Math HL Formulae IB PDF
Math HL Formulae IB PDF
Math HL Formulae IB PDF
Mathematics HL and
further mathematics HL
formula booklet
For use during the course and in the examinations
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
Formulae
Prior learning
Area of a triangle 1
A (b h) , where b is the base, h is the height
2
Area of a trapezium 1
A (a b) h , where a and b are the parallel sides, h is the height
2
Volume of a pyramid 1
V (area of base vertical height)
3
Area of the curved surface of A 2rh , where r is the radius, h is the height
a cylinder
Volume of a sphere 4
V r 3 , where r is the radius
3
Volume of a cone 1
V r 2 h , where r is the radius, h is the height
3
Topic 1: Algebra
1.1 The nth term of an un u1 (n 1)d
arithmetic sequence
a x e x ln a
loga a x x aloga x
log c a
logb a
log c b
Permutations n P n!
r (n r )!
Area of a sector 1
A r 2 , where is the angle measured in radians, r is the
2
radius
tan A tan B
tan( A B)
1 tan A tan B
Sine rule a b c
sin A sin B sin C
Area of a triangle 1
A ab sin C
2
Topic 4: Vectors
4.1 Magnitude of a vector v1
v v v2 v3 , where v v2
1
2 2 2
v
3
Coordinates of the x1 x2 y1 y2 z1 z2
midpoint of a line segment , ,
2 2 2
with endpoints ( x1 , y1 , z1 ) ,
( x2 , y2 , z2 )
v1 w1
v w v1w1 v2 w2 v3 w3 , where v v2 , w w2
v w
3 3
Cartesian equations of a x x0 y y0 z z0
line
l m n
4.5 Vector product v2 w3 v3 w2 v1 w1
v w v3 w1 v1w3 where v v2 , w w2
v w v w v w
1 2 2 1 3 3
v w v w sin , where is the angle between v and w
Area of a triangle 1
A v w where v and w form two sides of a triangle
2
Cartesian equation of a ax by cz d
plane
Mean k
fx i i
i 1
Variance 2 k k
f x fx
2 2
i i i i
2 i 1
i 1
2
n n
Standard deviation k
f x
2
i i
i 1
Independent events P( A B) P( A) P( B)
P( Bi ) P( A Bi )
P( Bi | A)
P( B1 ) P( A | B1 ) P( B2 ) P( A | B2 ) P( B3 ) P( A | B3 )
Expected value of a
E( X ) x f ( x)dx
continuous random
variable X
Variance of a continuous
Var( X ) ( x )2 f ( x)dx x 2 f ( x)dx 2
random variable X
Poisson distribution m x e m
X ~ Po(m) P( X x) , x 0,1, 2,
Mean x!
Variance E( X ) m
Var( X ) m
Derivative of e x f ( x) e x f ( x) e x
Derivative of ln x 1
f ( x) ln x f ( x)
x
Derivative of sec x f ( x) sec x f ( x) sec x tan x
Derivative of a x f ( x) a x f ( x) a x (ln a)
Derivative of log a x 1
f ( x) log a x f ( x)
x ln a
Derivative of arcsin x 1
f ( x) arcsin x f ( x)
1 x2
Derivative of arccos x 1
f ( x) arccos x f ( x)
1 x2
Derivative of arctan x 1
f ( x) arctan x f ( x)
1 x2
Chain rule dy dy du
y g (u) , where u f ( x)
dx du dx
Product rule dy dv du
y uv u v
dx dx dx
Quotient rule du dv
v u
u dy dx dx
y 2
v dx v
6.4 Standard integrals x n 1
x dx C , n 1
n
n 1
1
x dx ln x C
sin x dx cos x C
cos x dx sin x C
e dx e x C
x
1 x
a dx a C
x
ln a
1 1 x
a 2
x2
dx arctan C
a a
1 x
a x
2
dx arcsin C , x a
2
a
Volume of revolution
(rotation) b b
V πy 2 dx or V πx 2 dy
a a
Variance sn2 k k
f (x x ) f x
i i
2
i i
2
sn2 i 1
i 1
x2
n n
Standard deviation sn k
f (x x )i i
2
sn i 1
Unbiased estimate of k k
n 2
sn21 sn i 1
i 1
x
n 1 n 1 n 1 n 1
(3.7)
correlation coefficient x y i i nx y
r i 1
n 2 2
n
2
i yi n y
2
x nx
i 1 i 1
i 1
i 1
Topic 9: Calculus
Further mathematics HL topic 5
e
Integrating factor for P ( x )dx
y P( x) y Q( x)
9.6 Maclaurin series x2
(5.6) f ( x) f (0) x f (0) f (0)
2!
10.7 Euler’s formula for v e f 2 , where v is the number of vertices, e is the number
(6.7) connected planar graphs of edges, f is the number of faces
Discrete distributions
Distribution Notation Probability mass Mean Variance
function
Negative binomial X ~ NB (r , p) x 1 r x r r rq
p q p2
r 1 p
for x r , r 1,...
Continuous distributions
Distribution Notation Probability Mean Variance
density function
Normal X ~ N ( , 2 ) 1 x
2 2
1
2
e
2π
Further mathematics
Inverse of a 2 2 matrix a b 1 1 d b
A A , ad bc
c d det A c a
Determinant of a 3 3 a b c
matrix e f d f d e
A d e f det A a b c
g h h k g k g h
k