SL Formulabooklet

IB MATH SL formula booklet

1. Mathematical studies SL: Formula booklet 1
Published March 2012
© International Baccalaureate Organization 2012 5045







Mathematics SL formula booklet

For use during the course and in the examinations
First examinations 2014
Diploma Programme

3. Mathematics SL f ormula booklet 1
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

4. Mathematics SL f ormula booklet 2
Formulae
Prior learning
Area of a parallelogram Abh= ×
Area of a triangle 1
()
2
= ×A bh
Area of a trapezium 1
()
2
= +A a bh
Area of a circle
2
= πAr
Circumference of a circle 2= πCr
Volume of a pyramid 1
(area of base vertical height)
3
= ×V
Volume of a cuboid (rectangular prism) =××V lwh
Volume of a cylinder
2
= πV rh
Area of the curved surface of a cylinder 2= πA rh
Volume of a sphere
34
3
= πVr
Volume of a cone
21
3
= πV rh

Distance between two points
1 11
(, ,)xyz and
2 22(, ,)xyz
2 22
12 1 2 12
( )( )( )= − +− +−d xx yy zz

Coordinates of the midpoint of a line segment
with endpoints
1 11(, ,)xyz and
2 22(, ,)xyz
1 2 1 21 2
, , 
222
+++


x xy yz z

5. Mathematics SL f ormula booklet 3
Topics
Topic 1—Algebra
1.1 The n
th
term of an
arithmetic sequence
1( 1)=+−
nuu n d
The sum of n terms of an
arithmetic sequence
11
(2 ( 1) ) ( )
22
= +− = +
nn
nn
S u n d uu
The n
th
term of a
geometric sequence
1
1
−
=
n
n
u ur

The sum of n terms of a
finite geometric sequence
11
( 1) (1 )
11
−−
= =
−−
nn
n
ur u r
S
rr
, 1≠r
The sum of an infinite
geometric sequence
1
1
u
S
r
∞
=
−
, 1r<
1.2 Exponents and logarithms log
x
a
ab x b=⇔=

Laws of logarithms log log log
ccc
a b ab+=
log log log
ccc
a
ab
b
−=
log log
r
cc
ar a=

Change of base log
log
log
c
b
c
a
a
b
=

1.3 Binomial coefficient
()!!
!
rnr
n
r
n
−
=









Binomial theorem
1
()
1
−− 
+ = + ++ ++
 
 

n n n nr r n
nn
ab a ab ab b
r

6. Mathematics SL f ormula booklet 4
Topic 2—Functions and equations
2.4 Axis of symmetry of
graph of a quadratic
function
2
( ) axis of symmetry
2
b
f x ax bx c x
a
= ++ ⇒ =−
2.6 Relationships between
logarithmic and
exponential functions
ln
e
x xa
a=
log
log
a
xx
a
a xa= =
2.7 Solutions of a quadratic
equation
2
2
4
0 ,0 2
b b ac
ax bx c x a
a
−± −
+ += ⇒ = ≠

Discriminant
2
4b ac∆= −
Topic 3—Circular functions and trigonometry
3.1 Length of an arc lrθ=
Area of a sector
21
2
Arθ=
3.2 Trigonometric identity sin
tan
cosθ
θ
θ
=

3.3 Pythagorean identity
2 2
sin 1cosθθ+=
Double angle formulae 2sinsin 2 cosθ θθ=
22 2 2
cos sin 2cos 1 1 2cs io s2 nθ θθ θ θ= − = −=−
3.6 Cosine rule
2 22
2 cosc a b ab C=+− ;
2 22
cos
2
abc
C
ab
+−
=
Sine rule
sin sin sin
abc
ABC
= =
Area of a triangle 1
sin
2
A ab C=

7. Mathematics SL f ormula booklet 5
Topic 4—Vectors
4.1 Magnitude of a vector 222
123
vvv
= ++v
4.2 Scalar product cosθ⋅=vw vw
11 2 2 3 3⋅= + +vw vw vwvw
Angle between two
vectors
cosθ
⋅
=
vw
vw

4.3 Vector equation of a line =+tra b
Topic 5—Statistics and probability
5.2 Mean of a set of data
1
1
n
ii
i
n
i
i
fx
x
f
=
=
=
∑
∑

5.5 Probability of an event A ()
P( )
()
=
nA
A
nU

Complementary events P( ) P( ) 1′+=AA
5.6 Combined events P( ) P( ) P( ) P( )∪= + − ∩AB A B AB
Mutually exclusive events P( ) P( ) P( )∪= +AB A B
Conditional probability P( ) P( )P( | )A B A BA∩=
Independent events P( ) P( ) P( )∩=AB A B
5.7 Expected value of a discrete
random variable X
E( ) P( )µ= = =∑
x
X xXx
5.8 Binomial distribution
~ B( , ) P( ) (1 ) , 0, 1, ,
r nr
n
Xnp Xr ppr n
r
−
⇒== − =




Mean E( )=X np
Variance Var( ) (1 )X np p= −
5.9 Standardized normal
variable
µ
σ−
=
x
z

8. Mathematics SL f ormula booklet 6
Topic 6—Calculus
6.1 Derivative of ()fx
0
d ( ) ()
() () lim
d
h
y fx h fx
y fx f x
xh
→
+−
′= ⇒= =



6.2 Derivative of
n
x
1
() ()
nn
f x x f x nx
−
′=⇒=
Derivative of sinx ( ) sin ( ) cosfx x f x x ′=⇒=
Derivative of cosx ( ) cos ( ) sinfx x f x x ′=⇒=−
Derivative of tanx
2
1
() tan ()
cos
fx x f x
x
′=⇒=
Derivative of e
x
() e () e
xx
fx f x ′=⇒=
Derivative of lnx 1
() ln ()fx x f x
x
′=⇒=
Chain rule
()=y gu,
d dd
()
ddd
y yu
u fx
xux
= ⇒=×
Product rule ddd
ddd
yvu
y uv u v
xxx
=⇒=+
Quotient rule
2
dd
d dd
d
uv
vu
uy xx
y
v xv
−
=⇒=
6.4 Standard integrals
1
d ,1
1
n
n x
xx C n
n
+
= + ≠−
+
∫

1
d ln , 0x xCx
x
=+>
∫

sin d cosxx x C=−+∫

cos d sinxx x C= +∫

ed e= +∫
xx
xC
6.5 Area under a curve
between x = a and x = b d
b
a
A yx=∫

Volume of revolution
about the x -axis from x = a
to x = b
2
πd
b
a
V yx=∫

6.6 Total distance travelled
from
1t to
2t
distance
2
1
()d
t
t
vt t=∫

9. Mathematics SL f ormula booklet 7