Example: "After getting the indeterminate form 0/0, the student applied          and differentiated numerator and denominator separately."

Lagrange Error Bound

An upper bound on the error |f(x) - Pₙ(x)| when approximating f with the nth-degree Taylor polynomial: error ≤ M|x-a|ⁿ⁺¹/(n+1)!, where M bounds |f⁽ⁿ⁺¹⁾|.

Example: "Using the         , the student confirmed the third-degree Taylor polynomial approximated the function within 0.001 on the given interval."

Left Riemann Sum

An approximation of a definite integral using rectangles whose heights are determined by the function's value at the left endpoint of each subinterval.

Example: "The          overestimated the integral because the function was decreasing on the interval."

Leibniz Notation

A notation for derivatives written as dy/dx (first derivative), d²y/dx² (second derivative), and so on; emphasizes the ratio of infinitesimal changes and is used in the chain rule form dy/dx = (dy/du)(du/dx).

Example: "When applying the chain rule, the student used          to write dy/dx = (dy/du)(du/dx) and clearly track each substitution."

Limit

The value that a function f(x) approaches as the input x approaches a given value c; written lim_{x→c} f(x) = L.

Example: "The          of (x²-1)/(x-1) as x approaches 1 equals 2, even though the function is undefined at x = 1."

Limit Comparison Test

If lim(n→∞) aₙ/bₙ = L where 0 < L < ∞, then Σaₙ and Σbₙ either both converge or both diverge.

Example: "The          was used to compare Σ(2n+1)/(n³+1) to the convergent p-series Σ(1/n²)."

Limit Definition of Derivative

The formal definition of the derivative: f'(x) = lim_{h→0} [(f(x+h) - f(x))/h]; also written as lim_{x→a} [(f(x)-f(a))/(x-a)].

Example: "To rigorously find f'(x) for f(x) = x², the student used the          and simplified the difference quotient."

Limits at Infinity

The behavior of a function as x → +∞ or x → -∞; used to identify horizontal asymptotes and understand end behavior.

Example: "Evaluating          for a rational function involves dividing numerator and denominator by the highest power of x."

Linear Approximation

Using the tangent line at a point a to estimate nearby function values: L(x) = f(a) + f'(a)(x - a); valid when x is close to a.

Similar definitions: local linearization, tangent line approximation

Example: "The student used          near x = 0 to estimate sin(0.1) ≈ 0.1."

Local Maximum

A point x = c where f(c) ≥ f(x) for all x near c; the function reaches a peak relative to its immediate surroundings.

Similar definitions: relative maximum

Example: "The function had a          at x = 1 because the first derivative changed from positive to negative there."

Local Minimum

A point x = c where f(c) ≤ f(x) for all x near c; the function reaches a valley relative to its immediate surroundings.

Similar definitions: relative minimum

Example: "The          at x = 4 was confirmed using the second derivative test, which yielded a positive value."

Logarithmic Differentiation

A technique for differentiating complicated products, quotients, or power functions by first taking the natural log of both sides and then differentiating implicitly.

Example: "To differentiate y = x^x, the student used          by writing ln y = x ln x and differentiating both sides."

Logistic Differential Equation

A model for bounded population growth: dP/dt = kP(1 - P/L), where L is the carrying capacity; the solution follows an S-shaped logistic curve.

Example: "The          predicted that the population would grow quickly at first, then slow as it approached the carrying capacity of 500."

Maclaurin Series

A Taylor series centered at a = 0: f(x) = Σ [f⁽ⁿ⁾(0)/n!] xⁿ; common examples include those for eˣ, sin x, cos x, and 1/(1-x).

Example: "The          for eˣ is 1 + x + x²/2! + x³/3! + …, valid for all real x."

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a); the instantaneous rate of change equals the average rate of change at some point.

Example: "By the         , there is a point where the car's instantaneous speed equals its average speed over the trip."

Mean Value Theorem for Integrals

If f is continuous on [a, b], then there exists c in (a, b) such that f(c) = (1/(b-a)) ∫ₐᵇ f(x) dx; the function attains its average value at some point.

Example: "The          guaranteed that the temperature function equaled its average value at some specific time during the day."

Midpoint Riemann Sum

An approximation of a definite integral using rectangles whose heights are determined by the function's value at the midpoint of each subinterval.

Example: "The          is generally more accurate than the left or right Riemann sum for the same number of subintervals."

Monotonic Sequence

A sequence that is either entirely non-decreasing (aₙ₊₁ ≥ aₙ) or entirely non-increasing (aₙ₊₁ ≤ aₙ); a monotonic bounded sequence always converges.

Example: "The student proved the sequence was          and bounded, guaranteeing convergence by the Monotone Convergence Theorem."

Natural Logarithm

The inverse of the natural exponential function; ln x = log_e(x); its derivative is d/dx[ln x] = 1/x and its antiderivative is ∫(1/x) dx = ln|x| + C.

Example: "The student used the          antiderivative rule to evaluate ∫(1/x) dx = ln|x| + C."

Normal Line

The line perpendicular to the tangent line of a curve at a given point; its slope is the negative reciprocal of f'(a), so m_normal = -1/f'(a).

Example: "Since the tangent line had slope 2, the          at that point had slope -1/2."

nth-Term Divergence Test

If lim_{n→∞} aₙ ≠ 0, then the series Σaₙ diverges; note this test cannot prove convergence — only divergence.

Similar definitions: divergence test, test for divergence

Example: "The          immediately showed divergence because the terms approached 1/2, not zero."

Oblique (Slant) Asymptote

A non-horizontal, non-vertical asymptote of the form y = mx + b that the graph of a function approaches as x → ±∞; occurs when the degree of the numerator exceeds the degree of the denominator by exactly one.

Similar definitions: slant asymptote

Example: "After performing polynomial long division, the student identified y = 2x + 1 as an          of the rational function."

One-Sided Limit

The limit of a function as the input approaches a value from only one direction; the left-hand limit (x → c⁻) or the right-hand limit (x → c⁺).

Example: "The          from the right was 5, but from the left was 3, so the two-sided limit did not exist."

Optimization

The process of finding the maximum or minimum value of a function subject to given constraints; typically involves setting the derivative to zero and applying a derivative test.

Example: "In the          problem, the student minimized the cost function by finding where the first derivative equaled zero."

p-Series

A series of the form Σ(1/nᵖ); converges if p > 1 and diverges if p ≤ 1; the harmonic series is the special case p = 1.

Example: "Since p = 2 > 1, the          Σ(1/n²) converges; its sum is π²/6."

Parametric Derivative

The slope of a parametric curve: dy/dx = (dy/dt)/(dx/dt); the second derivative d²y/dx² = [d(dy/dx)/dt] / (dx/dt).

Example: "The          of the curve defined by x = t², y = t³ was found as dy/dx = (3t²)/(2t) = 3t/2."

Parametric Equations

A set of equations expressing the coordinates of a curve as functions of a third variable (parameter) t: x = f(t), y = g(t).

Example: "The motion of the projectile was described using         : x = v₀t and y = -16t² + v₀t."

Partial Fractions

A technique for decomposing a rational function into simpler fractions to facilitate integration; requires factoring the denominator and solving for unknown numerators.

Example: "Using         , the integrand (2x+1)/((x-1)(x+2)) was split into A/(x-1) + B/(x+2) before integrating."

Particle Motion

The analysis of an object's position s(t), velocity v(t) = s'(t), and acceleration a(t) = v'(t) along a line or a parametric path; includes determining direction of motion, speed, and total distance traveled.

Example: "Given v(t) = t² - 4t, the student analyzed          to determine when the particle changed direction and computed its total distance over [0, 5]."

Particular Solution

A specific solution to a differential equation satisfying given initial conditions; obtained from the general solution by solving for the constant C.

Example: "Using the initial condition y(0) = 2, the student found the          y = 2eˣ from the general solution y = Ceˣ."