# Mathematics SL formula booklet

```Diploma Programme
Mathematics SL formula booklet
For use during the course and in the examinations
First examinations 2014
Edited in 2015 (version 2)
&copy; 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
A= b &times; h
Area of a parallelogram
Area of a triangle
=
A
1
(b &times; h)
2
Area of a trapezium
=
A
1
( a + b) h
2
Area of a circle
A = πr 2
Circumference of a circle
C = 2πr
Volume of a pyramid
=
V
1
(area of base &times; vertical height)
3
Volume of a cuboid (rectangular prism)
V =l &times; w &times; h
Volume of a cylinder
V = πr 2 h
Area of the curved surface of a cylinder
A= 2πrh
Volume of a sphere
V=
4 3
πr
3
Volume of a cone
V=
1 2
πr h
3
Distance between two points ( x1 , y1 , z1 ) and
d=
( x1 − x2 ) 2 + ( y1 − y2 ) 2 + ( z1 − z2 ) 2
( x2 , y2 , z2 )
Coordinates of the midpoint of a line segment
with endpoints ( x1 , y1 , z1 ) and ( x2 , y2 , z2 )
Mathematics SL formula booklet
 x1 + x2 y1 + y2 z1 + z2 
,
,


2
2 
 2
2
Topics
Topic 1—Algebra
1.1
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 )= (u1 + un )
2
2
The sum of n terms of a
u1 (r n − 1) u1 (1 − r n )
, r ≠1
=
S
=
n
finite geometric sequence
r −1
1.2
1.3
1− r
The sum of an infinite
geometric sequence
S∞ =
Exponents and logarithms
ax = b ⇔
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
n
n!
 =
 r  r !(n − r )!
Binomial theorem
 n
 n
(a + b) n = a n +   a n −1b + +   a n − r b r + + b n
1
r
Mathematics SL formula booklet
u1
, r &lt;1
1− r
x = log a b
log c a
log c b
3
Topic 2—Functions and equations
2.4
Axis of symmetry of graph
b
f ( x) =
ax 2 + bx + c ⇒ axis of symmetry x =
−
2a
2.6
Relationships between
logarithmic and
exponential functions
a x = e x ln a
log a a x= x= a loga x
2.7
equation
ax 2 + bx + c= 0 ⇒
Discriminant
∆= b 2 − 4ac
x=
−b &plusmn; b 2 − 4ac
, a≠0
2a
Topic 3—Circular functions and trigonometry
3.1
3.2
3.3
Length of an arc
l =θr
Area of a sector
1
A = θ r2
2
Trigonometric identity
tan θ =
Pythagorean identity
cos 2 θ + sin 2 θ =
1
Double angle formulae
sin 2θ = 2sin θ cos θ
sin θ
cos θ
cos 2θ = cos 2 θ − sin 2 θ = 2cos 2 θ − 1 = 1 − 2 sin 2 θ
3.6
Cosine rule
c 2 = a 2 + b 2 − 2ab cos C ; cos C =
Sine rule
a
b
c
= =
sin A sin B sin C
Area of a triangle
1
A = ab sin C
2
Mathematics SL formula booklet
a 2 + b2 − c2
2ab
4
Topic 4—Vectors
4.1
4.2
Magnitude of a vector
Scalar product
v =
v12 + v2 2 + v32
v⋅w =
v w cos θ
v ⋅ w= v1w1 + v2 w2 + v3 w3
4.3
v⋅w
v w
Angle between two
vectors
cos θ =
Vector equation of a line
r = a + tb
Topic 5—Statistics and probability
5.2
Mean of a set of data
n
∑fx
i i
x=
i =1
n
∑f
i
i =1
5.5
5.6
5.7
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) =
P (A) P (B | A)
Independent events
P ( A ∩ B) =
P ( A) P ( B)
Expected value of a
discrete random variable X
E(X =
) &micro;=
∑ x P ( X=
x)
x
Binomial distribution
n r
n−r
X ~ B(n , p ) ⇒ P ( X =
r) =
0,1,  , n
  p (1 − p ) , r =
r
 
Mean
E ( X ) = np
Variance
Var (=
X ) np (1 − p )
Standardized normal
variable
z=
5.8
5.9
n ( A)
n (U )
Probability of an event A
Mathematics SL formula booklet
x−&micro;
σ
5
Topic 6—Calculus
6.1
6.2
y = f ( x) ⇒
Derivative of x n
f ( x) =
xn ⇒
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) =
ex ⇒
f ′( x) =
ex
Derivative of ln x
f ( x) =
ln x ⇒
1
f ′( x) =
x
Chain rule
y = g (u ) , u =f ( x) ⇒
Product rule
y =uv ⇒
Quotient rule
6.4
dy
 f ( x + h) − f ( x ) 
= f ′( x) = lim 

h
→
0
dx
h


Derivative of f ( x)
u
y=
v
n
dx
∫x=
1
cos 2 x
dy dy du
= &times;
dx du dx
dy
dv
du
=u + v
dx
dx
dx
du
dv
v −u
dy
= dx 2 dx
dx
v
⇒
Standard integrals
f ′( x) =
nx n −1
x n +1
+ C , n ≠ −1
n +1
1
∫ x dx =ln x + C ,
x&gt;0
− cos x + C
∫ sin x dx =
dx
∫ cos x=
∫e
6.5
Area under a curve
between x = a and x = b
x
Total distance travelled
from t1 to t 2
Mathematics SL formula booklet
x ex + C
d=
b
A = ∫ y dx
a
Volume
of
revolution
V=
about the x-axis from x = a
to x = b
6.6
sin x + C
∫
b
a
πy 2 dx
distance =
∫
t2
t1
v(t ) dt
6
```