AP Calculus BC Vocabulary

Absolute Convergence

A series $\sum a_n$ converges absolutely if the series of absolute values $\sum |a_n|$ 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; $f(c) \ge f(x)$ 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; $f(c) \leq f(x)$ 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 $\sum(-1)^{n}b_{n}$ or $\sum(-1)^{n+1}b_{n}$ where $b_{n} > 0$.

Example: "The series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots$ 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 $|a_{n+1}|$, 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 $\sum(-1)^{n}b_{n}$ converges if the terms $b_{n}$ are positive, eventually decreasing, and approach zero as $n \to \infty$.

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 $\frac{x^{3}}{3}$ is $x^{2}$, the expression $\frac{x^{3}}{3} + C$ is the          of $x^{2}$."

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 $[a,b]$: $L = \int_a^b \sqrt{1 + \left[f'(x)\right]^2} \, dx$; for a parametric curve: $L = \int \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$.

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 $r = f(\theta)$ from $\theta = \alpha$ to $\theta = \beta$ is $A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^{2} d\theta$.

Example: "Finding the          of the rose curve required setting up an integral from 0 to $\frac{\pi}{4}$ 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 $\frac{1}{b-a} \int_a^b f(x) \, dx$.

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

Bounded Sequence

A sequence $\{a_n\}$ that is bounded above if $a_n \leq M$ for all n and bounded below if $a_n \geq m$ for all n; a sequence that is both monotonic and bounded must converge.

Example: "The student showed the sequence $\{\frac{1}{n}\}$ was a          because all terms satisfied $0 < a_n \leq 1$."

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: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$; the derivative of the outside times the derivative of the inside.

Example: "To differentiate $\sin(x^2)$, the student applied the          to get $2x \cdot \cos(x^2)$."

Comparison Test

A convergence test: if $0 \le a_n \le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges; if $\sum a_n$ diverges, then $\sum b_n$ 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 $\Sigma(-1)^{n+1}/n$ 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, $\lim_{x \to c} f(x)$ 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, $\lim_{x \to c} f(x) = f(c)$ 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 $(-\infty, 2)$ and decreasing on $(2, \infty)$, confirming a local maximum at $x = 2$."

Definite Integral

An integral evaluated between two bounds a and b, written $\int_a^b f(x) \, dx$; 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 $\int_{a}^{x} f(t) dt$ 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 $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.

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: $V = \pi \int_{a}^{b} [f(x)]^{2} dx$, where each cross-section perpendicular to the axis is a full disc.

Similar definitions: disk method

Example: "To find the volume formed by rotating $y = \sqrt{x}$ 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
^{a}_{b} 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 largest value of a function on a given interval or its entire domain; $f(c) \ge f(x)$ 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."

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: $y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x$.

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 $F(x) = \int_a^x f(t) \, dt$ where $f$ is continuous, then $F'(x) = f(x)$; differentiation and integration are inverse operations.

Example: "Using the         , the derivative of $\int_0^x \sin(t) \, dt$ was immediately found to be $\sin(x)$."

Fundamental Theorem of Calculus, Part 2

If $F$ is any antiderivative of $f$ on [a, b], then $\int_{a}^{b} f(x) dx = F(b) - F(a)$; used to evaluate definite integrals exactly.

Example: "By applying the         , the student evaluated $\int_{1}^{3} 2x dx$ 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 $\frac{dy}{dx} = 2x$ was $y = x^2 + C$, 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 $x \to +\infty$ or $x \to -\infty$; determined by evaluating limits at infinity.

Example: "The function had a          at $y = 3$ because the limit of $f(x)$ as $x \to \infty$ equaled 3."

Horizontal Tangent Line

A tangent line with slope zero; occurs where $f'(x) = 0$; for parametric curves, occurs where $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} \neq 0$.

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 $\frac{dy}{dx}$ 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 $x^2 + y^2 = 25$, the student used          and solved for $\frac{dy}{dx}$."

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 \to ∞}
^{1}_{b} f(x) dx.

Example: "The         
^{1}_{∞} (1/x^2) 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 $\int f(x) \, dx = F(x) + C$; represents the entire family of antiderivatives.

Similar definitions: antiderivative, primitive

Example: "Evaluating the          of $3x^2$ yielded $x^3 + C$, 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; $\lim_{x \to c} f(x) = \infty$ (or $-\infty$); indicates a vertical asymptote at x = c.

Example: "The          as $x \to 0^+$ of $\frac{1}{x}$ equals +$\infty$, 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 $a_n = f(n)$, the series $\sum a_n$ and the integral $\int_{1}^{\infty} f(x) dx$ either both converge or both diverge.

Example: "The          was applied to $\sum(\frac{1}{n^2})$ by evaluating $\int_{1}^{\infty} (\frac{1}{x^2}) dx$ 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 $u = x^2 + 1$, the student used          to evaluate $\int 2x(x^2+1)^5 \, dx$."

Integration of Trigonometric Functions

Standard antiderivative formulas for trig functions: $\int \sin x \, dx = -\cos x + C$, $\int \cos x \, dx = \sin x + C$, $\int \sec^2x \, dx = \tan x + C$, and related formulas; essential for AP BC integration problems.

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

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, $\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1 - x^2}}$ and $\frac{d}{dx}[\arctan(x)] = \frac{1}{1 + x^2}$.

Example: "Recognizing the integrand as a          form, the student integrated $\frac{1}{1 + x^2}$ 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 $\lim \frac{f(x)}{g(x)}$ yields 0/0 or $\pm\infty/\pm\infty$, then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$, 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) - P_n(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 $\frac{dy}{dx}$ (first derivative), $\frac{d^2y}{dx^2}$ (second derivative), and so on; emphasizes the ratio of infinitesimal changes and is used in the chain rule form $\frac{dy}{dx} = \left(\frac{dy}{du}\right)\left(\frac{du}{dx}\right)$.

Example: "When applying the chain rule, the student used          to write $\frac{dy}{dx} = \left(\frac{dy}{du}\right)\left(\frac{du}{dx}\right)$ and clearly track each substitution."

Limit

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

Limit Comparison Test

If $\lim_{n \to ∞} \frac{a_n}{b_n} = L$ where $0 < L < ∞$, then $\Sigma a_n$ and $\Sigma b_n$ either both converge or both diverge.

Example: "The          was used to compare $\Sigma\frac{2n+1}{n^3+1}$ to the convergent p-series $\Sigma\frac{1}{n^2}.$"

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 $x \to +\infty$ or $x \to -\infty$; 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

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 $u = x^2 + 1$, the student used          to evaluate
{2}x(x^2+1)^{5} dx."

Local Maximum

A point $x = c$ where $f(c) \ge f(x)$ 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 $f(c) \le f(x)$ 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: $\frac{dP}{dt} = kP(1 - \frac{P}{L})$, 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 $a = 0$: $f(x) = \sum \frac{f^{(n)}(0)}{n!} x^n$; common examples include those for $e^x$, $\sin x$, $\cos x$, and $\frac{1}{1-x}$.

Example: "The          for $e^x$ is $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + …$, 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)) $\int_a^b f(x) \,dx$; 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 ($a_{n+1} \geq a_n$) or entirely non-increasing ($a_{n+1} \leq a_n$); 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 $\frac{d}{dx}[\ln x] = \frac{1}{x}$ and its antiderivative is $\int(\frac{1}{x}) \, dx = \ln|x| + C$.

Example: "The student used the          antiderivative rule to evaluate $\int(\frac{1}{x}) \, dx = \ln|x| + C$."

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 $m_{normal} = -\frac{1}{f'(a)}$.

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

nth-Term Divergence Test

If $\lim_{n \to \infty} a_n \ne 0$, then the series $\sum a_n$ 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 $\frac{1}{2}$, 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 $x \to \pm \infty$; 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: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$; the second derivative $\frac{d^2y}{dx^2} = \frac{d(\frac{dy}{dx})/dt}{dx/dt}$.

Example: "The          of the curve defined by $x = t^2$, $y = t^3$ was found as $\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}$."

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         : $x = v_0t$ and $y = -16t^{2} + v_0t$."

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          $y = 2e^{x}$ from the general solution $y = Ce^{x}$."