Diploma Programme
Mathematics: applications and interpretation
SL formula booklet
For use during the course and in the examinations
First examinations 2021
Version 1.0
STANDARD LEVEL
© International Baccalaureate Organization 2023
Contents
Topic 1: Number and algebra – SL
2
Topic 2: Functions – SL
2
Topic 3: Geometry and trigonometry – SL
3
Topic 4: Statistics and probability – SL
5
Topic 5: Calculus – SL
6
Topic 1: Number and algebra – SL
1.2
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
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
1.5
Exponents and logarithms
a x = b ⇔ x = log a b , where a > 0, b > 0, a ≠ 1
Percentage error
ε=
1.6
kn
vA − vE
× 100% , where vE is the exact value and vA is
vE
the approximate value of v
Topic 2: Functions – SL
2.1
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 = −
Mathematics: applications and interpretation SL formula booklet
b
2a
2
Topic 3: Geometry and trigonometry – SL
Prior learning – SL
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
3.1
( x1 − x2 ) 2 + ( y1 − y2 ) 2
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
( x1 − x2 ) 2 + ( y1 − y2 ) 2 + ( z1 − z2 ) 2
( x2 , y2 , z2 )
and ( x2 , y2 , z2 )
Mathematics: applications and interpretation SL formula booklet
3
3.2
3.4
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
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 SL formula booklet
4
Topic 4: Statistics and probability – SL
4.2
Interquartile range
IQR
= Q3 − Q1
4.3
k
Mean, x , of a set of data
4.5
4.6
4.7
4.8
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)
Expected value of a
E(X )
discrete random variable X=
P ( A ∩ B)
P ( B)
k
x P(X
∑=
i =1
i
xi )
Binomial distribution
X ~ B (n , p)
Mean
E ( X ) = np
Variance
Var (=
X ) np (1 − p )
Mathematics: applications and interpretation SL formula booklet
5
Topic 5: Calculus – SL
5.3
5.5
Derivative of x n
f ( x) =
x n ⇒ f ′( x) =
nx n −1
Integral of x n
n
dx
∫x=
Area of region enclosed by
a curve y = f ( x) and the
A = ∫ y dx
x n +1
+ C , n ≠ −1
n +1
b
a
x-axis, where f ( x) > 0
5.8
The trapezoidal rule
1
y dx ≈ h ( ( y0 + yn ) + 2( y1 + y2 + ... + yn −1 ) ) ,
2
b−a
where h =
n
∫
b
a
Mathematics: applications and interpretation SL formula booklet
6