Download SLAA Formula Booklet

Diploma Programme
Mathematics: analysis and approaches
SL formula booklet
For use during the course and in the examinations
First examinations 2021
Version 1.0
STANDARD LEVEL
© International Baccalaureate Organization 2023
Contents
Topic 1: Number and algebra – SL
2
Topic 2: Functions – SL
3
Topic 3: Geometry and trigonometry – SL
4
Topic 4: Statistics and probability – SL
6
Topic 5: Calculus – SL
7
Topic 1: Number and algebra – SL
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 SL formula booklet
2
Topic 2: Functions – SL
2.1
2.6
2.7
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
Mathematics: analysis and approaches SL formula booklet
x=
b
2a
−b ± b 2 − 4ac
, a≠0
2a
3
Topic 3: Geometry and trigonometry – SL
Prior learning – SL
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
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 )
Mathematics: analysis and approaches SL formula booklet
4
3.2
3.4
Volume of a right-pyramid
V=
1
Ah , where A is the area of the base, h is the height
3
Volume of a right cone
V=
1 2
πr h , where r is the radius, h is the height
3
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
Mathematics: analysis and approaches SL formula booklet
5
Topic 4: Statistics and probability – SL
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
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=
x−µ
Mathematics: analysis and approaches SL formula booklet
σ
6
Topic 5: Calculus – SL
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
f ( x) =
e x ⇒ f ′( x) =
ex
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
y= ⇒
= dx 2 dx
v
dx
v
5.9
5.5
dy
dv
du
=u + v
dx
dx
dx
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,
dy dy du
= ×
dx du dx
n
dx
∫x=
∫
t2
t1
v(t ) dt
∫
t2
t1
v (t )dt
x n +1
+ C , n ≠ −1
n +1
b
A = ∫ y dx
a
where f ( x) > 0
Mathematics: analysis and approaches SL formula booklet
7
5.10
Standard integrals
1
dx
∫ x=
ln x + C
− cos x + C
∫ sin x dx =
dx
∫ cos x=
∫e
5.11
Area of region enclosed
by a curve and x-axis
x
sin x + C
d=
x ex + C
b
A = ∫ y dx
Mathematics: analysis and approaches SL formula booklet
a
8