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Mathematics: Applications and Interpretation, first assessment 2021. This is the first examinations 2021, version 1.0 edition of the booklet, the one printed on its cover sheet above.

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Booklet: Mathematics: applications and interpretation formula booklet, first examinations 2021, version 1.0
Syllabus: Mathematics: Applications and Interpretation, first assessment 2021
Levels: SL and HL, switch between them above the booklet
In the exam: You get the official printed copy, not this page and not our explanations

/

Mathematics: applications and interpretation formula booklet

First examinations 2021, Version 1.0

Reproduced from the IB Mathematics: applications and interpretation formula booklet, First examinations 2021, Version 1.0. Explanations are ours.

Prior learningsheet 3
Topic 1: Number and algebrasheet 4
Topic 2: Functionssheet 5
Topic 3: Geometry and trigonometrysheet 6
Topic 4: Statistics and probabilitysheet 7
Topic 5: Calculussheet 8

Prior learning

SL and HL

Area of a parallelogram

A = bh

, where

b

is the base,

h

is the height

Area of a triangle

A = \frac{1}{2}(bh)

, where

b

is the base,

h

is the height

Area of a trapezoid

A = \frac{1}{2}(a + b)h

, where

a

and

b

are the parallel sides,

h

is the height

Area of a circle

A = \pi r^2

, where

r

is the radius

Circumference of a circle

C = 2\pi r

, where

r

is the radius

Volume of a cuboid

V = lwh

, where

l

is the length,

w

is the width,

h

is the height

Volume of a cylinder

V = \pi r^2 h

, where

r

is the radius,

h

is the height

Volume of prism

V = Ah

, where

A

is the area of cross-section,

h

is the height

Area of the curved surface of a cylinder

A = 2\pi rh

, where

r

is the radius,

h

is the height

Distance between two points

(x_1, y_1)

and

(x_2, y_2)

d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}

Coordinates of the midpoint of a line segment with endpoints

(x_1, y_1)

and

(x_2, y_2)

\left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)

HL only

Solutions of a quadratic equation

\text{The solutions of } ax^2 + bx + c = 0 \text{ are } x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

a \neq 0

Topic 1: Number and algebra

SL and HL

SL 1.2

The

n

th term of an arithmetic sequence

u_n = u_1 + (n - 1)d

SL 1.2

The sum of

n

terms of an arithmetic sequence

S_n = \frac{n}{2}\left(2u_1 + (n - 1)d\right)

S_n = \frac{n}{2}(u_1 + u_n)

SL 1.3

The

n

th term of a geometric sequence

u_n = u_1 r^{n-1}

SL 1.3

The sum of

n

terms of a finite geometric sequence

S_n = \frac{u_1(r^n - 1)}{r - 1} = \frac{u_1(1 - r^n)}{1 - r}

r \neq 1

SL 1.4

Compound interest

FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}

, where

FV

is the future value,

PV

is the present value,

n

is the number of years,

k

is the number of compounding periods per year,

r\%

is the nominal annual rate of interest

SL 1.5

Exponents and logarithms

a^x = b \Leftrightarrow x = \log_a b

, where

a > 0, b > 0, a \neq 1

SL 1.6

Percentage error

\varepsilon = \left|\frac{v_{\mathrm{A}} - v_{\mathrm{E}}}{v_{\mathrm{E}}}\right| \times 100\%

, where

v_{\mathrm{E}}

is the exact value and

v_{\mathrm{A}}

is the approximate value of

v

HL only

AHL 1.9

Laws of logarithms

\log_a xy = \log_a x + \log_a y

\log_a \frac{x}{y} = \log_a x - \log_a y

\log_a x^m = m\log_a x

for

a, x, y > 0

AHL 1.11

The sum of an infinite geometric sequence

S_\infty = \frac{u_1}{1 - r}

|r| < 1

AHL 1.12

Complex numbers

z = a + b\mathrm{i}

AHL 1.12

Discriminant

\Delta = b^2 - 4ac

AHL 1.13

Modulus-argument (polar) and exponential (Euler) form

z = r(\cos\theta + \mathrm{i}\sin\theta) = r\mathrm{e}^{\mathrm{i}\theta} = r\operatorname{cis}\theta

AHL 1.14

Determinant of a

2 \times 2

matrix

\boldsymbol{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow \det\boldsymbol{A} = |\boldsymbol{A}| = ad - bc

AHL 1.14

Inverse of a

2 \times 2

matrix

\boldsymbol{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow \boldsymbol{A}^{-1} = \frac{1}{\det\boldsymbol{A}}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

ad \neq bc

AHL 1.15

Power formula for a matrix

\boldsymbol{M}^n = \boldsymbol{PD}^n\boldsymbol{P}^{-1}

, where

\boldsymbol{P}

is the matrix of eigenvectors and

\boldsymbol{D}

is the diagonal matrix of eigenvalues

Topic 2: Functions

SL and HL

SL 2.1

Equations of a straight line

y = mx + c

ax + by + d = 0

y - y_1 = m(x - x_1)

SL 2.1

Gradient formula

m = \frac{y_2 - y_1}{x_2 - x_1}

SL 2.5

Axis of symmetry of the graph of a quadratic function

f(x) = ax^2 + bx + c \Rightarrow \text{axis of symmetry is } x = -\frac{b}{2a}

HL only

AHL 2.9

Logistic function

f(x) = \frac{L}{1 + C\mathrm{e}^{-kx}}

L, k, C > 0

Topic 3: Geometry and trigonometry

SL and HL

SL 3.1

Distance between two points

(x_1, y_1, z_1)

and

(x_2, y_2, z_2)

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

SL 3.1

Coordinates of the midpoint of a line segment with endpoints

(x_1, y_1, z_1)

and

(x_2, y_2, z_2)

\left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2},\ \frac{z_1 + z_2}{2}\right)

SL 3.1

Volume of a right-pyramid

V = \frac{1}{3}Ah

, where

A

is the area of the base,

h

is the height

SL 3.1

Volume of a right cone

V = \frac{1}{3}\pi r^2 h

, where

r

is the radius,

h

is the height

SL 3.1

Area of the curved surface of a cone

A = \pi rl

, where

r

is the radius,

l

is the slant height

SL 3.1

Volume of a sphere

V = \frac{4}{3}\pi r^3

, where

r

is the radius

SL 3.1

Surface area of a sphere

A = 4\pi r^2

, where

r

is the radius

SL 3.2

Sine rule

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

SL 3.2

Cosine rule

c^2 = a^2 + b^2 - 2ab\cos C

\cos C = \frac{a^2 + b^2 - c^2}{2ab}

SL 3.2

Area of a triangle

A = \frac{1}{2}ab\sin C

SL 3.4

Length of an arc

l = \frac{\theta}{360} \times 2\pi r

, where

\theta

is the angle measured in degrees,

r

is the radius

SL 3.4

Area of a sector

A = \frac{\theta}{360} \times \pi r^2

, where

\theta

is the angle measured in degrees,

r

is the radius

HL only

AHL 3.7

Length of an arc

l = r\theta

, where

r

is the radius,

\theta

is the angle measured in radians

AHL 3.7

Area of a sector

A = \frac{1}{2}r^2\theta

, where

r

is the radius,

\theta

is the angle measured in radians

AHL 3.8

Identities

\cos^2\theta + \sin^2\theta = 1

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

AHL 3.9

Transformation matrices

\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}

, reflection in the line

y = (\tan\theta)x

\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}

, horizontal stretch / stretch parallel to

x

-axis with a scale factor of

k

\begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix}

, vertical stretch / stretch parallel to

y

-axis with a scale factor of

k

\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}

, enlargement, with a scale factor of

k

, centre

(0, 0)

\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

, anticlockwise/counter-clockwise rotation of angle

\theta

about the origin (

\theta > 0

)

\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}

, clockwise rotation of angle

\theta

about the origin (

\theta > 0

)

AHL 3.10

Magnitude of a vector

|\boldsymbol{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}

, where

\boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}

AHL 3.11

Vector equation of a line

\boldsymbol{r} = \boldsymbol{a} + \lambda\boldsymbol{b}

AHL 3.11

Parametric form of the equation of a line

x = x_0 + \lambda l,\ y = y_0 + \lambda m,\ z = z_0 + \lambda n

AHL 3.13

Scalar product

\boldsymbol{v} \cdot \boldsymbol{w} = v_1 w_1 + v_2 w_2 + v_3 w_3

, where

\boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}

\boldsymbol{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}

\boldsymbol{v} \cdot \boldsymbol{w} = |\boldsymbol{v}||\boldsymbol{w}|\cos\theta

, where

\theta

is the angle between

\boldsymbol{v}

and

\boldsymbol{w}

AHL 3.13

Angle between two vectors

\cos\theta = \frac{v_1 w_1 + v_2 w_2 + v_3 w_3}{|\boldsymbol{v}||\boldsymbol{w}|}

AHL 3.13

Vector product

\boldsymbol{v} \times \boldsymbol{w} = \begin{pmatrix} v_2 w_3 - v_3 w_2 \\ v_3 w_1 - v_1 w_3 \\ v_1 w_2 - v_2 w_1 \end{pmatrix}

, where

\boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}

\boldsymbol{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}

|\boldsymbol{v} \times \boldsymbol{w}| = |\boldsymbol{v}||\boldsymbol{w}|\sin\theta

, where

\theta

is the angle between

\boldsymbol{v}

and

\boldsymbol{w}

AHL 3.13

Area of a parallelogram

A = |\boldsymbol{v} \times \boldsymbol{w}|

where

\boldsymbol{v}

and

\boldsymbol{w}

form two adjacent sides of a parallelogram

Topic 4: Statistics and probability

SL and HL

SL 4.2

Interquartile range

\mathrm{IQR} = Q_3 - Q_1

SL 4.3

Mean,

\bar{x}

, of a set of data

\bar{x} = \frac{\sum\limits_{i=1}^{k} f_i x_i}{n}

, where

n = \sum\limits_{i=1}^{k} f_i

SL 4.5

Probability of an event

A

\mathrm{P}(A) = \frac{n(A)}{n(U)}

SL 4.5

Complementary events

\mathrm{P}(A) + \mathrm{P}(A') = 1

SL 4.6

Combined events

\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B)

SL 4.6

Mutually exclusive events

\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B)

SL 4.6

Conditional probability

\mathrm{P}(A \mid B) = \frac{\mathrm{P}(A \cap B)}{\mathrm{P}(B)}

SL 4.6

Independent events

\mathrm{P}(A \cap B) = \mathrm{P}(A)\,\mathrm{P}(B)

SL 4.7

Expected value of a discrete random variable

X

\mathrm{E}(X) = \sum x\,\mathrm{P}(X = x)

SL 4.8

Binomial distribution

X \sim \mathrm{B}(n, p)

SL 4.8

Mean

\mathrm{E}(X) = np

SL 4.8

Variance

\mathrm{Var}(X) = np(1 - p)

HL only

AHL 4.14

Linear transformation of a single random variable

\mathrm{E}(aX + b) = a\mathrm{E}(X) + b

\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)

AHL 4.14

Linear combinations of

n

independent random variables,

X_1, X_2, ..., X_n

\mathrm{E}(a_1X_1 \pm a_2X_2 \pm \ldots \pm a_nX_n) = a_1\mathrm{E}(X_1) \pm a_2\mathrm{E}(X_2) \pm \ldots \pm a_n\mathrm{E}(X_n)

\begin{aligned} &\mathrm{Var}(a_1X_1 \pm a_2X_2 \pm \ldots \pm a_nX_n) \\ &\qquad = a_1^2\,\mathrm{Var}(X_1) + a_2^2\,\mathrm{Var}(X_2) + \ldots + a_n^2\,\mathrm{Var}(X_n) \end{aligned}

AHL 4.14

Sample statistics

AHL 4.14

Unbiased estimate of population variance

s_{n-1}^2

s_{n-1}^2 = \frac{n}{n - 1}s_n^2

AHL 4.17

Poisson distribution

X \sim \mathrm{Po}(m)

AHL 4.17

Mean

\mathrm{E}(X) = m

AHL 4.17

Variance

\mathrm{Var}(X) = m

AHL 4.19

Transition matrices

\boldsymbol{T}^n\boldsymbol{s}_0 = \boldsymbol{s}_n

, where

\boldsymbol{s}_0

is the initial state

Topic 5: Calculus

SL and HL

SL 5.3

Derivative of

x^n

f(x) = x^n \Rightarrow f'(x) = nx^{n-1}

SL 5.5

Integral of

x^n

\int x^n\,\mathrm{d}x = \frac{x^{n+1}}{n + 1} + C

n \neq -1

SL 5.5

Area of region enclosed by a curve

y = f(x)

and the

x

-axis, where

f(x) > 0

A = \int_a^b y\,\mathrm{d}x

SL 5.8

The trapezoidal rule

\int_a^b y\,\mathrm{d}x \approx \frac{1}{2}h\left((y_0 + y_n) + 2(y_1 + y_2 + \ldots + y_{n-1})\right)

, where

h = \frac{b - a}{n}

HL only

AHL 5.9

Derivative of

\sin x

f(x) = \sin x \Rightarrow f'(x) = \cos x

AHL 5.9

Derivative of

\cos x

f(x) = \cos x \Rightarrow f'(x) = -\sin x

AHL 5.9

Derivative of

\tan x

f(x) = \tan x \Rightarrow f'(x) = \frac{1}{\cos^2 x}

AHL 5.9

Derivative of

\mathrm{e}^x

f(x) = \mathrm{e}^x \Rightarrow f'(x) = \mathrm{e}^x

AHL 5.9

Derivative of

\ln x

f(x) = \ln x \Rightarrow f'(x) = \frac{1}{x}

AHL 5.9

Chain rule

y = g(u)\text{, where } u = f(x) \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}u} \times \frac{\mathrm{d}u}{\mathrm{d}x}

AHL 5.9

Product rule

y = uv \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = u\frac{\mathrm{d}v}{\mathrm{d}x} + v\frac{\mathrm{d}u}{\mathrm{d}x}

AHL 5.9

Quotient rule

y = \frac{u}{v} \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{v\dfrac{\mathrm{d}u}{\mathrm{d}x} - u\dfrac{\mathrm{d}v}{\mathrm{d}x}}{v^2}

AHL 5.11

Standard integrals

\int \frac{1}{x}\,\mathrm{d}x = \ln|x| + C

\int \sin x\,\mathrm{d}x = -\cos x + C

\int \cos x\,\mathrm{d}x = \sin x + C

\int \frac{1}{\cos^2 x} = \tan x + C

\int \mathrm{e}^x\,\mathrm{d}x = \mathrm{e}^x + C

AHL 5.12

Area of region enclosed by a curve and

x

y

-axes

A = \int_a^b |y|\,\mathrm{d}x

A = \int_a^b |x|\,\mathrm{d}y

AHL 5.12

Volume of revolution about

x

y

-axes

V = \int_a^b \pi y^2\,\mathrm{d}x

V = \int_a^b \pi x^2\,\mathrm{d}y

AHL 5.13

Acceleration

a = \frac{\mathrm{d}v}{\mathrm{d}t} = \frac{\mathrm{d}^2 s}{\mathrm{d}t^2} = v\frac{\mathrm{d}v}{\mathrm{d}s}

AHL 5.13

Distance travelled from

t_1

t_2

\text{distance} = \int_{t_1}^{t_2} |v(t)|\,\mathrm{d}t

AHL 5.13

Displacement from

t_1

t_2

\text{displacement} = \int_{t_1}^{t_2} v(t)\,\mathrm{d}t

AHL 5.16

Euler’s method

y_{n+1} = y_n + h \times f(x_n, y_n)

x_{n+1} = x_n + h

, where

h

is a constant (step length)

AHL 5.16

Euler’s method for coupled systems

\begin{aligned} &x_{n+1} = x_n + h \times f_1(x_n, y_n, t_n) \\ &y_{n+1} = y_n + h \times f_2(x_n, y_n, t_n) \\ &t_{n+1} = t_n + h \end{aligned}

where

h

is a constant (step length)

AHL 5.17

Exact solution for coupled linear differential equations

\boldsymbol{x} = A\mathrm{e}^{\lambda_1 t}\boldsymbol{p}_1 + B\mathrm{e}^{\lambda_2 t}\boldsymbol{p}_2

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Contents

Prior learning

SL and HL

HL only

Topic 1: Number and algebra

SL and HL

HL only

Topic 2: Functions

SL and HL

HL only

Topic 3: Geometry and trigonometry

SL and HL

HL only

Topic 4: Statistics and probability

SL and HL

HL only

Topic 5: Calculus

SL and HL

HL only

Questions

Is this the official IB booklet?

Which syllabus does it go with?

What changes between SL and HL?

Can I download it as a PDF?

Can I use it while I practice?

Practice with the booklet open beside you