Diploma Programme
Mathematics: analysis and approaches
formula booklet
For use during the course and in the examinations
First examinations 2021
Version 1.3
© International Baccalaureate Organization 2019
Contents
Prior learning
SL and HL
2
Topic 1: Number and algebra
SL and HL
3
HL only
4
Topic 2: Functions
SL and HL
5
HL only
5
Topic 3: Geometry and trigonometry
SL and HL
6
HL only
7
Topic 4: Statistics and probability
SL and HL
HL only
9
10
Topic 5: Calculus
SL and HL
11
HL only
12
Prior learning – SL and 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
1
A = (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 = ( x1 − x2 ) 2 + ( y1 − y2 ) 2
Coordinates of the midpoint of
a line segment with endpoints
( x1 , y1 ) and ( x2 , y2 )
x1 + x2 y1 + y2
,
2
2
Mathematics: analysis and approaches formula booklet
2
Topic 1: Number and algebra – SL and HL
SL
1.2
SL
1.3
SL
1.4
The nth term of an
arithmetic sequence
un = u1 + (n − 1) d
The sum of n terms of an
arithmetic sequence
Sn =
The nth term of a
geometric sequence
un = u1r n −1
The sum of n terms of a
finite geometric sequence
Sn =
Compound interest
FV = PV × 1 +
n
n
( 2u1 + (n − 1) d ) ; Sn = (u1 + un )
2
2
u1 (r n − 1) u1 (1 − r n )
, r ≠1
=
r −1
1− r
r
100k
kn
, where FV is the future value,
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
r%=r/100
SL
1.5
Exponents and logarithms
a x = b ⇔ x = log a b , where a > 0, b > 0, a ≠ 1
SL
1.7
Exponents and logarithms
log a xy = log a x + log a y
x
log a = log a x − log a y
y
log a x m = m log a x
log a x =
log b x
log b a
SL
1.8
The sum of an infinite
geometric sequence
S∞ =
SL
1.9
Binomial theorem n ∈
(a + b) n = a n + n C a n −1b +
1
u1
, r <1
1− r
+ nC a n−r br +
r
+ bn
n!
nC =
r r !(n − r )!
Mathematics: analysis and approaches formula booklet
3
Topic 1: Number and algebra – HL only
AHL
1.10
Combinations
n!
nC =
r r !(n − r )!
Permutations
n P = n!
r (n − r )!
Extension of binomial
theorem, n ∈
(a + b)
n
= an 1 + n
n ( n − 1) b
b
+
a
2!
a
AHL
1.12
Complex numbers
z = a + bi
AHL
1.13
Modulus-argument (polar)
and exponential (Euler)
form
z = r (cos θ + isin θ ) = reiθ = r cis θ
AHL
1.14
De Moivre’s theorem
[ r (cosθ + isin θ )]
Mathematics: analysis and approaches formula booklet
n
2
+ ...
= r n (cos nθ + isin nθ ) = r n einθ = r n cis nθ
4
Topic 2: Functions – SL and HL
SL
2.1
SL
2.6
SL
2.7
SL
2.9
Equations of a straight line
y = mx + c ; ax + by + d = 0 ; y − y1 = m ( x − 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
Exponential and
logarithmic functions
a x = e x ln a ; log a a x = x = a loga x where a , x > 0, a ≠ 1
y2 − y1
x2 − x1
x=
b
2a
−b ± b 2 − 4ac
, a≠0
2a
Topic 2: Functions – HL only
AHL
2.12
Sum and product of the
roots of polynomial
equations of the form
n
∑a x
r =0
r
r
( −1) a0
− an −1
; product is
Sum is
an
an
n
=0
Mathematics: analysis and approaches formula booklet
5
Topic 3: Geometry and trigonometry – SL and HL
SL
3.1
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 )
x1 + x2 y1 + y2 z1 + z2
,
,
2
2
2
and ( x2 , y2 , z2 )
V=
Volume of a right cone
1
V = πr 2 h , where r is the radius, h is the height
3
Area of the curved surface
of a cone
SL
3.2
SL
3.4
1
Ah , where A is the area of the base, h is the height
3
Volume of a right-pyramid
A = πrl , where r is the radius, l is the slant height
Volume of a sphere
4
V = πr 3 , where r is the radius
3
Surface area of a sphere
A = 4 r 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
a 2 + b2 − c2
2ab
radians
Mathematics: analysis and approaches formula booklet
6
sin θ
cos θ
SL
3.5
Identity for tan θ
tan θ =
SL
3.6
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
Topic 3: Geometry and trigonometry – HL only
AHL
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
AHL
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 ± B) =
Double angle identity
for tan
AHL
3.12
Magnitude of a vector
tan 2θ =
tan A ± tan B
1 tan A tan B
2 tan θ
1 − tan 2 θ
v1
v = v + v2 + v3 , where v = v2
v3
Mathematics: analysis and approaches formula booklet
2
1
2
2
7
AHL
3.13
Scalar product
v1
w1
v ⋅ w = v1w1 + v2 w2 + v3 w3 , where v = v2 , w = w2
v3
w3
v ⋅ w = v w cos θ , where θ is the angle between v and w
AHL
3.14
AHL
3.16
v1w1 + v2 w2 + v3 w3
v w
Angle between two
vectors
cos θ =
Vector equation of a line
r=a+ b
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
v2 w3 − v3 w2
v1
w1
v × w = v3 w1 − v1w3 , where v = v2 , w = w2
v1w2 − v2 w1
w3
v3
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
AHL
3.17
Vector equation of a plane
r = a + b + µc
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 formula booklet
8
Topic 4: Statistics and probability – SL and HL
SL
4.2
SL
4.3
SL
4.5
SL
4.6
Interquartile range
IQR = Q3 − Q1
k
Mean, x , of a set of data
x=
∑fx
i i
i =1
, where n =
n
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
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)
SL
4.7
Expected value of a
discrete random variable X
E ( X ) = ∑ x P ( X = x)
SL
4.8
Binomial distribution
SL
4.12
k
∑f
X ~ B (n , p)
Mean
E ( X ) = np
Variance
Var ( X ) = np (1 − p )
Standardized normal
variable
z=
x−µ
Mathematics: analysis and approaches formula booklet
σ
9
Topic 4: Statistics and probability – HL only
AHL
4.13
AHL
4.14
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 )
k
Variance σ 2
σ2 =
∑ f (x
i =1
i
− µ)
i
k
Standard deviation σ
Linear transformation of a
single random variable
σ=
=
n
∑ f (x
i =1
i
i
k
2
− µ)
∑fx
i i
i =1
n
2
− µ2
2
n
E ( aX + b ) = aE ( X ) + b
Var ( aX + b ) = a 2 Var ( X )
Expected value of a
continuous random
variable 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 − µ ) 2 P ( X = x) = ∑ x 2 P ( X = x) − µ 2
Variance of a continuous
random variable X
Var ( X ) = ∫ ( x − µ ) 2 f ( x) dx = ∫ x 2 f ( x) dx − µ 2
∞
−∞
2
Mathematics: analysis and approaches formula booklet
∞
∞
−∞
−∞
10
Topic 5: Calculus – SL and HL
SL
5.3
SL
5.5
Derivative of x n
f ( x) = x n ⇒ f ′( x) = nx n −1
Integral of x n
n
∫ x dx =
Area between a curve
y = f ( x) and the x-axis,
where f ( x) > 0
SL
5.6
SL
5.9
x n +1
+ C , n ≠ −1
n +1
b
A = ∫ y dx
a
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
f ( x) = e x ⇒ f ′( x) = e x
Derivative of ln x
f ( x) = ln x ⇒ f ′( x) =
Chain rule
y = g (u ) , where u = f ( x) ⇒
Product rule
y = uv ⇒
Quotient rule
du
dv
v −u
u
dy
y= ⇒
= dx 2 dx
dx
v
v
Acceleration
a=
Distance travelled from
t1 to t 2
distance =
Displacement from
t1 to t 2
displacement =
1
x
dy dy du
=
×
dx du dx
dy
dv
du
=u +v
dx
dx
dx
dv d 2 s
=
dt dt 2
Mathematics: analysis and approaches formula booklet
∫
t2
t1
v(t ) dt
∫
t2
t1
v (t )dt
11
SL
5.10
Standard integrals
1
∫ x dx = ln
x +C
∫ sin x dx = − cos x + C
∫ cos x dx = sin x + C
∫e
SL
5.11
Area of region enclosed
by a curve and x-axis
x
dx = e x + C
b
A = ∫ y dx
a
△y = change in y
△x = change in x = h
Topic 5: Calculus – HL only
AHL
5.12
Derivative of f ( x) from
first principles
AHL
5.15
Standard derivatives
y = f ( x) ⇒
f ( x + h) − f ( x )
dy
= f ′( x) = lim
h →0
h
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
f ( x) = arccos x ⇒ f ′( x) = −
arctan x
f ( x) = arctan x ⇒ f ′( x) =
Mathematics: analysis and approaches formula booklet
1
x ln a
1
1 − x2
1
1 − x2
1
1 + x2
12
AHL
5.15
Standard integrals
∫a
x
∫a
∫
AHL
5.16
AHL
5.17
Integration by parts
Area of region enclosed
by a curve and y-axis
Volume of revolution
about the x or y-axes
AHL
5.18
Euler’s method
1 x
a +C
ln a
1
1
x
dx = arctan
+C
2
a
a
+x
1
a −x
2
x
+C,
a
dx = arcsin
2
dv
x <a
du
∫ u dx dx = uv − ∫ v dx dx or ∫ u dv = uv − ∫ v du
b
A = ∫ x dy
a
V =∫
b
a
y 2 x or V = ∫
b
a
x2 y
yn +1 = yn + h × f ( xn , yn ) ; xn +1 = xn + h , where h is a constant
(step length)
Integrating factor for
e∫
Maclaurin series
f ( x) = f (0) + x f ′(0) +
Maclaurin series for
special functions
ex = 1 + x +
y ′ + P ( x) y = Q ( x)
AHL
5.19
2
dx =
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 formula booklet
x2
f ′′(0) +
2!
x3 x5
+ − ...
3 5
13