AP Calculus AB Vocabulary

Absolute Maximum

The largest value a function attains over its entire domain or a specified interval; also called the global maximum.

Similar definitions: Global Maximum

Example: "On [0, $2\pi$], $f(x) = \sin(x)$ has an          of 1 at $x = \frac{\pi}{2}$."

Absolute Minimum

The smallest value a function attains over its entire domain or a specified interval; also called the global minimum.

Similar definitions: Global Minimum

Example: "f(x) = $x^2$ has an          of 0 at $x = 0$."

Acceleration

The instantaneous rate of change of velocity with respect to time: a(t) = v'(t) = s''(t). Positive acceleration indicates increasing velocity; negative acceleration indicates decreasing velocity.

Example: "If v(t) = 3t², then          is a(t) = 6t, so at t = 2, a = 12 m/s²."

Accumulation Function

A function defined by a definite integral with a variable upper limit, F(x) = ∫ₐˣ f(t) dt; by the Fundamental Theorem of Calculus, F'(x) = f(x).

Similar definitions: Integral function

Example: "F(x) = ∫₀ˣ t² dt is an          whose derivative equals x²."

Antiderivative

A function F(x) such that F'(x) = f(x); the reverse operation of differentiation. Because the derivative of any constant is zero, antiderivatives include an arbitrary constant C.

Similar definitions: Indefinite integral, Primitive function

Example: "An          of f(x) = 2x is F(x) = x² + C."

Area Between Curves

The definite integral of the difference of two functions over an interval, $\int_a^b [f(x) - g(x)] \, dx$, where $f(x) \ge g(x)$, representing the enclosed area between the curves.

Example: "The          $y = x^2$ and $y = x$ on [0,1] is $\int_0^1 (x - x^2) \, dx = \frac{1}{6}$."

Asymptote

A line that a curve approaches arbitrarily closely but may never reach. Asymptotes can be horizontal, vertical, or oblique.

Example: "The line y = 0 is a horizontal          of f(x) = 1/x."

Average Rate of Change

Leibniz notation for the derivative of y with respect to x, representing the instantaneous rate of change of y as x changes. Equivalent to $f'(x)$ when $y = f(x)$.

Similar definitions: Leibniz notation for derivative

Example: "For $y = x³$,          = $3x²$."

Average Value of a Function

The mean output of a function over a closed interval [a, b], computed as $\frac{1}{b-a} \int_a^b f(x) \,dx$.

Example: "The          of f(x) = x² on [0, 3] is $\frac{1}{3} \int_0^3 x^2 \,dx = 3$."

Chain Rule

A differentiation rule for composite functions: if $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \, \cdot \, g'(x)$. In Leibniz notation: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

Example: "By the         , $\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \, \cdot \, 2x$."

Closed Interval

An interval that includes both its endpoints, written [a, b]. The Extreme Value Theorem guarantees absolute extrema for continuous functions on closed intervals.

Example: "The EVT applies to f(x) = x² on the          [−2, 3]."

Composite Function

A function formed by applying one function to the output of another: h(x) = f(g(x)). The Chain Rule is used to differentiate composite functions.

Example: "h(x) = (3x + 1)⁵ is a          where f(u) = u⁵ and g(x) = 3x + 1."

Concave Downward

A function is concave downward (concave down) on an interval where its second derivative is negative ($f''(x) < 0$), meaning the curve bends like an upside-down bowl.

Similar definitions: Concave down

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

Concave Upward

A function is concave upward (concave up) on an interval where its second derivative is positive ($f''(x) > 0$), meaning the curve bends like a bowl opening upward.

Similar definitions: Concave up

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

Concavity

The property describing whether a curve bends upward (f'' > 0) or downward (f'' < 0). Changes in concavity occur at inflection points.

Example: "Analyzing          of f(x) = x³ reveals it is concave down for x < 0 and concave up for x > 0."

Constant of Integration

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

Example: "$\int 2x \,dx = x^2 + C$, where $C$ is the         ."

Constant Rule

The derivative of a constant function is zero: d/dx[c] = 0.

Example: "By the         , d/dx[7] = 0."

Continuity

A function $f$ is continuous at $x = c$ if: $f(c)$ is defined, $\lim_{x\rightarrow c} f(x)$ exists, and $\lim_{x\rightarrow c} f(x) = f(c)$. A continuous function has no holes, jumps, or vertical asymptotes at that point.

Example: "f(x) = $x^2$ has          at every real number because no breaks exist."

Continuous on an Interval

A function is continuous on an interval if it is continuous at every point in that interval (including one-sided continuity at endpoints for closed intervals).

Example: "f(x) = $\sqrt{x}$ is          [0, $\infty$) since it has no breaks on that interval."

Critical Number

A value $x = c$ in the domain of f where $f'(c) = 0$ or $f'(c)$ is undefined. Critical numbers are candidates for local extrema.

Similar definitions: Critical value

Example: "f(x) = x³ − 3x has         s at $x = 1$ and $x = -1$, where $f'(x) = 0$."

Critical Point

A point $(c, f(c))$ on the graph of $f$ where $f'(c) = 0$ or $f'(c)$ is undefined; these are the locations where local maxima, local minima, or inflection points may occur.

Decreasing Function

A function is decreasing on an interval if f'(x) < 0 for all x in that interval; as x increases, f(x) decreases.

Example: "f(x) = −x is          on (−∞, ∞) because f'(x) = −1 < 0."

Definite Integral

The signed area under a curve f(x) from x = a to x = b, denoted $\int_a^b f(x) \,dx$. Evaluated using the Fundamental Theorem of Calculus as F(b) − F(a), where F is any antiderivative of f.

Example: "The          $\int_0^2 x^2 \,dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3}$."

Derivative

The instantaneous rate of change of a function at a point, defined as $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Geometrically, it is the slope of the tangent line to the curve at that point.

Example: "The          of f(x) = x³ is $f'(x) = 3x²$."

Derivative of Inverse Function

If f and g are inverses, then $g'(x) = \frac{1}{f'(g(x))}$, provided $f'(g(x)) \neq 0$. This allows finding the derivative of an inverse without explicitly solving for it.

Example: "Using the          formula, if $f(x) = x³$ and $g = f^{-1}$, then $g'(8) = \frac{1}{f'(2)} = \frac{1}{12}$."

Difference Quotient

The expression [f(x+h) − f(x)] / h, which represents the average rate of change of f over [x, x+h] and whose limit as h→0 defines the derivative.

Example: "The          of f(x) = x² simplifies to 2x + h, which approaches 2x as h → 0."

Difference Rule

The derivative of the difference of two functions equals the difference of their derivatives: $\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$.

Example: "By the         , $\frac{d}{dx}[x^3 - \text{sin }x] = 3x^2 - \text{cos }x.$"

Differentiable

A function is differentiable at x = c if its derivative exists at that point; the function must be continuous and have no corner, cusp, or vertical tangent at c.

Example: "f(x) = |x| is not          at x = 0 because there is a corner."

Differential Equation

An equation relating a function to one or more of its derivatives; in AP Calculus AB, separable differential equations and exponential growth/decay models are primary focuses.

Similar definitions: ODE (Ordinary Differential Equation)

Example: "$\frac{dy}{dx} = ky$ is a          whose solution is $y = Ce^{kx}$."

Differentials

Infinitesimally small changes in variables, written dx and dy, related by dy = f'(x) dx. Used in linear approximations and integral notation.

Example: "For y = x², dy = 2x dx; if x = 3 and dx = 0.01, then dy ≈ 0.06."

Differentiation

The process of computing the derivative of a function using differentiation rules such as the power rule, product rule, quotient rule, and chain rule.

Example: "         of $f(x) = x^4 + 3x^2$ yields $f'(x) = 4x^3 + 6x$."

Discontinuous

A function is discontinuous at x = c if it fails any of the three conditions for continuity: f(c) undefined, limit does not exist, or limit ≠ f(c).

Example: "f(x) = 1/x is          at x = 0 because f(0) is undefined."

Displacement

The net change in position of a particle over a time interval: $\text{displacement} = ∫_a^b v(t) \, dt$. Unlike total distance, displacement accounts for direction and can be negative.

Example: "If $v(t) = t - 2$ on [0, 4], the          is $∫_0^4 (t - 2) \, dt = 0$."

Domain

The set of all input values (x-values) for which a function is defined.

Example: "The          of $f(x) = \sqrt{x}$ is $[0, \infty]$ since the square root requires a non-negative input."

dy/dx

The line perpendicular to the tangent line at a point on a curve; if the tangent slope is $m \neq 0$, the normal line has slope $-\frac{1}{m}$.

Example: "At $x = 1$ on f(x) = x², the tangent slope is $2$, so the          has slope $-\frac{1}{2}$."

Epsilon-Delta Definition of a Limit

The formal definition: $\lim_{x \to c} f(x) = L$ means that for every $\varepsilon > 0$, there exists $\delta > 0$ such that $0 < |x - c| < \delta$ implies $|f(x) - L| < \varepsilon$.

Example: "Using the         , one can rigorously prove that $\lim_{x \to 2} (3x) = 6$."

Euler's Method

A numerical procedure for approximating solutions to differential equations using the tangent line: $y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x$, stepping from an initial value.

Example: "Using          with step size 0.1 on $\frac{dy}{dx} = y$, $y(0) = 1$ gives $y(0.1) \approx 1.1$."

Exponential Decay

A decrease modeled by $y = Ce^{kt}$ with $k < 0$; the quantity decreases at a rate proportional to its current value. Governed by the differential equation $\frac{dy}{dt} = ky$.

Example: "Radioactive substances undergo         , halving at regular time intervals."

Exponential Function

A function of the form f(x) = aˣ (a > 0, a ≠ 1); the most important case in calculus is f(x) = eˣ, whose derivative equals itself: d/dx[eˣ] = eˣ.

Example: "The          f(x) = eˣ is its own derivative and its own antiderivative (up to a constant)."

Exponential Growth

An increase modeled by y = Ce^(kt) with k > 0; the quantity grows at a rate proportional to its current value, governed by dy/dt = ky.

Example: "A population exhibiting          doubles at regular time intervals."

Exponential Growth and Decay

A family of models y = Ce^(kt) satisfying dy/dt = ky. When k > 0 the quantity grows; when k < 0 it decays. C is the initial value at t = 0.

Example: "Newton's Law of Cooling is an          model: dT/dt = k(T − T_env)."

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 [a, b].

Example: "The          guarantees f(x) = sin x achieves a max and min on [0, $2\pi$]."

Extremum

A general term for a maximum or minimum value of a function, either local or global.

Similar definitions: Extreme value

Example: "Every local maximum and local minimum is an          of the function."

f'(x) Notation

Prime notation for the derivative of a function f with respect to x, read as "f prime of x." Equivalent to $\frac{dy}{dx}$ in Leibniz notation.

Similar definitions: Prime notation

Example: "If $f(x) = x^4$, then          = $4x^3$."

Fermat's Theorem

If f has a local extremum at an interior point c and f is differentiable at c, then f'(c) = 0. This is the basis for finding extrema using critical numbers.

Example: "By         , any differentiable local max or min occurs where f'(c) = 0."

First Derivative Test

A method to classify critical numbers: if $f'$ changes from positive to negative at $c$, then $f$ has a local maximum at $c$; if $f'$ changes from negative to positive, there is a local minimum; if $f'$ does not change sign, there is neither.

Example: "The          shows $f(x) = x^3 - 3x$ has a local max at $x = -1$ and local min at $x = 1$."

First Fundamental Theorem of Calculus

If $f$ is continuous on $[a, b]$ and $F(x) = \textstyle \bigg( \frac{F(b) - F(a)}{b - a} \bigg)$, then $F'(x) = f(x)$. This theorem links differentiation and integration as inverse processes.

Fundamental Theorem of Calculus

The theorem linking differentiation and integration. Part 1: $\frac{d}{dx}[\int_a^x f(t) \text{dt}] = f(x)$. Part 2: $\int_a^b f(x) \, dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$.

Example: "The          allows evaluating $\int_1^4 2x \, dx$ as [$x^2$]$_1^4 = 16 - 1 = 15$."

General Solution

The complete family of solutions to a differential equation, expressed with an arbitrary constant $C$. A specific solution is obtained by applying an initial condition.

Example: "The          of $\frac{dy}{dx} = 2x$ is $y = x^2 + C$."

Higher-Order Derivative

A derivative of a derivative; the second derivative $f''(x)$ measures concavity, the third derivative $f'''(x)$ measures the rate of change of concavity, and so on.

Example: "The          $f''(x)$ of $f(x) = x^4$ is $12x^2$."

Hole (Removable Discontinuity)

A removable discontinuity at $x = c$ where $\lim_{x\rightarrow c} f(x)$ exists but either $f(c)$ is undefined or $f(c) \neq \text{the limit}$. The graph has a "hole" at that point.

Example: "f(x) = $\frac{x^2 - 1}{x - 1}$ has a          at $x = 1$ because $f(1)$ is undefined but the limit is 2."

Horizontal Asymptote

A horizontal line $y = L$ that a function approaches as $x \to \infty$ or $x \to -\infty$, i.e., $\lim_{x \to ±∞} f(x) = L$.

Example: "f(x) = $\frac{2x}{x + 1}$ has a          at $y = 2$ as $x \to \infty$."

Implicit Differentiation

A technique for differentiating equations where y is not explicitly solved for in terms of x; differentiate both sides with respect to x, treating y as a function of x and applying the chain rule to y-terms.

Example: "Using          on x² + y² = 25 gives dy/dx = −x/y."

Increasing Function

A function is increasing on an interval if $f'(x) > 0$ for all $x$ in that interval; as $x$ increases, $f(x)$ increases.

Example: "f(x) = $e^x$ is an          on ($-\infty, \infty$) since its derivative $e^x$ is always positive."

Indefinite Integral

The general antiderivative of a function, written $\int f(x) \, dx = F(x) + C$, representing the family of all functions whose derivative is $f$.

Example: "The          of $f(x) = 3x^2$ is $x^3 + C$."

Indeterminate Form

A limit expression whose value cannot be determined by direct substitution alone; common types are $\frac{0}{0}$ and $\frac{\infty}{\infty}$, which can often be resolved with L'Hôpital's Rule or algebraic manipulation.

Infinite Limit

A limit in which the function grows without bound as x approaches a value or as x → ±∞; written lim_{x→c} f(x) = ∞ or lim_{x→∞} f(x) = L.

Example: "lim_{x→0⁺} (1/x) is an          equal to +∞."

Inflection Point

A point on a curve where the concavity changes (from concave up to concave down or vice versa); occurs where $f''(x) = 0$ or $f''(x)$ is undefined, provided the sign of $f''$ actually changes.

Example: "f(x) = x³ has an          at x = 0 where concavity changes from down to up."

Initial Condition

A known value of the solution to a differential equation at a specific input, used to determine the arbitrary constant C in the general solution.

Example: "Given dy/dx = 2x and the          y(0) = 3, the solution is y = x² + 3."

Initial Value Problem

A differential equation paired with an initial condition; the goal is to find the unique particular solution satisfying both the equation and the given starting value.

Example: "Solving the          dy/dx = 3x², y(1) = 2 gives y = x³ + 1."

Instantaneous Rate of Change

The derivative $f'(x)$ at a specific point; the limiting value of the average rate of change as the interval shrinks to zero. It equals the slope of the tangent line at that point.

Example: "The          of $f(x) = x^2$ at $x = 3$ is $f'(3) = 6$."

Integrand

The function being integrated in an integral expression; in $\int f(x) \,dx$, the integrand is f(x).

Example: "In $\int (x² + 1) \,dx$, the          is $x² + 1$."

Integration

The process of finding the antiderivative (indefinite integral) or computing the accumulated area (definite integral) of a function.

Example: "         of f(x) = 2x gives F(x) = $x^2 + C$."

Integration by Substitution

A technique for evaluating integrals by substituting u = g(x), du = g'(x) dx to simplify the integral into a basic form. Also called u-substitution.

Similar definitions: u-substitution, change of variables

Example: "Using          with u = x², ∫ 2x·cos(x²) dx = sin(x²) + 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 ($a, b$) with $f(c) = N$. Guarantees that continuous functions hit every intermediate value.

Example: "By the         , $f(x) = x^3 - x - 1$ has a root between 1 and 2 since $f(1) < 0$ and $f(2) > 0$."

Inverse Function

A function g such that g(f(x)) = x and f(g(x)) = x; the graph of the inverse is a reflection of the original graph across y = x. The derivative of an inverse can be found using the inverse function derivative formula.

Example: "The          of f(x) = eˣ is g(x) = ln x."

Jump Discontinuity

A discontinuity where the left-hand and right-hand limits both exist but are not equal; the function "jumps" from one value to another.

Example: "The greatest integer function has a          at every integer."

L'Hôpital's Rule

If $\lim_{x \to c} \frac{f(x)}{g(x)}$ is an indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$, provided the latter limit exists.

Example: "Using         , $\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1$."

Left Riemann Sum

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

Example: "A          with 4 equal subintervals overestimates $\int_0^4 (-x + 4) \, dx$ because the function is decreasing."

Left-Hand Limit

The value that $f(x)$ approaches as $x$ approaches $c$ from the left ($x < c$), written $\lim_{x \to c^-} f(x)$.

Example: "For $f(x) = \frac{|x|}{x}$, the          as $x \to 0$ is -1."

Leibniz Notation

The notation dy/dx for the derivative and ∫f(x) dx for the integral, emphasizing the relationship between infinitesimal changes. Widely used for implicit differentiation, related rates, and integration.

Example: "In         , the chain rule is written dy/dx = (dy/du)(du/dx)."

Limit

The value that a function f(x) approaches as x gets arbitrarily close to a particular value c; written $\lim_{x \to c} f(x) = L$. The function need not be defined at c itself.

Example: "$\lim_{x \to 2} \frac{(x^2 - 4)}{(x - 2)} = 4$, found by factoring; this is the          of the expression."

Limit at Infinity

The value a function approaches as x increases or decreases without bound: $\lim_{x \to \infty} f(x)$ or $\lim_{x \to -\infty} f(x)$. Used to identify horizontal asymptotes.

Example: "$\lim_{x \to \infty} \frac{3x^2}{x^2 + 1} = 3$ is a          that reveals the horizontal asymptote y = 3."

Linear Approximation

The approximation of f(x) near x = a using the tangent line: L(x) = f(a) + f'(a)(x − a). Valid for x close to a when f is differentiable at a.

Similar definitions: Linearization, Tangent line approximation

Example: "The          of f(x) = √x at x = 4 is L(x) = 2 + (1/4)(x − 4)."

Local Maximum

A function value $f(c)$ that is greater than or equal to $f(x)$ for all $x$ in some open interval around $c$; the function has a "peak" at that point.

Similar definitions: Relative maximum

Example: "f(x) = $-x^2 + 4$ has a          of 4 at $x = 0$."

Local Minimum

A function value f(c) that is less than or equal to f(x) for all x in some open interval around c; the function has a "valley" at that point.

Similar definitions: Relative minimum

Example: "f(x) = x² − 4 has a          of −4 at x = 0."

Logarithmic Differentiation

A technique that simplifies differentiation of products, quotients, or powers by taking the natural log of both sides, differentiating implicitly, and solving for dy/dx.

Example: "Using         , d/dx[xˣ] is found by writing ln y = x ln x and differentiating implicitly."

Lower Sum

A Riemann sum approximation that uses the minimum function value on each subinterval as the rectangle height, giving an underestimate for positive functions.

Example: "The          for an increasing function uses left endpoints for rectangle heights."

Mean Value Theorem

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c$ in $(a, b)$ where $f'(c) = \frac{f(b) - f(a)}{b - a}$; the instantaneous rate of change equals the average rate of change at some interior point.

Mean Value Theorem for Integrals

If $f$ is continuous on $[a, b]$, then there exists $c$ in $(a, b)$ such that $f(c) = \frac{1}{b - a} \int_a^b f(x) \, dx$; i.e., a continuous function attains its average value at some point in the interval.

Example: "The          guarantees $f(c) = 3$ for $f(x) = x^2$ on $[0, 3]$ at some $c$ in $(0, 3)$."

Midpoint Riemann Sum

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

Example: "A          generally gives a better approximation than left or right sums for the same number of subintervals."

Multiplication by Constant Rule

The derivative of a constant times a function equals the constant times the derivative: $\frac{d}{dx}[c \, f(x)] = c \, f'(x)$.

Example: "By the         , $\frac{d}{dx}[5x^3] = 5 \, 3x^2 = 15x^2$."

Natural Logarithm

The logarithm with base $e$, written \lnx = \log_e x; defined for $x > 0$. Its derivative is $\frac{d}{dx}[\ln x] = \frac{1}{x}$, and its antiderivative rule gives $\int \left(\frac{1}{x}\right) dx = \ln|x| + C$.

Example: "The          satisfies $\frac{d}{dx}[\ln x] = \frac{1}{x}$."

Net Area

The value of a definite integral interpreted as signed area: regions above the x-axis are positive, regions below are negative. Net area may be less than the total area.

Example: "$\int_0^{2\pi} \sin x \,dx = 0$ because the          above and below the x-axis cancel out."

Net Change Theorem

The integral of a rate of change gives the net change of the quantity: $∫_a^b F'(x) \, dx = F(b) - F(a)$. It connects the definite integral with total accumulation.

Example: "The          states that $∫_0^3 v(t) \, dt$ gives the net displacement of a particle from t = 0 to t = 3."

Newton's Method

An iterative numerical method for approximating zeros of a function using the formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, based on the tangent line approximation.

Example: "         can approximate $\sqrt2$ by applying the iteration to $f(x) = x^2 - 2$."

Normal Line

The line perpendicular to the tangent line at a point on a curve; if the tangent slope is $m \neq 0$, the normal line has slope $-\frac{1}{m}$.

Example: "At $x = 1$ on f(x) = x², the tangent slope is $2$, so the          has slope $-\frac{1}{2}$."

One-Sided Limit

A limit evaluated as x approaches c from only one direction: the left-hand limit $\lim_{x \to c^-} f(x)$ or the right-hand limit $\lim_{x \to c^+} f(x)$. The two-sided limit exists only if both one-sided limits are equal.

Example: "The          from the right of f(x) = |x|/x at x = 0 equals +1."

Optimization

The process of finding the absolute maximum or minimum of a function on a given domain by analyzing critical points and endpoints; widely applied in real-world problems.

Example: "         is used to find the dimensions of a rectangle with fixed perimeter that maximize area."

Particle Motion

The analysis of a moving particle's position s(t), velocity v(t) = s'(t), and acceleration a(t) = v'(t) using derivatives and integrals. Speed is |v(t)|.

Example: "In         , a particle moves right when v(t) > 0 and left when v(t) < 0."

Particular Solution

The unique solution to a differential equation that satisfies a given initial condition, obtained by substituting the initial condition into the general solution to find the value of C.

Example: "Given dy/dx = 2x and y(0) = 5, the          is y = x² + 5."

Piecewise Function

A function defined by different formulas on different parts of its domain; limits and continuity at the boundary points require special care.

Example: "f(x) = x² for x < 0 and f(x) = x for x ≥ 0 is a         ."

Position Function

A function s(t) giving the location of a particle at time t; its derivative is velocity v(t) = s'(t) and its second derivative is acceleration a(t) = s''(t).

Example: "If the          is s(t) = t³ − 6t², then v(t) = 3t² − 12t."

Power Rule (Differentiation)

The derivative of $x^n$ is $nx^{n-1}$: $\frac{d}{dx}[x^n] = nx^{n-1}$, valid for any real number $n$.

Example: "By the         , $\frac{d}{dx}[x^5] = 5x^4.$"

Power Rule (Integration)

The integral of $x^n$ (n ≠ −1) is $\frac{x^{n+1}}{n+1} + C$: $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$.

Example: "By the         , $\int x^3 \,dx = \frac{x^4}{4} + C$."

Product Rule

The derivative of a product of two functions: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$.

Example: "By the         , $\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x$."

Quotient Rule

The derivative of a quotient of two functions: d/dx[f(x)/g(x)] = [f'(x)g(x) − f(x)g'(x)] / [g(x)]².

Example: "Using the         , d/dx[sin x / x] = (x cos x − sin x) / x²."

Rate of Change

A measure of how quickly one quantity changes relative to another; instantaneous rate of change is the derivative, while average rate of change is the difference quotient over an interval.

Example: "The          of volume with respect to time describes how fast a tank fills."

Related Rates

Problems in which two or more quantities are related by an equation, and their rates of change with respect to time are linked by implicit differentiation of that equation.

Example: "A          problem: given a ladder sliding down a wall, find how fast the base moves using $x^2 + y^2 = L^2$."

Relative Extremum

A local maximum or local minimum; a function value that is the highest or lowest in some neighborhood of the point, though not necessarily over the entire domain.

Similar definitions: Local extremum

Example: "f(x) = x³ − 3x has         s at x = ±1."