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Diploma Programme
Mathematics SL formula booklet
For use during the course and in the examinations
First examinations 2014
Edited in 2015 (version 2)
© International Baccalaureate Organization 2012
5045
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
Formulae
Prior learning
Area of a parallelogram
A b h
Area of a triangle
A
1
(b h)
2
Area of a trapezium
A
1
( a b) h
2
Area of a circle
A
r2
Circumference of a circle
C
2 r
Volume of a pyramid
V
1
(area of base
3
Volume of a cuboid (rectangular prism)
V
l w h
Volume of a cylinder
V
Area of the curved surface of a cylinder
A 2 rh
Volume of a sphere
V
4 3
r
3
Volume of a cone
V
1 2
r h
3
Distance between two points ( x1 , y1 , z1 ) and
d
( x1
( x2 , y2 , z2 )
Coordinates of the midpoint of a line segment
with endpoints ( x1 , y1 , z1 ) and ( x2 , y2 , z2 )
Mathematics SL formula booklet
vertical height)
r 2h
x1
2
x2 ) 2
x2 y1
,
2
y2 ) 2
( y1
y2 z1
,
( z1
z2 ) 2
z2
2
2
Topics
Topic 1—Algebra
1.1
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
Sn
n
2u1 (n 1) d
2
The nth term of a
geometric sequence
un
u1r n
The sum of n terms of a
finite geometric sequence
Sn
u1 (r n 1)
r 1
The sum of an infinite
geometric sequence
S
Exponents and logarithms
ax
Laws of logarithms
log c a log c b log c ab
a
log c a log c b log c
b
r
log c a r log c a
Change of base
log b a
Binomial coefficient
Binomial theorem
Mathematics SL formula booklet
n
r
n
(u1 un )
2
1
u1
1 r
, r
b
u1 (1 r n )
, r 1
1 r
1
x log a b
log c a
log c b
n!
r !(n r )!
( a b) n
an
n n1
a b
1
n nr r
a b
r
bn
3
Topic 2—Functions and equations
2.4
Axis of symmetry of graph
of a quadratic function
f ( x) ax 2
bx c
2.6
Relationships between
logarithmic and
exponential functions
a x e x ln a
log a a x x
a loga x
Solutions of a quadratic
equation
ax 2
2.7
Discriminant
bx c 0
b2
b
2a
axis of symmetry x
x
b
b 2 4ac
, a
2a
0
4ac
Topic 3—Circular functions and trigonometry
3.1
3.2
3.3
3.6
Length of an arc
l
r
Area of a sector
A
1 2
r
2
Trigonometric identity
tan
Pythagorean identity
cos 2
sin 2
Double angle formulae
sin 2
2sin cos
cos 2
cos 2
Cosine rule
c2
b2
Sine rule
a
sin A
Area of a triangle
A
Mathematics SL formula booklet
sin
cos
a2
b
sin B
1
sin 2
2cos 2
2ab cos C ; cos C
1 1 2 si n 2
a2
b2 c2
2ab
c
sin C
1
ab sin C
2
4
Topic 4—Vectors
4.1
4.2
4.3
Magnitude of a vector
Scalar product
v12
v
v2 2
v32
v w
v w cos
v w
v1w1 v2 w2
v3 w3
v w
v w
Angle between two
vectors
cos
Vector equation of a line
r = a + tb
Topic 5—Statistics and probability
5.2
Mean of a set of data
n
fi xi
i 1
n
x
fi
i 1
5.5
5.6
5.7
5.8
5.9
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
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)
Expected value of a
discrete random variable X
E(X )
Binomial distribution
X ~ B(n , p )
Mean
E ( X ) np
Variance
Var ( X ) np (1 p )
Standardized normal
variable
z
Mathematics SL formula booklet
x P(X
B)
x)
x
P(X
r)
n r
p (1 p ) n r , r
r
0,1,
,n
x
5
Topic 6—Calculus
6.1
6.2
y
Derivative of x n
f ( x)
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)
f ( x)
1
cos 2 x
Derivative of e x
f ( x) e x
Derivative of ln x
f ( x) ln x
Chain rule
y
Product rule
y uv
Quotient rule
6.4
dy
dx
Derivative of f ( x)
f ( x)
xn
h
f ( x) nx n
tan x
f ( x)
u
v
dy
dx
f ( x)
1
dy
dx
dv
dx
v
du
dx
du
dx
u
dv
dx
u
v
dy
du
du
dx
v2
xn 1
C, n
n 1
x n dx
f ( x h)
h
1
x
f ( x)
dy
dx
Standard integrals
0
f ( x) e x
g (u ) , u
y
f ( x) lim
1
1
dx ln x C , x 0
x
sin x dx
cos x C
cos x dx sin x C
e x dx e x
6.5
Area under a curve
between x = a and x = b
A
Volume
of
revolution
V
about the x-axis from x = a
to x = b
6.6
Total distance travelled
from t1 to t 2
Mathematics SL formula booklet
b
a
b
a
C
y dx
y2 x
distance
t2
t1
v(t ) dt
6