Mathematics HL and further mathematics HL formula booklet
Diploma Programme
Mathematics HL and further mathematics HL
formula booklet
For use during the course and in the examinations First examinations 2014
Edited in 2015 (version 2)
? International Baccalaureate Organization 2012
5048
Contents
Prior learning
2
Core
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
6
Topic 6: Calculus
8
Options
10
Topic 7: Statistics and probability
10
Further mathematics HL topic 3
Topic 8: Sets, relations and groups
11
Further mathematics HL topic 4
Topic 9: Calculus
11
Further mathematics HL topic 5
Topic 10: Discrete mathematics
12
Further mathematics HL topic 6
Formulae for distributions
13
Topics 5.6, 5.7, 7.1, further mathematics HL topic 3.1
Discrete distributions
13
Continuous distributions
13
Further mathematics
14
Topic 1: Linear algebra
14
Mathematics HL and further mathematics formula booklet
1
Formulae
Prior learning
Area of a parallelogram
A= b ? h , where b is the base, h is the height
Area of a triangle
=A 1 (b ? h) , where b is the base, h is the height 2
Area of a trapezium
=A 1 (a + b) h , where a and b are the parallel sides, h is the height 2
Area of a circle
A = r2 , where r is the radius
Circumference of a circle
C= 2r , where r is the radius
Volume of a pyramid=V 1 (area of base ? vertical height) 3
Volume of a cuboid
V =l ? w ? h , where l is the length, w is the width, h is the height
Volume of a cylinder
V = r2h , where r is the radius, h is the height
Area of the curved surface of A= 2rh , where r is the radius, h is the height
a cylinder
Volume of a sphere
V= 4 r3 , where r is the radius 3
Volume of a cone
V= 1 r2h , where r is the radius, h is the height 3
Distance between two
points (x1, y1) and (x2 , y2 )
d = (x1 - x2 )2 + ( y1 - y2 )2
Coordinates of the midpoint of a line segment with endpoints
(x1, y1) and (x2 , y2 )
x1
+ 2
x2
, y1
+ 2
y2
Solutions of a quadratic equation
The solutions of ax2 + bx + c =0 are x = -b ? b2 - 4ac 2a
Mathematics HL and further mathematics formula booklet
2
Core
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
Sn=
n 2
(
2u1
+
(n
-
1)
d
)=
n 2
(u1
+
u n
)
The nth term of a
geometric sequence
un = u1r n-1
The sum of n terms of a finite geometric sequenc= e Sn
u= 1(rn -1) r -1
u1(1 - rn ) , r 1 1- r
The sum of an infinite geometric sequence
S
=
u1 1- r
,
r
0, b > 0, a 1
a x = exln a
loga a x= x= aloga x
logb
a
=
logc logc
a b
1.3 Combinations
n r
=
r
n! !(n -
r)!
Permutations
n Pr
=
n! (n - r)!
Binomial theorem 1.5 Complex numbers 1.7 De Moivre's theorem
(a + b)n
=
an
+
n 1
a n -1b
++
n r
an-rbr
+ + bn
z =a + ib =r (cos + isin ) =rei =r cis
[r(cos + i sin )]n =rn (cos n + i sin n ) =rnein =rn cis n
Mathematics HL and further mathematics formula booklet
3
Topic 2: Functions and equations
2.5 Axis of symmetry of the graph of a quadratic function
f (x) =ax2 + bx + c axis of symmetry x =- b 2a
2.6 Discriminant
= b2 - 4ac
Topic 3: Circular functions and trigonometry
3.1 Length of an arc
l = r , where is the angle measured in radians, r is the radius
Area of a sector
A = 1 r2 , where is the angle measured in radians, r is the 2
radius
3.2 Identities
tan = sin cos
sec = 1 cos
cosec = 1 sin
Pythagorean identities
cos2 + sin2 = 1 1 + tan2 = sec2 1 + cot2 = csc2
3.3 Compound angle identities sin ( A= ? B) sin Acos B ? cos Asin B cos ( A ? B) = cos Acos B sin Asin B tan ( A ? B) =tan A ? tan B 1 tan A tan B
Double angle identities
sin 2 = 2sin cos cos 2 = cos2 - sin2 = 2cos2 -1 = 1 - 2sin2 tan 2 = 2 tan
1 - tan2
Mathematics HL and further mathematics formula booklet
4
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