IB Maths AI Topic 5 Definitions

This page contains our IB Maths AI definitions for topic 5. By learning each one of these definitions, you will fully cover the content for IB Maths AI 'Calculus'.

All Topic 5 Sub-topics 5.1: Introduction To Derivatives 5.2: Applying Derivatives 5.3: Fundamentals Of Integration (HL)5.4: Advanced derivatives (HL)5.5: Further integration (HL)5.6: Applying integration (HL)5.7: Kinematics (HL)5.8: Introduction to differential equations (HL)5.9: Advanced differential equations

acceleration

The rate of change of velocity with respect to time, given by

a=\frac{dv}{dt}=\frac{d^2s}{dt^2}

and typically measured in m s^-2.

anti-differentiation

Finding a function

F(x)

such that

F'(x)=f(x)

, so that

\int f(x)\,dx=F(x)+C

antiderivative

A function

F(x)

whose derivative is

f(x)

, meaning

F'(x)=f(x)

antiderivatives

Functions whose derivative is the given integrand; any two differ by a constant because differentiation removes constant terms.

chain rule

A differentiation rule for a composite function, used in implicit differentiation so that differentiating a term in $y$ introduces a factor of $\frac{dy}{dx}$ , for example $\frac{d}{dx}\bigl(y^2\bigr)=2y\frac{dy}{dx}$ .

composite

Describes a function formed by putting one function inside another, for example '

f\left(g\left(x\right)\right)

', which typically requires the chain rule to differentiate.

composites

Functions formed by applying one function to the output of another, such as '

f\left(g(x)\right)

', often requiring a chain-rule structure when integrating.

concavity

The way a graph curves; it is concave up where

f''(x)\gt0

and concave down where

f''(x)\lt0

cos

A trigonometric function whose derivative is '

-\sin x

cosine

A trigonometric function whose derivative is

\frac{d}{dx}(\cos x)=-\sin x

derivative

Measures how quickly a function changes; geometrically it gives the slope of the tangent to the curve, and in applications it represents an instantaneous rate of change.

differential equation

An equation involving a function and one of its derivatives, so it describes how the function changes rather than giving the function directly.

disk

Circular cross-section produced when a thin strip is rotated about an axis; its area is $\pi r^2$ , leading to volume integrals of the form $V=\pi \int r^2\,d\left(\text{variable}\right)$ .

displacement

The vector that represents the change in position from one point to another, for example $\rightarrow{AB}=\textbf{b}-\textbf{a}$ .

distance

The length of the straight line segment joining two points; in three-dimensional space it can be found using

$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}$ .

eigenvalues

Numbers associated with the matrix $A=\begin{pmatrix}a&&b\\c&&d\end{pmatrix}$ that determine the qualitative behaviour of solutions, such as whether paths move toward, away from, or around an equilibrium.

eigenvectors

Non-zero direction vectors corresponding to eigenvalues that give the principal directions of motion in a linear phase portrait.

equilibrium

A point where the system does not move because both derivatives are zero, so $\frac{dx}{dt}=0$ and $\frac{dy}{dt}=0$ .

Euler

A numerical method that generates successive approximations to a solution by stepping forward with a fixed step size

h

using update rules such as

x_{n+1}=x_n+h y_n

Euler's method

A numerical technique that approximates the solution of '

\frac{dy}{dx}=f(x,y)

' from an initial point '

(x_0,y_0)

' using step size '

h

' and the updates '

x_{n+1}=x_n+h

' and '

y_{n+1}=y_n+hf(x_n,y_n)

first-order differential equation

A differential equation that involves only the first derivative, typically written as '

\frac{dy}{dx}

', and can be expressed in the form '

\frac{dy}{dx}=f(x,y)

gradient

The coefficient

a

y=ax+b

, giving the predicted change in

y

when

x

increases by

1

homogeneous

Able to be written in the form '

\frac{dy}{dx}=f\left(\frac{y}{x}\right)

', so the right-hand side depends only on the ratio '

\frac{y}{x}

Indefinite integration

Describing an integral that represents a family of antiderivatives and therefore includes an arbitrary constant '

C

inflexion

Describes a point where the graph changes concavity, which happens when $f''(x)$ changes sign (not just when $f''(x)=0$ ).

integrals

Expressions of the form '

\int f(x)\,dx

' that represent antiderivatives of a function, reversing differentiation.

integrating factor

A function '

\mu(x)=e^{\int P(x)dx}

' used to solve '

y'+P(x)y=Q(x)

by making the left-hand side become '

\frac{d}{dx}\left(\mu y\right)

' after multiplication.

integration

The reverse of differentiation, used to rebuild a function from its derivative and to calculate accumulated change such as area.

integration by inspection

Recognising an antiderivative directly by reversing a known derivative pattern, often using the reverse chain rule when the integrand matches '

g'\left(x\right)f\left(g\left(x\right)\right)

integration by substitution

Replacing one expression with another to simplify a differential equation, such as using '

y=vx

' so that '

\frac{dy}{dx}=v+x\frac{dv}{dx}

limit

The value that an expression approaches as the input approaches a specified number or tends to infinity, used to describe behaviour near a point or at an asymptote.

The natural logarithm function, with derivative $\frac{1}{x}$ for $x\gt0$ .

maximum

Describes a local highest point where the graph changes from increasing to decreasing; at a stationary point $x=c$ , this is indicated by $f''(c)\lt 0$ .

minimum

Describes a local lowest point where the graph changes from decreasing to increasing; at a stationary point $x=c$ , this is indicated by $f''(c)\gt 0$ .

normal

A line through a point on a curve that is perpendicular to the tangent at that point, with gradient equal to the negative reciprocal of the tangent gradient.

phase

The angle that determines the horizontal shift of a sinusoidal function; in

a\cos\left(\omega t+\alpha\right)

it is

\alpha

point-gradient

A form of a straight-line equation using a point

(x_1,y_1)

and gradient

m

y-y_1=m\left(x-x_1\right)

quotient

A result of division; in a logarithmic identity this refers to combining a fraction inside a single log, such as

\log_a\left(\frac{x}{y}\right)=\log_ax-\log_ay

for suitable positive

x

and

y

quotient rule

A differentiation rule for a quotient

\frac{u}{v}

, giving

\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2}

radius

The distance from the centre of a circle to any point on the circle.

Reverse integration

Describes using the chain rule in reverse to integrate expressions matching the pattern '

g'(x)f\left(g(x)\right)

', giving '

F\left(g(x)\right)+C

' where '

F'(u)=f(u)

revolution

Rotation of a planar region around an axis to generate a three-dimensional solid.

saddle

An unstable equilibrium where one eigenvalue is positive and the other is negative, so some trajectories approach the equilibrium while others move away.

separation of variables

A solving technique that rewrites an equation like '

\frac{dy}{dx}=g(x)h(y)

' as '

\frac{1}{h(y)}\,dy=g(x)\,dx

' and then integrates both sides.

shell

Hollow cylindrical layer formed when a vertical strip is rotated about the $y$ -axis; its volume contribution is based on circumference $2\pi x$ times height $f(x)$ and thickness $dx$ .

sin

A trigonometric function whose derivative is '

\cos x

speed

The magnitude of velocity, given by

|v|

, so it is always non-negative even when velocity is negative.

spiral

A phase-portrait pattern produced by complex eigenvalues, where trajectories curve around the equilibrium while moving inward or outward depending on the sign of the real part.

stationary point

Describes a point on a graph where the gradient is zero, so it occurs when $f'(x)=0$ .

step size

The fixed increment '

h

' in the independent variable used in Euler's method to move from '

x_n

' to '

x_{n+1}=x_n+h

tangent

A line that touches a curve at a point and has the same local slope as the curve there, used to interpret the derivative at that point.

trajectories

Curves in a phase portrait that show how the state

(x,y)

changes over time for a particular solution.

velocity

The rate of change of displacement with respect to time, given by

v=\frac{ds}{dt}

and typically measured in m s^-1.

x-axis

Horizontal coordinate axis used as a line of rotation; distances to it determine radii when using the disk method with

y=f(x)

y-axis

Vertical coordinate axis used as a boundary for area with horizontal strips, or as a line of rotation where radii are measured horizontally.

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