Download 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
Published June 2012
© 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
Formulae
Prior learning
Area of a parallelogram
Area of a triangle
Area of a trapezium
A b h , where b is the base, h is the height
A
1
(b h) , where b is the base, h is the height
2
A
1
(a b) h , where a and b are the parallel sides, h is the height
2
Area of a circle
A
r 2 , where r is the radius
Circumference of a circle
C
2 r , where r is the radius
V
1
(area of base
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
Area of the curved surface of
a cylinder
A 2 rh , where r is the radius, h is the height
Volume of a pyramid
Volume of a sphere
Volume of a cone
Distance between two
points ( x1, y1 ) and ( x2 , y2 )
Coordinates of the midpoint of
a line segment with endpoints
( x1, y1 ) and ( x2 , y2 )
Solutions of a quadratic
equation
vertical height)
r 2h , where r is the radius, h is the height
V
4 3
r , where r is the radius
3
V
1 2
r h , where r is the radius, h is the height
3
( x1 x2 )2 ( y1
d
x1
x2
2
,
y1
y2 )2
y2
2
The solutions of ax2 bx c 0 are x
b
b2 4ac
2a
Core
Topic 1: Algebra
1.1
1.2
The nth term of an
arithmetic sequence
un
u1 (n 1)d
The sum of n terms of an
arithmetic sequence
Sn
n
(2u1 (n 1)d )
2
The nth term of a
geometric sequence
un
u1r n
The sum of n terms of a
finite geometric sequence
Sn
u1 (r n 1)
r 1
The sum of an infinite
geometric sequence
S
Exponents and logarithms
ax
b
ax
ex ln a
1
u1
1 r
u1 (1 r n )
, r 1
1 r
, r
x
log a a x
1
log a b , where a 0, b 0, a 1
a loga x
x
log c a
log c b
log b a
1.3
Combinations
n
r
n!
r !(n r )!
Permutations
nP
r
n!
(n r )!
Binomial theorem
(a b)n
1.5
Complex numbers
z
1.7
De Moivre’s theorem
r (cos
n
(u1 un )
2
a ib
an
n n1
a b
1
r (cos
isin )
isin )
n
r n (cos n
n nr r
a b
r
rei
bn
r cis
isin n ) r nein
r n cis n
Topic 2: Functions and equations
2.5
Axis of symmetry of the
graph of a quadratic
function
2.6
Discriminant
f ( x) ax2 bx c
axis of symmetry x
b
2a
b2 4ac
Topic 3: Circular functions and trigonometry
3.1
3.2
Length of an arc
l
Area of a sector
1 2
r , where
2
radius
Identities
Pythagorean identities
3.3
r , where
is the angle measured in radians, r is the radius
A
tan
sin
cos
sec
1
cos
cosec
1
sin
cos 2
sin 2
is the angle measured in radians, r is the
1
1 tan
2
2
sec
1 cot
2
csc 2
Compound angle identities sin( A B) sin Acos B cos Asin B
cos( A B) cos Acos B sin Asin B
tan( A B )
Double angle identities
tan A tan B
1 tan A tan B
sin 2
2sin cos
cos2
cos2
tan 2
2tan
1 tan 2
sin2
2cos2
1 1 2sin2
3.7
Cosine rule
Sine rule
Area of a triangle
c2
a2 b2 2ab cos C ; cos C
a
sin A
b
sin B
a2
b2 c2
2ab
c
sin C
A
1
ab sin C
2
v
2
1
v
v2
d
( x1
x2 )2
Topic 4: Vectors
4.1
Magnitude of a vector
Distance between two
points ( x1, y1, z1 ) and
( x2 , y2 , z2 )
Coordinates of the
midpoint of a line segment
with endpoints ( x1, y1, z1 ) ,
( x2 , y2 , z2 )
4.2
Scalar product
x1
x2
2
v w
2
y1
,
2
2
v3 , where v
y2 )2
( y1
y2 z1
,
v1
v2
v3
( z1
z2 )2
z2
2
v w cos , where
is the angle between v and w
v1
v2 , w
v3
v w v1w1 v2 w2 v3w3 , where v
4.3
Angle between two
vectors
cos
Vector equation of a line
r = a + λb
Parametric form of the
equation of a line
x x0
l, y
Cartesian equations of a
line
x x0
l
y
v1w1 v2 w2 v3 w3
v w
y0
m
y0
m, z z0
z z0
n
n
w1
w2
w3
4.5
Vector product
Area of a triangle
4.6
v w
v2 w3 v3 w2
v3 w1 v1w3
v1w2 v2 w1
v w
v w sin , where
A
where v
v1
v2 , w
v3
is the angle between v and w
1
v w where v and w form two sides of a triangle
2
Vector equation of a plane
r = a + λb+ c
Equation of a plane
(using the normal vector)
r n
Cartesian equation of a
plane
ax by cz d
a n
Topic 5: Statistics and probability
5.1
Population parameters
k
Let n
fi
i 1
k
Mean
f i xi
i 1
n
Variance
k
2
f i xi
2
2
i 1
k
f i xi 2
Standard deviation
2
i 1
n
k
fi xi
n
2
i 1
n
5.2
5.3
Probability of an event A
w1
w2
w3
P( A)
n( A)
n(U )
Complementary events
P( A) P( A ) 1
Combined events
P( A
B) P( A) P(B) P( A
Mutually exclusive events
P( A
B) P( A) P(B)
B)
5.4
Conditional probability
Independent events
P A B
P( A
P( A B)
P( B)
B) P( A) P(B)
Bayes’ theorem
P B| A
P( Bi | A)
5.5
P( B)P A | B
P( B1 ) P( A | B1 ) P( B2 ) P( A | B2 ) P( B3 ) P( A | B3 )
x P( X
E( X )
x)
x f ( x)dx
Variance
Var( X ) E( X
)2
Variance of a discrete
random variable X
Var( X )
(x
)2 P( X
Variance of a continuous
random variable X
Var( X )
(x
) 2 f ( x)dx
E( X 2 )
Binomial distribution
X ~ B(n, p)
P( X
Mean
E( X ) np
Variance
Var( X ) np(1 p)
Poisson distribution
X ~ Po( m)
Mean
Variance
E( X ) m
Var( X ) m
5.7
P( B )P A | B
P( Bi ) P( A Bi )
Expected value of a
E( X )
discrete random variable X
Expected value of a
continuous random
variable X
5.6
P( B)P A | B
Standardized normal
variable
z
x
P( X
x)
x)
x)
E(X )
2
x 2 P( X
x 2 f ( x)dx
2
x)
2
n x
p (1 p)n x , x 0,1,
x
mxe m
, x 0,1, 2,
x!
,n
Topic 6: Calculus
6.1
6.2
Derivative of f ( x)
y
dy
dx
f ( x)
xn
h
nx n
0
f ( x)
Derivative of sin x
f (x) sin x
f (x) cos x
Derivative of cos x
f (x) cos x
f (x)
Derivative of tan x
f ( x)
f ( x) sec 2 x
Derivative of ex
f ( x) e x
f ( x)
tan x
sin x
f ( x) e x
f ( x) ln x
1
x
f ( x)
Derivative of sec x
f (x) sec x
f (x) sec x tan x
Derivative of csc x
f (x) csc x
f (x)
csc x cot x
Derivative of cot x
f ( x) cot x
f ( x)
csc 2 x
Derivative of a x
f ( x)
Derivative of loga x
Derivative of arcsin x
Derivative of arccos x
Derivative of arctan x
Chain rule
Product rule
ax
f ( x)
f ( x) loga x
a x (ln a )
f ( x)
f ( x) arcsin x
f ( x)
f ( x) arccos x
f ( x)
f ( x) arctan x
f ( x)
y
g (u) , where u
y uv
dy
dx
Quotient rule
y
u
v
f ( x)
1
Derivative of xn
Derivative of ln x
f ( x h)
h
f ( x) lim
dy
dx
u
v
1
x ln a
f ( x)
dv
du
v
dx
dx
du
dv
u
dx
dx
v2
1
1 x2
1
1 x2
1
1 x2
dy
dx
dy du
du dx
6.4
Standard integrals
xn 1
C, n
n 1
x n dx
1
dx ln x
x
sin x dx
1
C
cos x C
cos x dx sin x C
ex dx ex
1 x
a C
ln a
a x dx
1
a2
x2
dx
1
a
6.5
Area under a curve
Volume of revolution
(rotation)
6.7
Integration by parts
A
2
x
b
a
b
V
a
u
C
2
1
x
arctan
a
a
x
a
dx arcsin
b
y dx or A
a
πy 2 dx or V
dv
dx uv
dx
v
C
C, x
a
x dy
b
a
πx 2 dy
du
dx or u dv uv
dx
v du
Options
Topic 7: Statistics and probability
Further mathematics HL topic 3
7.1
(3.1)
Probability generating
function for a discrete
random variable X
G (t )
7.2
(3.2)
Linear combinations of two
independent random
variables X1, X 2
E a1 X 1
7.3
(3.3)
Sample statistics
E (t X )
x
a2 E X 2
2
1
a2 X 2
a2 2 Var X 2
a Var X 1
f i xi
i 1
n
Variance sn2
k
k
x )2
fi ( xi
sn2
f i xi 2
i 1
x2
i 1
n
Standard deviation sn
n
k
fi ( xi
x )2
i 1
sn
7.6
(3.6)
a1E X 1
k
x
7.5
(3.5)
a2 X 2
Var a1 X 1
Mean x
Unbiased estimate of
population variance sn2 1
x )t x
P( X
n
k
n 2
sn
n 1
sn2 1
Confidence intervals
Mean, with known
variance
x
z
Mean, with unknown
variance
x
t
n
sn
1
n
Test statistics
Mean, with known
variance
z
Mean, with unknown
variance
t
x
/ n
x
sn 1 / n
k
f i ( xi
i 1
x )2
fi xi 2
i 1
n 1
n 1
n 2
x
n 1
7.7
(3.7)
Sample product moment
correlation coefficient
n
xi yi
nx y
i 1
r
n
n
xi 2
nx 2
i 1
Test statistic for H0:
ρ=0
t
r
yi 2
ny2
(y
y)
i 1
n 2
1 r2
Equation of regression line
of x on y
n
xi yi
nx y
i 1
n
x x
yi
2
ny
2
i 1
Equation of regression line
of y on x
n
y
xi yi
nx y
xi 2
nx 2
i 1
n
y
(x x )
i 1
Topic 8: Sets, relations and groups
Further mathematics HL topic 4
8.1
(4.1)
De Morgan’s laws
(A
B)
A
B
(A
B)
A
B
Topic 9: Calculus
Further mathematics HL topic 5
9.5
(5.5)
Euler’s method
Integrating factor for
y P(x) y Q(x)
yn 1 yn h f (xn , yn ) ; xn
(step length)
e
P ( x )dx
1
xn h , where h is a constant
9.6
(5.6)
Maclaurin series
Taylor series
Taylor approximations
(with error term Rn ( x) )
Lagrange form
Maclaurin series for
special functions
f (0)
f ( x)
f (a) ( x a) f (a)
f ( x)
f (a ) ( x a ) f (a ) ...
( x a)2
f (a ) ...
2!
( x a)n ( n )
f (a ) Rn ( x )
n!
f ( n 1) (c)
( x a) n 1 , where c lies between a and x
(n 1)!
Rn ( x)
ex
x f (0)
x2
f (0)
2!
f ( x)
x2
...
2!
1 x
x2
2
x3
3
ln(1 x)
x
sin x
x
x3
3!
x5
...
5!
cos x 1
x2
2!
x4
...
4!
arctan x
x
x3
3
x5
5
...
...
Topic 10: Discrete mathematics
Further mathematics HL topic 6
10.7
(6.7)
Euler’s formula for
connected planar graphs
v e f 2 , where v is the number of vertices, e is the number
of edges, f is the number of faces
Planar, simple, connected
graphs
e 3v 6 for v 3
e 2v 4 if the graph has no triangles
Formulae for distributions
Topics 5.6, 5.7, 7.1, further mathematics HL topic 3.1
Discrete distributions
Distribution
Geometric
Notation
Probability mass
function
pq x
X ~ Geo p
1
for x 1,2,...
Negative binomial
X ~ NB r , p
x 1 r x
pq
r 1
r
Mean
Variance
1
p
q
p2
r
p
rq
p2
Mean
Variance
for x r, r 1,...
Continuous distributions
Distribution
Normal
Notation
X ~N
,
Probability
density function
2
1
2π
e
1 x
2
2
2
Further mathematics
Topic 1: Linear algebra
1.2
Determinant of a 2 2
matrix
A
a b
c d
det A
A
a b
c d
A1
A
a b
d e
g h
Inverse of a 2 2 matrix
Determinant of a 3 3
matrix
c
f
k
A
ad bc
1
det A
det A a
d
c
b
, ad
a
e
f
h
k
b
bc
d
f
g
k
c
d
e
g
h