Download Mathematics SL formula booklet

Diploma Programme
Mathematics SL formula booklet
For use during the course and in the examinations
First examinations 2014
Published March 2012
© International Baccalaureate Organization 2012
Mathematical studies SL: Formula booklet
5045
1
Contents
Prior learning
2
Topics
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
5
Topic 6—Calculus
6
Mathematics SL formula booklet
1
Formulae
Prior learning
A= b × h
Area of a parallelogram
Area of a triangle
Area of a trapezium
=
A
1
(b × h)
2
=
A
1
( a + b) h
2
Area of a circle
A = πr 2
Circumference of a circle
C = 2πr
Volume of a pyramid
=
V
1
(area of base × vertical height)
3
Volume of a cuboid (rectangular prism)
V =l × w × h
Volume of a cylinder
V = πr 2 h
Area of the curved surface of a cylinder
A= 2πrh
Volume of a sphere
Volume of a cone
V=
4 3
πr
3
V=
1 2
πr h
3
Distance between two points ( x1 , y1 , z1 ) and
( x2 , y2 , z2 )
d=
Coordinates of the midpoint of a line segment
with endpoints ( x1 , y1 , z1 ) and ( x2 , y2 , z2 )
 x1 + x2 y1 + y2 z1 + z2 
, , 

2
2 
 2
Mathematics SL formula booklet
( x1 − x2 ) 2 + ( y1 − y2 ) 2 + ( z1 − z2 ) 2
2
Topics
Topic 1—Algebra
1.1
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 )=
(u1 + un )
2
2
The sum of n terms of a
u1 (r n − 1) u1 (1 − r n )
, r ≠1
=
S
=
n
finite geometric sequence
r −1
1− r
1.2
The sum of an infinite
geometric sequence
S∞ =
Exponents and logarithms
ax = b ⇔
Laws of logarithms
log c a + log c b =
log c ab
a
log c a − log c b =
log c
b
log c a r = r log c a
Change of base
1.3
u1
, r <1
1− r
log b a =
x = log a b
log c a
log c b
Binomial coefficient
n
n!
  =
 r  r ! (n − r )!
Binomial theorem
 n
 n
(a + b) n = a n +   a n −1b + +   a n − r b r + + b n
1
r
Mathematics SL formula booklet
3
Topic 2—Functions and equations
2.4
Axis of symmetry of
graph of a quadratic
function
b
f ( x) =
ax 2 + bx + c ⇒ axis of symmetry x =
−
2a
2.6
Relationships between
logarithmic and
exponential functions
a x = e x ln a
log a a x= x= a loga x
2.7
Solutions of a quadratic
equation
ax 2 + bx + c= 0 ⇒
Discriminant
∆= b 2 − 4ac
x=
−b ± b 2 − 4ac
, a≠0
2a
Topic 3—Circular functions and trigonometry
3.1
Length of an arc
l =θr
Area of a sector
1
A = θ r2
2
3.2
Trigonometric identity
3.3
Pythagorean identity
cos 2 θ + sin 2 θ =
1
Double angle formulae
sin 2θ = 2sin θ cos θ
tan θ =
sin θ
cos θ
cos 2θ = cos 2 θ − sin 2 θ = 2cos 2 θ − 1 = 1 − 2 sin 2 θ
3.6
Cosine rule
c 2 = a 2 + b 2 − 2ab cos C ; cos C =
Sine rule
a
b
c
= =
sin A sin B sin C
Area of a triangle
1
A = ab sin C
2
Mathematics SL formula booklet
a 2 + b2 − c2
2ab
4
Topic 4—Vectors
4.1
Magnitude of a vector
4.2
Scalar product
v12 + v2 2 + v32
v =
v⋅w =
v w cos θ
v ⋅ w= v1w1 + v2 w2 + v3 w3
4.3
Angle between two
vectors
cos θ =
Vector equation of a line
r = a + tb
v⋅w
v w
Topic 5—Statistics and probability
5.2
Mean of a set of data
n
x=
∑fx
i i
i =1
n
∑f
i =1
5.5
5.6
5.7
5.8
5.9
Probability of an event A
P( A) =
i
n( A)
n(U )
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 )
Conditional probability
P(A ∩ B ) =
P(A) P(B | A)
Independent events
P( A ∩ B ) =
P( A) P( B )
Expected value of a discrete E( X =
) µ=
random variable X
∑ x P( X=
x)
x
Binomial distribution
n r
n−r
0,1, , n
X ~ B(n, p ) ⇒ P( X =
r) =
  p (1 − p ) , r =
r
 
Mean
E( X ) = np
Variance
Var(=
X ) np (1 − p )
Standardized normal
variable
z=
Mathematics SL formula booklet
x−µ
σ
5
Topic 6—Calculus
6.1
Derivative of f ( x)
6.2
Derivative of x n
f ( x) =
xn ⇒
Derivative of sin x
f ( x) =sin x ⇒
f ′( x) =cos x
Derivative of cos x
f ( x) =⇒
cos x
f ′( x) =
− sin x
f ( x) =tan x ⇒
f ′( x) =
Derivative of tan x
Derivative of e x
Derivative of ln x
Chain rule
Product rule
Quotient rule
6.4
Standard integrals
dy
 f ( x + h) − f ( x ) 
= f ′( x) = lim 

h
→
0
dx
h


y = f ( x) ⇒
f ′( x) =
nx n −1
1
cos 2 x
f ( x) =
ex ⇒
f ′( x) =
ex
f ( x) =
ln x ⇒
1
f ′( x) =
x
y = g (u ) , u =f ( x) ⇒
dy
dv
du
=u + v
dx
dx
dx
y =uv ⇒
u
y=
v
du
dv
v −u
dy
= dx 2 dx
dx
v
⇒
n
dx
∫x=
dy dy du
= ×
dx du dx
x n +1
+ C , n ≠ −1
n +1
1
∫ x dx =ln x + C ,
x>0
− cos x + C
∫ sin x dx =
dx
∫ cos x=
∫e
6.5
Area under a curve
between x = a and x = b
x
sin x + C
d=
x ex + C
b
A = ∫ y dx
a
b
Volume of revolution
V = ∫ πy 2 dx
a
about the x-axis from x = a
to x = b
6.6
Total distance travelled
from t1 to t 2
Mathematics SL formula booklet
distance =
∫
t2
t1
v(t ) dt
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Mathematics SL formula booklet
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