Diploma Programme
Mathematics: analysis and approaches
HL formula booklet
For use during the course and in the examinations
First examinations 2021
Version 1.0
HIGHER LEVEL
© International Baccalaureate Organization 2023
Contents
Topic 1: Number and algebra – HL
2
Topic 2: Functions – HL
3
Topic 3: Geometry and trigonometry – HL
4
Topic 4: Statistics and probability – HL
7
Topic 5: Calculus – HL
9
Topic 1: Number and algebra – HL
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
S n=
The nth term of a
geometric sequence
un = u1r n −1
n
n
( 2u1 + (n − 1) d ) ; Sn= (u1 + un )
2
2
The sum of n terms of a
u1 (r n − 1) u1 (1 − r n )
, r ≠1
=
Sn =
finite geometric sequence
r −1
1.8
1.4
The sum of an infinite
=
S∞
geometric sequence
Compound interest
1− r
u1
, r <1
1− r
kn
r 

FV = PV × 1 +
 , where FV is the future value,
 100k 
PV is the present value, n is the number of years,
k is the number of compounding periods per year,
r% is the nominal annual rate of interest
1.5
Exponents and logarithms
a x = b ⇔ x = log a b , where a > 0, b > 0, a ≠ 1
1.7
Exponents and logarithms
log=
log a x + log a y
a xy
x
log
log a x − log a y
=
a
y
log a x m = m log a x
log a x =
1.9
log b x
log b a
Exponential and
logarithmic functions
a x = e x ln a ; log a a x= x= a loga x where a , x > 0, a ≠ 1
Binomial theorem n ∈ 
(a + b) n = a n + n C a n −1b + + n C a n − r b r + + b n
1
r
n!
nC =
r r !(n − r )!
Mathematics: analysis and approaches HL formula booklet
2
1.10
Combinations
n!
nC =
r r !(n − r )!
Permutations
n P = n!
r (n − r )!
Extension of binomial
theorem, n ∈ 
2


n
 b  n ( n − 1)  b 
n
1
a
b
a
n
+
=
+
+
(
)

 
  + ... 
2!  a 
a


1.12
Complex numbers
z= a + bi
1.13
Modulus-argument (polar)
and exponential (Euler)
form
z=
r (cos θ + isin θ ) =
re iθ =
r cis θ
1.14
De Moivre’s theorem
[ r (cosθ + isin θ )]
n
=
r n (cos nθ + isin nθ ) =
r n einθ =
r n cis nθ
Topic 2: Functions – HL
2.1
2.6
2.7
2.12
Equations of a straight line
0 ; y − y1= m ( x − x1 )
=
y mx + c ; ax + by + d =
y2 − y1
x2 − x1
Gradient formula
m=
Axis of symmetry of the
graph of a quadratic
function
f ( x) =
ax 2 + bx + c ⇒ axis of symmetry is x =
−
Solutions of a quadratic
equation
ax 2 + bx + c= 0 ⇒
Discriminant
∆= b 2 − 4ac
Sum and product of the
roots of polynomial
equations of the form
( −1) a0
− an −1
; product is
Sum is
an
an
n
∑a x
r =0
r
r
x=
b
2a
−b ± b 2 − 4ac
, a≠0
2a
n
=0
Mathematics: analysis and approaches HL formula booklet
3
Topic 3: Geometry and trigonometry – HL
Prior learning – HL
Area of a parallelogram
A = bh , where b is the base, h is the height
Area of a triangle
1
A = (bh) , where b is the base, h is the height
2
Area of a trapezoid
=
A
1
(a + b) h , where a and b are the parallel sides, h is the height
2
Area of a circle
A = πr 2 , where r is the radius
Circumference of a circle
C = 2πr , where r is the radius
Volume of a cuboid
V = lwh , where l is the length, w is the width, h is the height
Volume of a cylinder
V = πr 2 h , where r is the radius, h is the height
Volume of a prism
V = Ah , where A is the area of cross-section, h is the height
Area of the curved surface of
a cylinder
A= 2πrh , where r is the radius, h is the height
Distance between two
points ( x1 , y1 ) and ( x2 , y2 )
d=
Coordinates of the midpoint of
a line segment with endpoints
( x1 , y1 ) and ( x2 , y2 )
 x1 + x2 y1 + y2 
, 

2 
 2
3.1
Distance between two
points ( x1 , y1 , z1 ) and
( x1 − x2 ) 2 + ( y1 − y2 ) 2
( x1 − x2 ) 2 + ( y1 − y2 ) 2 + ( z1 − z2 ) 2
d=
( x2 , y2 , z2 )
Coordinates of the
midpoint of a line segment
with endpoints ( x1 , y1 , z1 )
 x1 + x2 y1 + y2 z1 + z2 
,
,


2
2 
 2
Volume of a right-pyramid
V=
and ( x2 , y2 , z2 )
1
Ah , where A is the area of the base, h is the height
3
Mathematics: analysis and approaches HL formula booklet
4
3.2
3.4
1 2
πr h , where r is the radius, h is the height
3
Volume of a right cone
V=
Area of the curved surface
of a cone
A = πrl , where r is the radius, l is the slant height
Volume of a sphere
V=
Surface area of a sphere
A = 4πr 2 , where r is the radius
Sine rule
a
b
c
= =
sin A sin B sin C
Cosine rule
c 2 = a 2 + b 2 − 2ab cos C ; cos C =
Area of a triangle
1
A = ab sin C
2
Length of an arc
l = rθ , where r is the radius, θ is the angle measured in radians
Area of a sector
1
A = r 2θ , where r is the radius, θ is the angle measured in
2
4 3
πr , where r is the radius
3
a 2 + b2 − c2
2ab
radians
3.5
3.6
sin θ
cos θ
Identity for tan θ
tan θ =
Pythagorean identity
cos 2 θ + sin 2 θ =
1
Double angle identities
sin 2θ = 2sin θ cos θ
cos 2θ = cos 2 θ − sin 2 θ = 2cos 2 θ − 1 = 1 − 2sin 2 θ
3.9
Reciprocal trigonometric
identities
secθ =
1
cos θ
cosecθ =
Pythagorean identities
1
sin θ
1 + tan 2 θ =
sec 2 θ
1 + cot 2 θ =
cosec 2θ
Mathematics: analysis and approaches HL formula booklet
5
3.10
Compound angle identities
sin ( A=
± B ) sin A cos B ± cos A sin B
cos ( A ± B ) =
cos A cos B  sin A sin B
tan A ± tan B
tan ( A ± B ) =
1  tan A tan B
Double angle identity
for tan
tan 2θ =
3.12
Magnitude of a vector
3.13
Scalar product
2 tan θ
1 − tan 2 θ
 v1 
 
v + v2 + v3 , where v =  v2 
v 
 3
2
1
v =
2
2
 v1 
 w1 
 
 
v ⋅ w= v1w1 + v2 w2 + v3 w3 , where v =  v2  , w =  w2 
v 
w 
 3
 3
v⋅w =
v w cos θ , where θ is the angle between v and w
3.14
v1w1 + v2 w2 + v3 w3
v w
Angle between two
vectors
cos θ =
Vector equation of a line
r = a + λb
Parametric form of the
equation of a line
x =+
x0 λ l , y =+
y0 λ m, z =+
z0 λ n
Cartesian equations of a
line
x − x0 y − y0 z − z0
= =
l
m
n
Vector product
 w1 
 v1 
 v2 w3 − v3 w2 
 
 


v ×=
w  v3 w1 − v1w3  , where v =  v2  , w =  w2 
w 
v 
v w −v w 
 3
 3
 1 2 2 1
3.16
v×w =
v w sin θ , where θ is the angle between v and w
Area of a parallelogram
A=
v × w where v and w form two adjacent sides of a
parallelogram
3.17
Vector equation of a plane
r = a + λb + µ c
Equation of a plane
(using the normal vector)
r ⋅n =a⋅n
Cartesian equation of a
plane
ax + by + cz =
d
Mathematics: analysis and approaches HL formula booklet
6
Topic 4: Statistics and probability – HL
4.2
Interquartile range
IQR
= Q3 − Q1
4.3
k
Mean, x , of a set of data
4.5
4.6
4.8
4.12
4.13
i =1
i i
, where n =
n
k
∑f
i =1
i
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 ∩ B)
Mutually exclusive events
4.7
x=
∑fx
P ( A ∪ B )= P ( A) + P ( B)
P ( A ∩ B)
P ( B)
Conditional probability
P ( A B) =
Independent events
P ( A ∩ B) =
P ( A) P ( B)
Expected value of a
E(X )
discrete random variable X=
k
x P(X
∑=
i =1
i
xi )
Binomial distribution
X ~ B (n , p)
Mean
E ( X ) = np
Variance
Var (=
X ) np (1 − p )
Standardized normal
variable
z=
Bayes’ theorem
P ( B | A) =
P ( B) P ( A | B)
P ( B ) P ( A | B) + P ( B′) P ( A | B′)
P ( Bi | A) =
P( Bi ) P( A | Bi )
P( B1 ) P( A | B1 ) + P( B2 ) P( A | B2 ) + P( B3 ) P( A | B3 )
x−µ
Mathematics: analysis and approaches HL formula booklet
σ
7
4.14
k
Variance σ 2
k
2
i
i
2
=
i 1 =i 1
=
σ
∑ f (x
− µ)
=
n
k
∑ f (x
i
i
− µ)
∑fx
σ=
Linear transformation of a
single random variable
E ( aX +=
b ) aE ( X ) + b
Expected value of a
continuous random
variable X
n
2
− µ2
2
Standard deviation σ
i =1
i i
n
Var ( aX + b ) =
a 2 Var ( X )
E(X =
) µ=
∫
∞
−∞
x f ( x) dx
Variance
Var ( X ) = E ( X − µ ) 2  = E ( X 2 ) − [ E (X ) ]
Variance of a discrete
random variable X
Var ( X ) =
x) =
x) − µ 2
∑ ( x − µ )2 P ( X =
∑ x2 P ( X =
Variance of a continuous
random variable X
2
∞
∞
−∞
−∞
2
2
2
Var ( X ) =
∫ ( x − µ ) f ( x ) dx =
∫ x f ( x) dx − µ
Mathematics: analysis and approaches HL formula booklet
8
Topic 5: Calculus – HL
5.12
Derivative of f ( x) from
first principles
y = f ( x) ⇒
5.3
Derivative of x n
f ( x) =
x n ⇒ f ′( x) =
nx n −1
5.6
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 e x
e x ⇒ f ′( x) =
ex
f ( x) =
Derivative of ln x
1
f ( x) =
ln x ⇒ f ′( x) =
x
Chain rule
y = g (u ) , where u = f ( x) ⇒
Product rule
y =uv ⇒
Quotient rule
du
dv
v −u
u
dy
d
x
dx
y= ⇒
=
2
v
dx
v
5.15
Standard derivatives
dy
 f ( x + h) − f ( x ) 
= f ′( x)= lim 

h →0
dx
h


dy dy du
= ×
dx du dx
dy
dv
du
=u + v
dx
dx
dx
tan x
f ( x) =tan x ⇒ f ′( x) =sec 2 x
sec x
f ( x) =sec x ⇒ f ′( x) =sec x tan x
cosec x
f ( x) =
cosec x ⇒ f ′( x) =
−cosec x cot x
cot x
f ( x) =⇒
cot x
f ′( x) =
−cosec 2 x
ax
f ( x) =
a x ⇒ f ′( x) =
a x (ln a )
log a x
f ( x) = log a x ⇒ f ′( x) =
arcsin x
f ( x)= arcsin x ⇒ f ′( x)=
arccos x
1
f ( x) =
arccos x ⇒ f ′( x) =
−
1 − x2
arctan x
f ( x)= arctan x ⇒ f ′( x)=
Mathematics: analysis and approaches HL formula booklet
1
x ln a
1
1 − x2
1
1 + x2
9
5.9
5.5
dv d 2 s
=
dt dt 2
Acceleration
=
a
Distance travelled from
t1 to t 2
distance =
Displacement from
t1 to t 2
displacement =
Integral of x n
Area between a curve
y = f ( x) and the x-axis,
∫
t2
t1
v(t ) dt
∫
t2
t1
v (t )dt
x n +1
+ C , n ≠ −1
n +1
n
dx
∫x=
b
A = ∫ y dx
a
where f ( x) > 0
5.10
Standard integrals
1
dx
∫ x=
ln x + C
− cos x + C
∫ sin x dx =
dx
∫ cos x=
∫e
5.15
Standard integrals
x
sin x + C
d=
x ex + C
a dx
∫=
x
1 x
a +C
ln a
1
1
 x
=
∫ a 2 + x 2 dx a arctan  a  + C
∫
5.16
Integration by parts
1
 x
d=
x arcsin   + C ,
a
a −x
2
dv
∫ u dx d=x
Mathematics: analysis and approaches HL formula booklet
2
uv − ∫ v
x <a
du
dx or ∫ u d=
v uv − ∫ v du
dx
10
5.11
5.17
5.18
b
Area of region enclosed
by a curve and x-axis
A = ∫ y dx
Area of region enclosed
by a curve and y-axis
A = ∫ x dy
a
b
a
b
b
a
a
Volume of revolution
about the x or y-axes
V = ∫ πy 2 dx or V = ∫ πx 2 dy
Euler’s method
yn +=
yn + h × f ( xn , yn ) ; xn +=
xn + h , where h is a constant
1
1
(step length)
Integrating factor for
e∫
Maclaurin series
f ( x) =f (0) + x f ′(0) +
Maclaurin series for
special functions
e x =1 + x +
y ′ + P ( x) y =
Q ( x)
5.19
P ( x )dx
x2
+ ...
2!
ln (1 + x) =x −
x 2 x3
+ − ...
2
3
sin x =x −
x3 x5
+ − ...
3! 5!
cos x =−
1
x2 x4
+ − ...
2! 4!
arctan x =x −
Mathematics: analysis and approaches HL formula booklet
x2
f ′′(0) + 
2!
x3 x5
+ − ...
3 5
11