Maths AI terms in Calculus

The rate of change of an object's velocity with respect to time, $a=\frac{\mathrm{d}v}{\mathrm{d}t}=\frac{{\mathrm{d}}^{2}s}{\mathrm{d}{t}^2}$; a positive acceleration means the velocity is increasing.

The difference between a numerical estimate after many steps and the exact value of the solution, made up of the small error introduced at every step and carried forward through all later steps.

A function $F(x)$ whose derivative is a given function $f(x)$, so that $F^{\mathrm{\prime }}(x)=f(x)$.

The area of the region enclosed by two graphs between two values of $x$, found by subtracting the area under the lower graph from the area under the upper graph over the same interval.

The area of the region bounded by a curve $y=f(x)$ with $f(x)>0$, the $x$-axis and the vertical lines $x=a$ and $x=b$.

A first-order differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=f(y)$ whose right-hand side depends only on the unknown $y$ and not on the independent variable.

For the differential equation $\ddot{x}+a\dot{x}+bx=0$, the quadratic equation ${\lambda }^2+a\lambda +b=0$, whose roots are the eigenvalues of the matrix $\left(\begin{array}{cc}0& 1\\ -b& -a\end{array}\right)$ and give the solutions ${\mathrm{e}}^{\lambda t}$.

The change in a quantity divided by the change in the variable it depends on over an interval, $\frac{f(b)-f(a)}{b-a}$, which is the gradient of the secant between the two endpoints.

The straight line about which a region is rotated to form a solid of revolution; every cross-section of the solid perpendicular to this line is a circle centred on it.

A known pair of corresponding values of the variables, used to find the constant of integration and so select one function from the family of anti-derivatives.

An equilibrium point of a linear system whose eigenvalues are purely imaginary, $\pm \beta \mathrm{i}$; the trajectories around it are closed curves (ellipses or circles), so solutions repeat periodically without approaching or leaving it.

The rule for differentiating a composite function: if $y=g(u)$ where $u=f(x)$, then $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\times \frac{\mathrm{d}u}{\mathrm{d}x}$.

An instant at which an object's velocity changes sign, so that it reverses its direction of motion along the line.

A pair of coupled differential equations for two populations that use the same resources, in which each population grows logistically on its own and the presence of the other species reduces its growth rate.

Describes a curve, or a part of one, that bends downwards, so that its gradient decreases as $x$ increases and every chord joining two of its points lies below the curve.

Describes a curve, or a part of one, that bends upwards, so that its gradient increases as $x$ increases and every chord joining two of its points lies above the curve.

The direction in which a curve bends on an interval, either concave up, where $f^{\mathrm{\prime }\mathrm{\prime }}(x)>0$ and the gradient increases, or concave down, where $f^{\mathrm{\prime }\mathrm{\prime }}(x)<0$ and the gradient decreases.

If a function is multiplied by a constant $a$, its derivative is multiplied by the same constant, so the derivative of $af(x)$ is $a{f}^{\mathrm{\prime }}(x)$.

The arbitrary constant $C$ added to an anti-derivative in an indefinite integral, because a constant term disappears on differentiation and so cannot be recovered from the derivative alone.

A fixed condition in an optimisation problem, such as a given amount of material, volume, perimeter or length, that links the variables and so allows all but one of them to be eliminated.

A set of differential equations in which the rate of change of each variable depends on the other variables as well as its own, so that the equations must be solved together.

A resistance to motion, modelled by a term proportional to the velocity $\dot{x}$ in an equation of motion, that removes energy from an oscillating system so that its motion dies away.

A decrease in speed, which happens when the velocity and the acceleration have opposite signs.

A function is decreasing on an interval if, for any two values $x_1<x_2$ in the interval, $f(x_1)>f(x_2)$, so that its graph falls from left to right.

The number ${\int }_a^{b}f(x)\mathrm{d}x$, the limit of the sum of $f(x)$ times the strip width over the interval from $x=a$ to $x=b$ as the strips become infinitely thin; when $f(x)>0$ it equals the area under the curve between $x=a$ and $x=b$.

The function $f^{\mathrm{\prime }}(x)$, also written $\frac{\mathrm{d}y}{\mathrm{d}x}$, that gives the gradient of the graph of $y=f(x)$ at each value of $x$, equal to the instantaneous rate of change of $y$ with respect to $x$; at each point it is the limit of the difference quotient, $f^{\mathrm{\prime }}(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$.

The derivative of any constant $c$ is 0, because the graph of $y=c$ is a horizontal line, whose gradient is 0 everywhere.

The gradient of a secant written as $\frac{f(a+h)-f(a)}{h}$, the change in a function divided by the change in its input over an interval of width $h$; it gives an average rate of change and, for small $h$, an estimate of the gradient at $x=a$.

An equation that contains one or more derivatives of an unknown function; a solution is a function that makes the equation true for every value of the variable in its domain.

The process of finding the derivative of a function.

A way of finding a volume of revolution by treating the solid as a stack of thin circular discs perpendicular to the axis of rotation, each of volume $\pi r^2$ times its thickness, and integrating.

The position of an object relative to a fixed origin on a line, given as a signed number whose sign shows which side of the origin the object is on; it is usually written $s$ or $x$.

A way of writing derivatives with respect to time in which a dot over a variable denotes its first derivative and two dots denote its second derivative, so $\dot{x}=\frac{\mathrm{d}x}{\mathrm{d}t}$ and $\ddot{x}=\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}$.

The equation of the line through $(a,f(a))$ perpendicular to the tangent there, $y-f(a)=-\frac{1}{f^{\mathrm{\prime }}(a)}(x-a)$ when $f^{\mathrm{\prime }}(a)\mathrm{\ne }0$, and the vertical line $x=a$ when $f^{\mathrm{\prime }}(a)=0$.

The equation of the straight line that touches the curve $y=f(x)$ at $(a,f(a))$ with gradient $f^{\mathrm{\prime }}(a)$, which is $y-f(a)=f^{\mathrm{\prime }}(a)(x-a)$.

A point in the phase plane at which every rate of change of a system is zero ($\dot{x}=0$ and $\dot{y}=0$ simultaneously), so that a solution starting there stays there for all time.

A numerical method for a first-order differential equation that starts from a known point and repeatedly moves a fixed distance along a straight line whose gradient is given by the differential equation at the current point.

A method of finding the nature of a stationary point at $x=a$ from the sign of $f^{\mathrm{\prime }}(x)$ just to the left and just to the right of $a$: a change from positive to negative shows a local maximum, a change from negative to positive shows a local minimum, and no change of sign shows neither.

The solution of a differential equation that contains an arbitrary constant (one constant for a first-order equation), so that it describes the whole family of functions satisfying the equation.

The greatest value that a function takes on its whole domain, which may occur at a local maximum or at an end point of the domain.

The least value that a function takes on its whole domain, which may occur at a local minimum or at an end point of the domain.

The gradient of the tangent to a curve at a given point, equal to the limit of the gradients of secants through the point as the second point approaches it.

A tangent to a curve that is parallel to the $x$-axis, so its gradient is zero; the curve has a horizontal tangent at $x=a$ exactly when $f^{\mathrm{\prime }}(a)=0$.

A function is increasing on an interval if, for any two values $x_1<x_2$ in the interval, $f(x_1)<f(x_2)$, so that its graph rises from left to right.

The family of all anti-derivatives of a function, written $\int f(x)\mathrm{d}x=F(x)+C$, where $F^{\mathrm{\prime }}(x)=f(x)$ and $C$ is any constant.

In a composite function $y=g(f(x))$, the function $f$ that is applied first, directly to $x$; its output $u=f(x)$ becomes the input of the outer function.

The rate of change of a quantity at one particular value of the input, equal to the derivative at that value and to the gradient of the tangent there; it is the limit of the average rates of change over shorter and shorter intervals.

The state of a moving object at an instant when its velocity is zero, $v=0$.

The function that is being integrated, written between the integral sign and $\mathrm{d}x$.

The process of finding an anti-derivative of a function, or of finding the accumulated total of a function between two values of the variable.

Finding an integral by recognising the integrand as the result of differentiating a known function, writing that function down and adjusting it by a constant factor so that its derivative matches the integrand exactly.

A method that replaces an inner function $g(x)$ by a new variable $u=g(x)$ and $g^{\mathrm{\prime }}(x)\mathrm{d}x$ by $\mathrm{d}u$, so that $\int f(g(x))g^{\mathrm{\prime }}(x)\mathrm{d}x$ becomes the simpler integral $\int f(u)\mathrm{d}u$.

A curve along which every solution of a differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=f(x,y)$ has the same gradient $c$; its equation is $f(x,y)=c$.

The notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ for the derivative of $y$ with respect to $x$, which names both the quantity that changes (on top) and the variable it changes with (underneath); the alternative prime (Lagrange) notation writes the derivative of $f(x)$ as $f^{\mathrm{\prime }}(x)$.

The value that $f(x)$ gets arbitrarily close to as $x$ gets closer and closer to a given value $a$ (or as $x$ increases without bound), written ${\lim }_{x\to a}f(x)$.

The two values of the variable between which a definite integral is taken: the lower limit $a$, written at the bottom of the integral sign, and the upper limit $b$, written at the top.

A pair of coupled differential equations $\dot{x}=ax+by$, $\dot{y}=cx+dy$ with constant coefficients, written in matrix form as $\dot{\mathit{x}}=M\mathit{x}$ where $M=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$.

The property of a smooth curve that, close enough to any point on it, the curve is almost indistinguishable from its tangent at that point, so the tangent gives a good approximation to the curve there.

The derivative $C^{\mathrm{\prime }}(x)$ of a cost function with respect to the number of items produced, which gives the rate of increase of total cost per extra item and approximates the cost of producing one more item; marginal revenue $R^{\mathrm{\prime }}(x)$ and marginal profit $P^{\mathrm{\prime }}(x)$ are defined in the same way.

Whether a stationary point is a local maximum, a local minimum, or neither.

A curve has a negative gradient at a point if its derivative there is less than zero, $f^{\mathrm{\prime }}(x)<0$, so that the tangent slopes downwards from left to right and the function is decreasing.

A power $x^n$ in which $n$ is a negative integer, so that $x^{-k}=\frac{1}{x^k}$ for a positive integer $k$; a term $\frac{a}{x^k}$ written as $a{x}^{-k}$ is ready to be differentiated by the power rule.

The law that the rate of change of the temperature $T$ of an object is proportional to the difference between its temperature and the constant temperature $T_s$ of its surroundings, $\frac{\mathrm{d}T}{\mathrm{d}t}=-k(T-T_s)$ with $k>0$.

An equilibrium point of a linear system whose eigenvalues are real, distinct and of the same sign; trajectories approach it (stable node, both negative) or leave it (unstable node, both positive) without circling it, tangent to an eigenvector.

A point of inflexion at which the gradient is not zero, so that $f^{\mathrm{\prime }}(a)\mathrm{\ne }0$ while $f^{\mathrm{\prime }\mathrm{\prime }}$ changes sign at $x=a$.

The straight line through a point on a curve that is perpendicular to the tangent at that point; at $(a,f(a))$ on $y=f(x)$ its gradient is $-\frac{1}{f^{\mathrm{\prime }}(a)}$ when $f^{\mathrm{\prime }}(a)\mathrm{\ne }0$.

The isocline on which $\frac{\mathrm{d}y}{\mathrm{d}x}=0$, with equation $f(x,y)=0$; every local maximum and local minimum of a solution curve lies on it.

The estimation of a definite integral, or of the area it represents, from values of the function at a finite number of points instead of from an anti-derivative.

A table of approximate values of the solution of an equation, produced by a step-by-step calculation, instead of an exact formula.

The function that gives the quantity to be maximised or minimised, written in terms of a single variable.

The greatest or least value of the objective function that can be achieved within the constraints, together with the value of the variable at which it occurs.

The process of finding the value of a variable that makes a quantity as large as possible (a maximum) or as small as possible (a minimum), usually subject to conditions set by the context.

The order of the highest derivative that appears in a differential equation: first order if only the first derivative appears, second order if the second derivative is the highest, and so on.

The $y$-coordinate of a point on a curve; in the trapezoidal rule, the ordinates $y_0,y_1,\dots ,y_n$ are the values of $y$ at the edges of the strips.

In a composite function $y=g(f(x))$, the function $g$ that is applied last, to the output $u$ of the inner function, giving $y=g(u)$.

Describes a damped system $\ddot{x}+a\dot{x}+bx=0$ whose auxiliary equation has two distinct negative real roots ($a^2>4b$), so that it returns to equilibrium without oscillating.

An approximation that is greater than the true value it approximates.

The single solution of a differential equation that satisfies a given initial condition, found by substituting the condition into the general solution to fix the arbitrary constant.

The plane with the two dependent variables of a coupled system, such as $x$ and $y$, on its axes, in which each solution is drawn as a curve and time is not shown on either axis.

A sketch in the phase plane of a representative set of trajectories of a coupled system, with arrows showing the direction of motion, which displays its equilibrium points and long-term behaviour.

The point at which a tangent touches a curve; for the tangent to $y=f(x)$ at $x=a$, it is $(a,f(a))$.

A point on a curve at which the concavity changes, from concave up to concave down or from concave down to concave up.

A curve has a positive gradient at a point if its derivative there is greater than zero, $f^{\mathrm{\prime }}(x)>0$, so that the tangent slopes upwards from left to right and the function is increasing.

The rule that the derivative of $a{x}^n$, for a constant $a$ and an integer $n$, is $an{x}^{n-1}$, so the coefficient is multiplied by the power and the power is reduced by 1.

A pair of coupled differential equations of the form $\frac{\mathrm{d}x}{\mathrm{d}t}=ax-bxy$, $\frac{\mathrm{d}y}{\mathrm{d}t}=cxy-dy$ with positive constants $a,b,c,d$, where $x$ is the prey population and $y$ the predator population.

The rule for differentiating a product of two functions: if $y=uv$, where $u$ and $v$ are functions of $x$, then $\frac{\mathrm{d}y}{\mathrm{d}x}=u\frac{\mathrm{d}v}{\mathrm{d}x}+v\frac{\mathrm{d}u}{\mathrm{d}x}$.

The rule for differentiating a quotient of two functions: if $y=\frac{u}{v}$, where $u$ and $v$ are functions of $x$ and $v\mathrm{\ne }0$, then $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{v\frac{\mathrm{d}u}{\mathrm{d}x}-u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}$.

How fast one quantity changes in relation to another quantity on which it depends, measured in units of the first quantity per unit of the second.

A formula that defines each term of a sequence from the term or terms before it, so that the sequence is generated one term at a time from a starting value.

Rewriting a second-order differential equation $\ddot{x}=f(x,\dot{x},t)$ as the coupled first-order equations $\dot{x}=y$, $\dot{y}=f(x,y,t)$, by introducing the velocity $y=\dot{x}$ as a second variable, so that $\dot{y}=\ddot{x}$.

The rates of change with respect to time of two or more quantities that are linked by an equation; differentiating the equation with the chain rule connects the rates, so that an unknown rate can be found from a known one.

The result $\int f(g(x))g^{\mathrm{\prime }}(x)\mathrm{d}x=F(g(x))+C$, where $F$ is an anti-derivative of $f$, which integrates a composite function multiplied by the derivative of its inner function.

An equilibrium point of a linear system whose eigenvalues are real with opposite signs; trajectories approach it along the eigenvector of the negative eigenvalue and leave along the eigenvector of the positive eigenvalue, so it is unstable.

A straight line that passes through two points on a curve; the line segment between the two points is called a chord.

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

A test for the nature of a stationary point at $x=a$, where $f^{\mathrm{\prime }}(a)=0$: if $f^{\mathrm{\prime }\mathrm{\prime }}(a)<0$ the point is a local maximum, if $f^{\mathrm{\prime }\mathrm{\prime }}(a)>0$ it is a local minimum, and if $f^{\mathrm{\prime }\mathrm{\prime }}(a)=0$ the test gives no conclusion.

A differential equation in which the highest derivative of the unknown function is the second derivative, such as $\ddot{x}=f(x,\dot{x},t)$.

A method for solving a differential equation of the form $\frac{\mathrm{d}y}{\mathrm{d}x}=f(x)g(y)$ by rewriting it as $\int \frac{1}{g(y)}\mathrm{d}y=\int f(x)\mathrm{d}x$ and integrating each side with respect to its own variable.

A number line for $x$ marked with the values where a function (usually $f^{\mathrm{\prime }}$) is zero, showing whether the function is positive or negative on each interval between them.

The value of the definite integral ${\int }_a^{b}f(x)\mathrm{d}x$, in which regions above the $x$-axis count as positive and regions below the $x$-axis count as negative.

Motion in which the acceleration is proportional to the displacement from equilibrium and directed towards it, $\ddot{x}=-{\omega }^{2}x$ with $\omega >0$; it is a periodic oscillation with period $\frac{2\pi }{\omega }$.

A method of numerical integration that estimates ${\int }_a^{b}y\mathrm{d}x$ by fitting parabolas through groups of three neighbouring points, using an even number $n$ of strips of width $h$.

A coupled system of differential equations for an epidemic in which a population is divided into susceptible ($S$), infected ($I$) and recovered ($R$) groups, with new infections at a rate proportional to $SI$ and recoveries at a rate proportional to $I$.

A diagram of a first-order differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=f(x,y)$ made of short line segments drawn at the points of a grid, each with gradient equal to $f(x,y)$ at its point.

A three-dimensional shape formed by rotating a plane region through ${360}^{\circ }$ about a straight line in the same plane.

The graph of a particular solution of a differential equation; on a slope field it is a smooth curve that is tangent to the segments at every point it passes through.

The magnitude of velocity, $\mathrm{\mid }v\mathrm{\mid }$, which is never negative and carries no information about direction.

An equilibrium point of a linear system whose eigenvalues are complex, $\alpha \pm \beta \mathrm{i}$ with $\alpha \mathrm{\ne }0$ and $\beta \mathrm{\ne }0$; trajectories wind round it, inwards if the real part $\alpha$ is negative (stable spiral) and outwards if it is positive (unstable spiral).

An equilibrium point that every trajectory starting close to it approaches as $t\to \mathrm{\infty }$; for $\dot{\mathit{x}}=M\mathit{x}$ this happens when both eigenvalues are negative or are complex with negative real part.

The derivative of a basic function that is used as a known result, without further working, when differentiating more complicated functions.

An anti-derivative of a basic function that is known directly from a matching derivative and used as a building block for other integrals.

A point on the graph of $y=f(x)$ at which the gradient is zero, that is, a point $(a,f(a))$ with $f^{\mathrm{\prime }}(a)=0$.

A point of inflexion at which the gradient is zero, so that $f^{\mathrm{\prime }}(a)=0$ and $f^{\mathrm{\prime }\mathrm{\prime }}$ changes sign at $x=a$.

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

The equal width $h$ of each of the $n$ strips into which the interval $a\le x\le b$ is divided for the trapezoidal rule, given by $h=\frac{b-a}{n}$.

The derivative of a sum or difference of functions is the sum or difference of their derivatives, so $\frac{\mathrm{d}}{\mathrm{d}x}(f(x)\pm g(x))=f^{\mathrm{\prime }}(x)\pm g^{\mathrm{\prime }}(x)$ and each term can be differentiated separately.

The straight line that touches a curve at a point and has the same gradient as the curve at that point; it is the limiting position of the secants through that point.

The constant velocity that a falling object approaches when the resisting force on it balances its weight, so that its acceleration tends to zero.

The law that water flows out of a small hole at depth $h$ below the surface with speed $\sqrt{2gh}$; for a tank of constant cross-section it gives $\frac{\mathrm{d}h}{\mathrm{d}t}=-k\sqrt{h}$ for a constant $k>0$.

The area of the region between a curve and an axis over an interval, counting every part as positive, found as ${\int }_a^b\mathrm{\mid }y\mathrm{\mid }\mathrm{d}x$ or by adding the sizes of the integrals over the intervals between consecutive roots.

The length of the whole path an object covers in a time interval, counting movement in both directions as positive, given by ${\int }_{t_1}^{t_2}\mathrm{\mid }v(t)\mathrm{\mid }\mathrm{d}t$.

The path $(x(t),y(t))$ traced in the phase plane by one solution of a coupled system as $t$ increases, with arrows showing the direction of motion.

A method of estimating ${\int }_a^{b}y\mathrm{d}x$ by dividing the interval into $n$ strips of equal width $h$ and adding the areas of the trapezoids formed by joining neighbouring points on the curve with straight lines.

Describes a system $\ddot{x}+bx=0$ with $b>0$ and no damping term, whose auxiliary equation has purely imaginary roots $\pm \sqrt{b}\mathrm{i}$, so that it oscillates for ever with constant amplitude.

Describes a damped system $\ddot{x}+a\dot{x}+bx=0$ whose auxiliary equation has complex roots with negative real part ($a^2<4b$), so that it oscillates about equilibrium with decreasing amplitude.

An approximation that is less than the true value it approximates.

An equilibrium point from which trajectories starting close to it move away as $t$ increases; for $\dot{\mathit{x}}=M\mathit{x}$ this happens when at least one eigenvalue is positive or the eigenvalues are complex with positive real part.

The rate of change of an object's displacement with respect to time, $v=\frac{\mathrm{d}s}{\mathrm{d}t}$; its sign gives the direction of motion and its size gives how fast the object moves.

A graph of an object's velocity against time, whose gradient at any point is the acceleration and whose signed area over an interval is the displacement.

The volume of a solid of revolution, given by $V={\int }_a^b\pi y^2\mathrm{d}x$ for rotation about the $x$-axis and $V={\int }_a^b\pi x^2\mathrm{d}y$ for rotation about the $y$-axis.