Maths AA terms in Calculus

The rate of change of velocity with respect to time, $a=\frac{dv}{dt}=\frac{d^{2}s}{d{t}^2}$.

A function $F$ whose derivative is a given function $f$, so that $F^{\mathrm{\prime }}(x)=f(x)$. Every continuous function has infinitely many antiderivatives, which differ only by a constant.

The area of the region enclosed by two curves $y=f(x)$ and $y=g(x)$ between $x=a$ and $x=b$, found as ${\int }_a^b\mathrm{\mid }f(x)-g(x)\mathrm{\mid }dx$; the limits are often the $x$-coordinates of the points where the curves intersect.

The change in a function divided by the change in its input over an interval, $\frac{f(b)-f(a)}{b-a}$: the gradient of the chord joining the two points on the graph.

The fixed line about which a region is turned to form a solid of revolution. Every cross-section of the solid perpendicular to it is a circle or a ring.

A known value of a function, such as a point on its graph, used to find the constant of integration and so pick out one antiderivative from the family $F(x)+C$.

The largest population $K$ that an environment can support in the long term; in the logistic model every positive solution approaches $K$ as time increases.

The rule for differentiating a composite function: if $y=g(u)$ and $u=f(x)$, then $\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$.

Describes a curve on an interval where its gradient is decreasing, so it bends downwards and lies below its tangents. This happens where $f^{\mathrm{\prime }\mathrm{\prime }}(x)<0$.

Describes a curve on an interval where its gradient is increasing, so it bends upwards and lies above its tangents. This happens where $f^{\mathrm{\prime }\mathrm{\prime }}(x)>0$.

The direction in which a curve bends on an interval: concave up or concave down. It is decided by the sign of the second derivative.

The arbitrary constant $C$ added to an antiderivative to give every function with the same derivative. Its value can be found from a boundary condition, a known point on the curve.

A condition linking the variables in an optimisation problem, such as a fixed volume or a fixed length of material. It is used to write the quantity to be optimised in terms of a single variable.

A function $f$ is continuous at $x=a$ if $f(a)$ is defined, ${\lim }_{x\to a}f(x)$ exists, and the two are equal. Informally, its graph can be drawn through $x=a$ without lifting the pen.

A point where a continuous graph changes direction sharply, so the gradients from the left and the right are different (a corner) or tend to $\pm \mathrm{\infty }$ (a cusp, as for $y=x^{2\mathrm{/}3}$ at $0$). The function is not differentiable there.

A function is decreasing on an interval if larger inputs give smaller outputs there: $x_1<x_2$ implies $f(x_1)>f(x_2)$. If $f^{\mathrm{\prime }}(x)<0$ on an interval, $f$ is decreasing on it.

The number ${\int }_a^{b}f(x)dx$, the limit of sums of $f(x)\delta x$ from $x=a$ to $x=b$. When $f(x)>0$ on $[a,b]$ it equals the area between the curve and the $x$-axis.

The function $f^{\mathrm{\prime }}(x)$, also written $\frac{dy}{dx}$, that gives the gradient of the curve $y=f(x)$ at each value of $x$. It measures the instantaneous rate of change of $y$ with respect to $x$.

A function is differentiable at $x=a$ if ${\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$ exists, so its graph has a single, non-vertical tangent there. A differentiable function is continuous, but not every continuous function is differentiable.

An equation that relates an unknown function to one or more of its derivatives, such as $\frac{dy}{dx}=2xy$. It is first-order when the highest derivative it contains is the first derivative.

The process of finding the derivative of a function. Differentiating $y=f(x)$ with respect to $x$ gives $\frac{dy}{dx}=f^{\mathrm{\prime }}(x)$.

Finding a derivative directly from its definition $f^{\mathrm{\prime }}(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$, by simplifying the gradient of a chord and then letting $h$ tend to $0$.

Finding a volume of revolution by slicing the solid perpendicular to the axis of rotation into thin discs of volume $\pi r^2\delta x$ and adding them with an integral.

A point where a function is not continuous, such as a jump (the one-sided limits differ), a hole (the limit exists but is not $f(a)$) or a vertical asymptote.

The position of a particle relative to a fixed origin, measured along a line with a sign for direction, usually written $s$. A change in displacement is final position minus initial position.

The term of an expression that grows fastest as $x\to \mathrm{\infty }$ and so decides its behaviour; in a polynomial it is the term with the highest power of $x$. For example, $3x^2$ is the dominant term of $3x^2-5x+1$.

A numerical method that approximates the solution of $\frac{dy}{dx}=f(x,y)$ by short straight steps along the gradient: $y_{n+1}=y_n+hf(x_n,y_n)$, $x_{n+1}=x_n+h$.

A way to classify a stationary point at $x=a$ from the sign of $f^{\mathrm{\prime }}(x)$ just before and just after $a$: positive then negative gives a maximum, negative then positive a minimum, no change of sign a stationary point of inflexion.

The result linking differentiation and integration: if $F^{\mathrm{\prime }}(x)=f(x)$, then ${\int }_a^{b}f(x)dx=F(b)-F(a)$. Equivalently, the rate of change of the area under $y=f(t)$ from $a$ to $x$ is $f(x)$.

The family of all solutions of a differential equation, written with an arbitrary constant such as $C$; a first-order equation has one arbitrary constant in its general solution.

A derivative found by differentiating more than once. The $n$th derivative of $f$ is written $f^{(n)}(x)$ or $\frac{d^{n}y}{d{x}^n}$.

A first-order differential equation that can be written as $\frac{dy}{dx}=f\left(\frac{y}{x}\right)$. The substitution $y=vx$ turns it into a separable equation in $v$ and $x$.

Differentiating both sides of an equation connecting $x$ and $y$ with respect to $x$, treating $y$ as a function of $x$ (so that, for example, $\frac{d}{dx}(y^2)=2y\frac{dy}{dx}$), then rearranging to make $\frac{dy}{dx}$ the subject.

An equation connecting $x$ and $y$ that is not written in the form $y=f(x)$, such as $x^2+y^2=25$. An equation written as $y=f(x)$ defines $y$ explicitly.

A function is increasing on an interval if larger inputs give larger outputs there: $x_1<x_2$ implies $f(x_1)<f(x_2)$. If $f^{\mathrm{\prime }}(x)>0$ on an interval, $f$ is increasing on it.

The family of all antiderivatives of $f(x)$, written $\int f(x)dx=F(x)+C$, where $F^{\mathrm{\prime }}(x)=f(x)$ and $C$ is an arbitrary constant.

An expression such as $\frac{0}{0}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$ obtained by substituting into a limit, which gives no information about the value of the limit. The limit may be any number, or may not exist, and another method is needed to find it.

A known value of the solution at one point, such as $y=2$ when $x=0$ (written $y(0)=2$), used to find a particular solution.

The rate of change of a function at a single value of its input: the limit of the average rate of change over smaller and smaller intervals, equal to the derivative and to the gradient of the tangent.

The function being integrated: in $\int f(x)dx$ or ${\int }_a^{b}f(x)dx$, the integrand is $f(x)$.

The function $e^{\int P(x)dx}$ that multiplies $\frac{dy}{dx}+P(x)y=Q(x)$ so that the left-hand side becomes the derivative of a product, $\frac{d}{dx}\left(y{e}^{\int P(x)dx}\right)$.

Integrating by recognising the integrand as the result of the chain rule, $k{g}^{\mathrm{\prime }}(x)f(g(x))$, guessing the antiderivative (here $kF(g(x))$ where $F^{\mathrm{\prime }}=f$), then checking the guess by differentiating and adjusting the constant factor.

A method that reverses the product rule, used to integrate a product: $\int u\frac{dv}{dx}dx=uv-\int v\frac{du}{dx}dx$.

A method that reverses the chain rule: a new variable $u=g(x)$ replaces part of the integrand, $dx$ is replaced using $\frac{du}{dx}$, and the result is integrated with respect to $u$.

The set of values of $x$ for which a power series converges. A Maclaurin series can be used to evaluate or approximate its function only for $x$ in this interval, for example $-1<x\le 1$ for $\ln (1+x)$.

A curve on which a differential equation $\frac{dy}{dx}=f(x,y)$ gives the same gradient $k$ everywhere, so all slope-field segments on it are parallel; its equation is $f(x,y)=k$.

The study of motion without regard to its causes: describing how the position, velocity and acceleration of an object change with time.

A method for limits of the indeterminate forms $\frac{0}{0}$ and $\frac{\mathrm{\infty }}{\mathrm{\infty }}$: if ${\lim }_{x\to a}\frac{f(x)}{g(x)}$ is of either form, then ${\lim }_{x\to a}\frac{f(x)}{g(x)}={\lim }_{x\to a}\frac{f^{\mathrm{\prime }}(x)}{g^{\mathrm{\prime }}(x)}$, provided the second limit exists. It also holds as $x\to \mathrm{\infty }$.

A guide for choosing $u$ in integration by parts: take as $u$ the factor that comes first in the list Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.

The value $L$ that $f(x)$ gets as close as you like to as $x$ gets closer and closer to $a$ (without being equal to $a$), written ${\lim }_{x\to a}f(x)=L$. The value of $f(a)$ itself does not affect the limit.

The value $L$ that $f(x)$ approaches as $x$ increases without bound, written ${\lim }_{x\to \mathrm{\infty }}f(x)=L$ (similarly as $x\to -\mathrm{\infty }$). When it exists, the line $y=L$ is a horizontal asymptote of the graph.

A first-order differential equation that can be written as $\frac{dy}{dx}+P(x)y=Q(x)$, in which $y$ and $\frac{dy}{dx}$ appear only to the first power and are not multiplied together.

Taking the natural logarithm of both sides of $y=f(x)$ and then differentiating implicitly. It is used for functions such as $x^x$, where the variable is in both the base and the exponent, and to simplify long products and quotients.

A model of limited growth, $\frac{dP}{dt}=rP(1-\frac{P}{K})$: growth is close to exponential while $P$ is small and slows to zero as $P$ approaches the carrying capacity $K$.

The polynomial formed by the terms of a Maclaurin series up to a chosen power of $x$. It has the same value and the same first few derivatives as the function at $x=0$, so it approximates the function near $0$.

The infinite series $f(0)+x{f}^{\mathrm{\prime }}(0)+\frac{x^2}{2!}{f}^{\mathrm{\prime }\mathrm{\prime }}(0)+\frac{x^3}{3!}{f}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(0)+\dots$, which writes a function as a power series in $x$ built from its derivatives at $x=0$.

The model in which an object's temperature $T$ changes at a rate proportional to the difference between $T$ and the surrounding temperature $T_s$: $\frac{dT}{dt}=-k(T-T_s)$, with $k>0$.

The straight line through a point on a curve that is perpendicular to the tangent at that point. If the tangent has gradient $m\mathrm{\ne }0$, the normal has gradient $-\frac{1}{m}$.

The value $f(x)$ approaches as $x$ tends to $a$ from one side only: ${\lim }_{x\to a^{-}}f(x)$ from values less than $a$, and ${\lim }_{x\to a^{+}}f(x)$ from values greater than $a$.

Finding the greatest or least value of a quantity, such as an area, a cost or a time, by writing it as a function of one variable and locating its maximum or minimum on the allowed domain.

In a composite function $f(g(x))$, the function $f$ that is applied last, to the output of the inner function $g$. In $\sin (x^2)$ the outer function is sine.

The one solution of a differential equation that also satisfies a given condition, found by using the condition to fix the arbitrary constant in the general solution.

A function defined by different rules on different parts of its domain, for example $f(x)=x^2$ for $x<1$ and $f(x)=2x-1$ for $x\ge 1$.

A point on a curve where the concavity changes, from concave up to concave down or the reverse, so $f^{\mathrm{\prime }\mathrm{\prime }}(x)$ changes sign there. The gradient at the point may be zero or non-zero.

The point at which a tangent touches a curve. It lies on both the curve and the tangent, and the gradient of the tangent equals the derivative there.

The rule for differentiating a power of $x$: if $f(x)=a{x}^n$, then $f^{\mathrm{\prime }}(x)=an{x}^{n-1}$. Multiply by the power, then reduce the power by one.

The rule for differentiating a product of two functions of $x$: if $y=uv$, then $\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$.

The rule for differentiating one function of $x$ divided by another: if $y=\frac{u}{v}$, then $\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$.

How fast one quantity changes compared with another, measured in units of the first quantity per unit of the second (for example metres per second). On a graph it is a gradient.

Rates of change of two or more quantities that are linked by an equation, usually all with respect to time $t$. One rate is found from another with the chain rule, for example $\frac{dV}{dt}=\frac{dV}{dr}\times \frac{dr}{dt}$.

A sum of areas of thin rectangles, $\sum f(x_i)\delta x$, that approximates the area under a curve. As the rectangles become narrower, the sum tends to the definite integral.

The derivative of the derivative, written $f^{\mathrm{\prime }\mathrm{\prime }}(x)$ or $\frac{d^{2}y}{d{x}^2}$. It measures the rate of change of the gradient of $y=f(x)$.

A way to classify a stationary point at $x=a$: if $f^{\mathrm{\prime }\mathrm{\prime }}(a)<0$ it is a local maximum, if $f^{\mathrm{\prime }\mathrm{\prime }}(a)>0$ a local minimum. If $f^{\mathrm{\prime }\mathrm{\prime }}(a)=0$ the test gives no conclusion.

A method for solving a differential equation that can be written as $g(y)\frac{dy}{dx}=f(x)$: integrate both sides, giving $\int g(y)dy=\int f(x)dx$.

A number line showing where an expression such as $f^{\mathrm{\prime }}(x)$ is positive, negative or zero, with the values where it is zero (or undefined) marked as boundaries.

The value of a definite integral read as an area: regions above the $x$-axis count as positive and regions below count as negative, so ${\int }_a^{b}f(x)dx$ is the area above minus the area below.

A diagram of short line segments at the points of a grid, each with gradient equal to the value of $\frac{dy}{dx}$ given by a differential equation at that point. Solution curves follow the segments.

The magnitude of the velocity, $\mathrm{\mid }v\mathrm{\mid }$: how fast a particle moves, whatever its direction. Speed is never negative.

An integral whose result is known directly from a standard derivative read backwards, such as $\int {\sec }^{2}xdx=\tan x+C$ or $\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan \left(\frac{x}{a}\right)+C$.

A limit whose value is used so often that it is quoted as a known result, such as ${\lim }_{x\to 0}\frac{\sin x}{x}=1$ or ${\lim }_{x\to 0}\frac{e^x-1}{x}=1$.

A point on the graph of $y=f(x)$ where the gradient is zero, $f^{\mathrm{\prime }}(x)=0$, so the tangent is horizontal. It is a local maximum, a local minimum or a stationary point of inflexion.

A point of inflexion where the gradient is also zero. The sign of $f^{\mathrm{\prime }}(x)$ is the same on both sides, so the point is stationary but neither a maximum nor a minimum, as at the origin on $y=x^3$.

The constant amount $h$ by which $x$ increases at each step of a numerical method such as Euler's method.

The derivative of a sum or difference of functions is the sum or difference of their derivatives: if $f(x)=g(x)\pm h(x)$, then $f^{\mathrm{\prime }}(x)=g^{\mathrm{\prime }}(x)\pm h^{\mathrm{\prime }}(x)$.

The straight line that touches a curve at a point and has the same gradient as the curve there. It is the limit of chords through that point as the second point moves towards it.

The total length of path covered by a particle, counting motion in both directions as positive: ${\int }_{t_1}^{t_2}\mathrm{\mid }v\mathrm{\mid }dt$. It is at least as large as the size of the displacement.

A substitution such as $x=a\sin \theta$ or $x=a\tan \theta$ that turns $\sqrt{a^2-x^2}$ or $a^2+x^2$ into a single trigonometric term by using an identity.

The rate of change of displacement with respect to time, $v=\frac{ds}{dt}$. Its sign shows the direction of motion; its size is the speed.

The volume of the solid formed when a region is rotated through $2\pi$ radians (one full turn) about a line: $V={\int }_a^b\pi y^{2}dx$ about the $x$-axis or $V={\int }_c^d\pi x^{2}dy$ about the $y$-axis.

Finding the volume of a solid with a hole by slicing it into rings (washers) of outer radius $R$ and inner radius $r$: $V=\pi {\int }_a^b(R^2-r^2)dx$.