IB Maths AA Topic 5 Definitions

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

All Topic 5 Sub-topics 5.1: Introduction To Derivatives 5.2: Applying Derivatives 5.3: Advanced Derivatives 5.4: Introduction To Integration 5.5: Advanced Integration 5.6: Kinematics (HL)5.7: Advanced limits (HL)5.8: Implicit differentiation (HL)5.9: Challenging derivatives (HL)5.10: Challenging integration (HL)5.11: Applying integration (HL)5.12: Differential equations (HL)5.13: Maclaurin series

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.

approximation

Using a truncated Maclaurin series (a finite number of terms) to estimate a function value near

x=0

arccos

The inverse function of cosine that returns the principal angle '

y

' such that '

\cos y=x

', with domain '

-1\leq x\leq1

' and output values '

0\leq y\leq\pi

arcsin

An inverse trigonometric function whose derivative is

\frac{d}{dx}\left(\arcsin x\right)=\frac{1}{\sqrt{1-x^2}}

, and for an input

f(x)

becomes

\frac{f'(x)}{\sqrt{1-\left(f(x)\right)^2}}

arctan

The inverse function of tangent that returns the principal angle '

y

' such that '

\tan y=x

', with domain all real numbers and output values '

-\frac{\pi}{2}\lt y\lt\frac{\pi}{2}

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

continuity

A property where the function has no break, jump, or hole at a point, so the limit matches the function value: '

\lim_{x\to a}f(x)=f(a)

converges

Approaches a single finite value as the variable approaches the target value (including as '

x\to\infty

').

cos

A trigonometric function whose derivative is '

-\sin x

cosec

A reciprocal trigonometric ratio defined by '

\cosec x=\frac{1}{\sin x}

', so it is undefined where '

\sin x=0

cosine

A trigonometric function whose derivative is

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

cot

A reciprocal trigonometric function whose derivative is

\frac{d}{dx}\left(\cot x\right)=-\cosec^2 x

, and for an input

f(x)

becomes

-\cosec^2\left(f(x)\right)f'(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.

differentiability

A property where the function has a well-defined tangent at a point, meaning there is a unique gradient there and the derivative exists.

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.

differentiation

A process that can be applied term-by-term to a power series to create a new series for the derivative, within the interval where the original series is valid.

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

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

diverges

Does not approach a single finite value as the variable approaches the target value, so the limit does not exist as a finite number.

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}

Implicit differentiation

A method for differentiating an equation that is not written as $y=f(x)$ by differentiating both sides with respect to $x$ while treating $y$ as a function of $x$ and applying the chain rule.

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 parts

A technique for integrating a product by choosing '

u

' and '

dv

' and using '

\int u\,dv=uv-\int v\,du

' to replace the original integral with a simpler one.

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}

Refers to l'Hôpital's rule, which evaluates indeterminate limits of the forms '

\frac{0}{0}

' or '

\frac{\infty}{\infty}

' by differentiating numerator and denominator and taking the new limit, possibly repeatedly.

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 $x\gt0$ .

log

A logarithmic function with base

a

whose derivative is

\frac{d}{dx}\left(\log_a x\right)=\frac{1}{x\ln a}

, and for an input

f(x)

becomes

\frac{f'(x)}{f(x)\ln a}

Maclaurin

Describes expressing a function as a power series centred at '

x=0

', used for approximation and for evaluating limits using standard expansions such as '

e^x

', '

\sin x

', and '

\cos x

Maclaurin formula

An expression that builds the Maclaurin series using derivatives at

0

f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+\cdots

Maclaurin series

A power series representation of a function centred at

x=0

, written as

f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n

, giving a polynomial-style approximation near

x=0

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.

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)

power series

An infinite series of the form $\sum_{n=0}^{\infty} a_n x^n$ for constants $a_n$ , used to represent a function as powers of $x$ .

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.

Repeated integration

Describing applying the same integration method more than once, especially using integration by parts multiple times until the integral can be completed or solved.

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.

sec

A reciprocal trigonometric function whose derivative is

\frac{d}{dx}\left(\sec x\right)=\sec x\tan x

; for an input

f(x)

\frac{d}{dx}\left(\sec\left(f(x)\right)\right)=\sec\left(f(x)\right)\tan\left(f(x)\right)f'(x)

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.

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

tan

Defined by

\tan\theta=\frac{\sin\theta}{\cos\theta}

, so it is undefined whenever

\cos\theta=0

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.

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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