Formula Book

Diploma Programme
Mathematics: applications and interpretation
formula booklet
For use during the course and in the examinations
First examinations 2021
Version 1.1
© International Baccalaureate Organization 2019
Contents
Prior learning
SL and HL
2
HL only
2
Topic 1: Number and algebra
SL and HL
3
HL only
4
Topic 2: Functions
SL and HL
5
HL only
5
Topic 3: Geometry and trigonometry
SL and HL
6
HL only
7
Topic 4: Statistics and probability
SL and HL
HL only
9
10
Topic 5: Calculus
SL and HL
11
HL only
11
Prior learning – SL and HL
Area of a parallelogram
A = bh , where b is the base, h is the height
Area of a triangle
1
A = (bh) , where b is the base, h is the height
2
Area of a trapezoid
=
A
1
(a + b) h , where a and b are the parallel sides, h is the height
2
Area of a circle
A = πr 2 , where r is the radius
Circumference of a circle
C = 2πr , 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 = πr 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= 2πrh , where r is the radius, h is the height
Distance between two
points ( x1 , y1 ) and ( x2 , y2 )
d=
Coordinates of the midpoint of
a line segment with endpoints
( x1 , y1 ) and ( x2 , y2 )
 x1 + x2 y1 + y2 
, 

2 
 2
( x1 − x2 ) 2 + ( y1 − y2 ) 2
Prior learning – HL only
Solutions of a quadratic
equation
The solutions of ax 2 +=
bx + c =
0 are x
Mathematics: applications and interpretation formula booklet
−b ± b 2 − 4ac
,a≠0
2a
2
Topic 1: Number and algebra – SL and HL
SL
1.2
SL
1.3
The nth term of an
arithmetic sequence
un = u1 + (n − 1) d
The sum of n terms of an
arithmetic sequence
S n=
The nth term of a
geometric sequence
un = u1r n −1
n
n
( 2u1 + (n − 1) d ) ; Sn= (u1 + un )
2
2
u1 (r n − 1) u1 (1 − r n )
The sum of n terms of a
, r ≠1
=
Sn =
finite geometric sequence
r −1
1− r
SL
1.4
Compound interest
r 

FV = PV × 1 +
 , where FV is the future value,
 100k 
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 ⇔ x = log a b , where a > 0, b > 0, a ≠ 1
Percentage error
ε=
SL
1.6
kn
vA − vE
× 100% , where vE is the exact value and vA is
vE
the approximate value of v
Mathematics: applications and interpretation formula booklet
3
Topic 1: Number and algebra – HL only
AHL
1.9
Laws of logarithms
log=
log a x + log a y
a xy
x
log
=
log a x − log a y
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∞ =
AHL
1.12
Complex numbers
z= a + bi
Discriminant
∆= b 2 − 4ac
Modulus-argument (polar)
and exponential (Euler)
form
z=
r (cos θ + isin θ ) =
re iθ =
r cis θ
AHL
1.13
AHL
1.14
AHL
1.15
Determinant of a 2 × 2
matrix
u1
, r <1
1− r
a b
A=
A =
ad − bc

 ⇒ det A =
c d
Inverse of a 2 × 2 matrix
a b
1  d
−1
=
A 
 ⇒ A=

det A  −c
c d
−b 
 , ad ≠ bc
a
Power formula for a matrix
M n = PD n P −1 , where P is the matrix of eigenvectors and D is
the diagonal matrix of eigenvalues
Mathematics: applications and interpretation formula booklet
4
Topic 2: Functions – SL and HL
SL
2.1
SL
2.5
Equations of a straight line =
0 ; y − y1= m ( x − x1 )
y mx + c ; ax + by + d =
y2 − y1
x2 − x1
Gradient formula
m=
Axis of symmetry of the
graph of a quadratic
function
f ( x) = ax 2 + bx + c ⇒ axis of symmetry is x = −
b
2a
Topic 2: Functions – HL only
AHL
2.9
Logistic function
f ( x) =
L
, L , k,C > 0
1 + Ce − kx
Mathematics: applications and interpretation formula booklet
5
Topic 3: Geometry and trigonometry – SL and HL
SL
3.1
Distance between two
points ( x1 , y1 , z1 ) and
d=
Coordinates of the
midpoint of a line segment
with endpoints ( x1 , y1 , z1 )
 x1 + x2 y1 + y2 z1 + z2 
, , 

2
2 
 2
Volume of a right-pyramid
V=
1
Ah , where A is the area of the base, h is the height
3
Volume of a right cone
V=
1 2
πr h , where r is the radius, h is the height
3
Area of the curved surface
of a cone
A = πrl , where r is the radius, l is the slant height
Volume of a sphere
V=
Surface area of a sphere
A = 4πr 2 , where r is the radius
Sine rule
a
b
c
= =
sin A sin B sin C
Cosine rule
c 2 = a 2 + b 2 − 2ab cos C ; cos C =
Area of a triangle
1
A = ab sin C
2
( x1 − x2 ) 2 + ( y1 − y2 ) 2 + ( z1 − z2 ) 2
( x2 , y2 , z2 )
and ( x2 , y2 , z2 )
SL
3.2
SL
3.4
Length of an arc
=
l
4 3
πr , where r is the radius
3
θ
360
a 2 + b2 − c2
2ab
× 2πr , where θ is the angle measured in degrees, r is
the radius
Area of a sector
=
A
θ
360
× πr 2 , where θ is the angle measured in degrees, r is
the radius
Mathematics: applications and interpretation formula booklet
6
Topic 3: Geometry and trigonometry – HL only
AHL
3.7
Length of an arc
l = rθ , where r is the radius, θ is the angle measured in radians
Area of a sector
1
A = r 2θ , where r is the radius, θ is the angle measured in
2
radians
AHL
3.8
Identities
cos 2 θ + sin 2 θ =
1
tan θ =
AHL
3.9
Transformation matrices
 cos 2θ

 sin 2θ
sin θ
cos θ
sin 2θ 
 , reflection in the line y = (tan θ ) x
− cos 2θ 
 k 0

 , horizontal stretch / stretch parallel to x-axis with a scale
 0 1
factor of k
1 0

 , vertical stretch / stretch parallel to y-axis with a scale
0 k 
factor of k
k 0

 , enlargement, with a scale factor of k, centre (0, 0)
0 k
 cos θ

 sin θ
− sin θ 
 , anticlockwise/counter-clockwise rotation of
cos θ 
angle θ about the origin ( θ > 0 )
 cos θ

 − sin θ
(θ > 0 )
sin θ 
 , clockwise rotation of angle θ about the origin
cos θ 
Mathematics: applications and interpretation formula booklet
7
AHL
3.10
AHL
3.11
AHL
3.13
Magnitude of a vector
 v1 
 
v + v2 + v3 , where v =  v2 
v 
 3
2
1
v =
2
2
Vector equation of a line
r = a + λb
Parametric form of the
equation of a line
x =+
x0 λ l , y =+
y0 λ m, z =+
z0 λ n
Scalar product
 v1 
 w1 
 
 
v ⋅ w= v1w1 + v2 w2 + v3 w3 , where v =  v2  , w =  w2 
v 
w 
 3
 3
v⋅w =
v w cos θ , where θ is the angle between v and w
v1w1 + v2 w2 + v3 w3
v w
Angle between two
vectors
cos θ =
Vector product
 w1 
 v2 w3 − v3 w2 
 v1 
 


 
v ×=
w  v3 w1 − v1w3  , where v =  v2  , w =  w2 


w 
v 
 3
 v1w2 − v2 w1 
 3
v×w =
v w sin θ , where θ is the angle between v and w
Area of a parallelogram
A=
v × w where v and w form two adjacent sides of a
parallelogram
Mathematics: applications and interpretation formula booklet
8
Topic 4: Statistics and probability – SL and HL
SL
4.2
SL
4.3
SL
4.5
SL
4.6
Interquartile range
IQR
= Q3 − Q1
k
Mean, x , of a set of data
x=
∑fx
i =1
i i
, where n =
n
k
∑f
i =1
i
n ( A)
n (U )
Probability of an event A
P ( A) =
Complementary events
P ( A) + P ( A′) =
1
Combined events
P ( A ∪ B )= P ( A) + P ( B) − P ( A ∩ B)
Mutually exclusive events
P ( A ∪ B )= P ( A) + P ( B)
Conditional probability
P ( A B) =
Independent events
P ( A ∩ B) =
P ( A) P ( B)
SL
4.7
Expected value of a
E(X )
=
discrete random variable X
SL
4.8
Binomial distribution
P ( A ∩ B)
P ( B)
x P(X
∑=
x)
X ~ B (n , p)
Mean
E ( X ) = np
Variance
Var (=
X ) np (1 − p )
Mathematics: applications and interpretation formula booklet
9
Topic 4: Statistics and probability – HL only
AHL
4.14
Linear transformation of a
single random variable
E ( aX +=
b ) aE ( X ) + b
Linear combinations of n
independent random
variables, X 1 , X 2 , ..., X n
a1E ( X 1 ) ± a2 E ( X 2 ) ± ... ± an E ( X n )
E ( a1 X 1 ± a2 X 2 ± ... ± a=
n Xn )
Var ( aX + b ) =
a 2 Var ( X )
Var ( a1 X 1 ± a2 X 2 ± ... ± an X n )
= a12 Var ( X 1 ) + a2 2 Var ( X 2 ) + ... + an 2 Var ( X n )
Sample statistics
Unbiased estimate of
population variance sn2−1
AHL
4.17
AHL
4.19
sn2−1 =
n 2
sn
n −1
Poisson distribution
X ~ Po (m)
Mean
E(X ) = m
Variance
Var ( X ) = m
Transition matrices
T n s0 = sn , where s0 is the initial state
Mathematics: applications and interpretation formula booklet
10
Topic 5: Calculus – SL and HL
SL
5.3
SL
5.5
Derivative of x n
f ( x) =
x n ⇒ f ′( x) =
nx n −1
Integral of x n
dx
∫x=
Area of region enclosed by
a curve y = f ( x) and the
x-axis, where f ( x) > 0
SL
5.8
The trapezoidal rule
n
x n +1
+ C , n ≠ −1
n +1
b
A = ∫ y dx
a
1
y dx ≈ h ( ( y0 + yn ) + 2( y1 + y2 + ... + yn −1 ) ) ,
2
b−a
where h =
n
∫
b
a
Topic 5: Calculus – HL only
AHL
5.9
Derivative of sin x
f ( x) =sin x ⇒ f ′( x) =cos x
Derivative of cos x
f ( x) =⇒
cos x
f ′( x) =
− sin x
Derivative of tan x
f ( x) =tan x ⇒ f ′( x) =
Derivative of e x
f ( x) =
e x ⇒ f ′( x) =
ex
Derivative of ln x
1
f ( x) =
ln x ⇒ f ′( x) =
x
Chain rule
y = g (u ) , where u = f ( x) ⇒
Product rule
y =uv ⇒
Quotient rule
du
dv
v −u
u
dy
d
x
d
x
y= ⇒
=
2
v
dx
v
Mathematics: applications and interpretation formula booklet
1
cos 2 x
dy dy du
= ×
dx du dx
dy
dv
du
=u + v
dx
dx
dx
11
AHL
5.11
Standard integrals
1
dx
∫ x=
ln x + C
− cos x + C
∫ sin x dx =
dx
∫ cos x=
1
∫ cos=
x
2
∫e
AHL
5.12
AHL
5.13
AHL
5.16
x
sin x + C
tan x + C
x ex + C
d=
b
b
a
a
Area of region enclosed
by a curve and x or y-axes
A = ∫ y dx or A = ∫ x dy
Volume of revolution
about x or y-axes
V = ∫ πy 2 dx or V = ∫ πx 2 dy
Acceleration
=
a
Distance travelled from
t1 to t2
distance =
Displacement from
t1 to t2
displacement =
Euler’s method
b
b
a
a
dv d 2 s
dv
= =
v
2
dt dt
ds
∫
t2
t1
v(t ) dt
∫
t2
t1
v(t ) dt
xn + h , where h is a constant
yn +=
yn + h × f ( xn , yn ) ; xn +=
1
1
(step length)
Euler’s method for
coupled systems
xn +1 = xn + h × f1 ( xn , yn , tn )
yn +1 = yn + h × f 2 ( xn , yn , tn )
tn +1= tn + h
where h is a constant (step length)
AHL
5.17
Exact solution for coupled =
x Aeλ1t p1 + Beλ2t p2
linear differential equations
Mathematics: applications and interpretation formula booklet
12