Ib Math Sl Formula Booklet.pdf 91a

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 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 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

6

Mathematics SL formula booklet

7