This document provides a formula booklet for use in the Diploma Programme Mathematics SL course and examinations. It contains formulas for topics covered in the course, including algebra, functions, trigonometry, vectors, statistics, and calculus. The booklet is published by the International Baccalaureate Organization and approved for use from 2014 onwards.

1. Diploma Programme
Mathematics SL formula booklet
For use during the course and in the examinations
First examinations 2014
Published March 2012
© International Baccalaureate Organization 2012 5045
Mathematical studies SL: Formula booklet 1

3. Contents
Prior learning 2
Topics 3
Topic 1—Algebra 3
Topic 2—Functions and equations 4
Topic 3—Circular functions and trigonometry 4
Topic 4—Vectors 5
Topic 5—Statistics and probability 5
Topic 6—Calculus 6
Mathematics SL formula booklet 1

4. Formulae
Prior learning
Area of a parallelogram A = b× h
Area of a triangle A = 1( b× h
)
2
Area of a trapezium 1( )
A = a + b h
2
Area of a circle A = πr2
Circumference of a circle C = 2πr
Volume of a pyramid V = 1 (area of base ×
vertical height)
3
Volume of a cuboid (rectangular prism) V = l × w× h
Volume of a cylinder V = πr2h
Area of the curved surface of a cylinder A = 2πrh
Volume of a sphere 4 3
V = πr
3
Volume of a cone 1 2
V = πr h
3
Distance between two points 1 1 1 (x , y , z ) and
2 2 2 (x , y , z )
2 2 2
1 2 1 2 1 2 d = (x − x ) + ( y − y ) + (z − z )
Coordinates of the midpoint of a line segment
x x , y y , z z
with endpoints (x , y z 1 2 1 2 1 2
, ) and (x , y , z ) 1 1 1 2 2 2  + + + 
  2 2 2


Mathematics SL formula booklet 2

5. Topics
Topic 1—Algebra
1.1 The nth term of an
u = u + ( n −1) d
arithmetic sequence n 1 The sum of n terms of an
arithmetic sequence S = n (2 u + ( n − 1) d ) = n ( u + u
)
n 1 1 n
2 2
The nth term of a
geometric sequence
= n−
n u u r
1
1
The sum of n terms of a
S = u ( r n − 1) −
n
finite geometric sequence 1 =
u (1 r
)
1 r − 1 1
−
r
n
, r ≠1
The sum of an infinite
S u
geometric sequence 1
=
∞ 1
−
r , r <1
1.2 Exponents and logarithms x log
a a = b ⇔ x = b
Laws of logarithms log log log c c c a + b = ab
a b a
log log log c c c
b
− =
log r log
c c a = r a
Change of base log log
a a
log
c
b
c
b
=
1.3 Binomial coefficient
!
n
! ( )!
r n r
n
r
−

=  


 

Binomial theorem ( ) 1
 n   n

n n n −  n − r r  n a b a a b a b b
+ = +   + +   + +
1
r
   
Mathematics SL formula booklet 3

6. Topic 2—Functions and equations
2.4 Axis of symmetry of
graph of a quadratic
function
f ( x ) ax 2 bx c axis of symmetry
x b
2
a
= + + ⇒ =−
2.6 Relationships between
logarithmic and
exponential functions
ax = ex ln a
log x loga x
aa = x = a
2.7 Solutions of a quadratic
equation
2
ax bx c x b b ac a
2 − ± −
+ + = 0 ⇒ = 4 , ≠
0
2
a
Discriminant Δ = b2 − 4ac
Topic 3—Circular functions and trigonometry
3.1 Length of an arc l =θ r
Area of a sector 1 2
A = θ r
2
3.2 Trigonometric identity sin tan
θ
cos
θ
θ
=
3.3 Pythagorean identity cos2θ + sin2θ =1
Double angle formulae sin 2θ = 2sinθ cosθ
cos 2θ = cos2θ − sin2θ = 2cos2θ −1=1− 2sin2θ
3.6 Cosine rule c2 = a2 + b2 − 2abcosC ;
2 2 2
cos
+ −
C a b c
2
ab
=
Sine rule
a = b =
c
A B C
sin sin sin
Area of a triangle 1 sin
A = ab C
2
Mathematics SL formula booklet 4

7. Topic 4—Vectors
4.1 Magnitude of a vector 2 2 2
v = v1 + v2 + v3
4.2 Scalar product v ⋅w = v w cosθ
1 1 2 2 3 3 v ⋅w = v w + v w + v w
Angle between two
vectors cosθ
⋅
= v w
v w
4.3 Vector equation of a line r = a + tb
Topic 5—Statistics and probability
5.2 Mean of a set of data
n
Σ
i
1
Σ
n
1
f x
i i
i
=
i
x
f
=
=
5.5 Probability of an event A P( A )
= n ( A
) n U
( )
Complementary events P(A) + P(A′) =1
5.6 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) = P(A)P(B | A)
Independent events P(A∩ B) = P(A) P(B)
5.7 Expected value of a discrete
random variable X
E( X ) = μ Σx
= x P( X = x
) 5.8 Binomial distribution
 n

X ~ B( np , ) P( X r ) p r (1 p )n − r , r 0,1, , n
⇒ = =   − =
r
 

Mean E(X ) = np
Variance Var(X ) = np(1− p)
5.9 Standardized normal
variable
−
μ
σ
z = x
Mathematics SL formula booklet 5

8. Topic 6—Calculus
6.1 Derivative of f (x)
y f x y f x f x h f x
( ) d ( ) lim ( ) ( )
′  + −  = ⇒ = =  
x → 0
h
d h
 
6.2 Derivative of xn f (x) = xn ⇒ f ′(x) = nxn−1
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
( ) tan ( ) 1
2
cos
= ⇒ ′ =
f x x f x
x
Derivative of ex f (x) = ex ⇒ f ′(x) = ex
Derivative of ln x 1 f (x) ln x f (x)
x
= ⇒ ′ =
Chain rule
u f x y y u
y = g(u) , = ( ) ⇒ d = d ×
d
x u x
d d d
Product rule d d d
y = uv ⇒ y = u v +
v u
x x x
d d d
Quotient rule
u y v d u −
u d
v y x x
d d d
d
2
= ⇒ =
v x v
6.4 Standard integrals n
+
1
xn x x C n
+ ∫
1 dx ln x C, x 0
x
d = + , ≠ −
1
1
n
∫ = + >
∫sin xdx = −cos x + C
∫cos xdx = sin x + C
∫e d = e + x x x C
6.5 Area under a curve
between x = a and x = b d b
A = ∫ y x
a
Volume of revolution
about the x-axis from x = a
to x = b
π 2 d b
a
V = ∫ y x
6.6 Total distance travelled
from 1 t to 2 t
( ) d t
t
= ∫ v t t
distance 2
1
Mathematics SL formula booklet 6

9. Mathematics SL formula booklet 7