Ib Matematics Sl Formula Booklet 1e5li

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 Area of a parallelogram Area of a triangle

Area of a trapezium

A buh

A

1 (b u h) 2

A

1 ( a b) h 2

Area of a circle

A Sr 2

Circumference of a circle

C

2Sr

V

1 (area of base u vertical height) 3

Volume of a cuboid (rectangular prism)

V

l u wu h

Volume of a cylinder

V

Sr 2 h

Area of the curved surface of a cylinder

A 2Srh

Volume of a pyramid

Volume of a sphere

Volume of a cone

V

4 3 Sr 3

V

1 2 Sr 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 ¹ © 2

Mathematics SL formula booklet

( x1 x2 ) 2 ( y1 y2 ) 2 ( z1 z2 ) 2

2

Topics

Topic 1—Algebra 1.1

1.2

The nth term of an arithmetic sequence

un

u1 (n 1)d

The sum of n of an arithmetic sequence

Sn

n (2u1 (n 1)d ) 2

The nth term of a geometric sequence

un

u1r n 1

The sum of n of a finite geometric sequence

Sn

u1 (r n 1) r 1

The sum of an infinite geometric sequence

Sf

u1 , r 1 1 r

Exponents and logarithms

ax

b œ

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

Binomial coefficient

Binomial theorem

Mathematics SL formula booklet

u1 (1 r n ) , r z1 1 r

x log a b

log c a log c b

log b a

§n· ¨¨ ¸¸ ©r¹

n (u1 un ) 2

n! r ! n r !

( a b) n

§ n· § n· a n ¨ ¸ a n 1b ! ¨ ¸ a n r b r ! b n ©1¹ ©r¹

3

Topic 2—Functions and equations 2.4

Axis of symmetry of graph of a quadratic function

f ( x) ax 2 bx c Ÿ axis of symmetry x

2.6

Relationships between logarithmic and exponential functions

a x e x ln a log a a x x

Solutions of a quadratic equation

ax 2 bx c 0 Ÿ

Discriminant

' b 2 4ac

2.7



b 2a

a loga x

x

b r b 2 4ac , az0 2a

Topic 3—Circular functions and trigonometry 3.1

Length of an arc Area of a sector

l Tr A

1 2 Tr 2

sin T cos T

3.2

Trigonometric identity

tan T

3.3

Pythagorean identity

cos 2 T sin 2 T

Double angle formulae

sin 2T

2sin T cos T

cos 2T

cos 2 T sin 2 T

3.6

Cosine rule

Sine rule

Area of a triangle

Mathematics SL formula booklet

c2

2cos 2 T 1 1 2 sin 2 T

a 2 b 2 2ab cos C ; cos C

a sin A A

1

b sin B

a 2 b2 c2 2ab

c sin C

1 ab sin C 2

4

Topic 4—Vectors 4.1

Magnitude of a vector

4.2

Scalar product

4.3

v12 v2 2 v32

v v˜w

v w cos T

v˜w

v1w1 v2 w2 v3 w3

v˜w v w

Angle between two vectors

cos T

Vector equation of a line

r = a + tb

Topic 5—Statistics and probability 5.2

Mean of a set of data

n

¦fx

i i

x

i 1 n

¦f

i

i 1

5.5

5.6

5.7

5.8

5.9

Probability of an event A

n( A) n(U )

P( A)

Complementary events

P( A) P( Ac) 1

Combined events

P( A ‰ B ) P( A) P( B) P( A ˆ B)

Mutually exclusive events

P( A ‰ B ) P( A) P( B)

Conditional probability

P(A ˆ B ) P(A) P(B | A)

Independent events

P( A ˆ B ) P( A) P( B )

Expected value of a discrete E( X ) random variable X Binomial distribution

P

¦ x P( X

X ~ B(n, p ) Ÿ P( X

Mean

E( X ) np

Variance

Var( X ) np (1 p )

Standardized normal variable

z

Mathematics SL formula booklet

x)

x

§n· r ) ¨ ¸ p r (1 p) n r , r ©r¹

0,1, !, n

xP

V

5

Topic 6—Calculus § f ( x h) f ( x ) · f c( x) lim ¨ ¸ h o0 h © ¹

6.1

Derivative of f ( x)

6.2

Derivative of x n

f ( x)

Derivative of sin x

f ( x) sin x Ÿ

f c( x) cos x

Derivative of cos x

f ( x) cos x Ÿ

f c( x) sin x

Derivative of tan x

dy dx

f ( x) Ÿ

y

f c( x) nx n 1

Ÿ

xn

f c( x) e x

Ÿ

Derivative of e x

f ( x) e x

Derivative of ln x

f ( x) ln x Ÿ

Chain rule

Product rule

dy dx

y uv Ÿ

Quotient rule u v

y

6.4

1

6.6

sin x C

dx e x C

³

b

Volume of revolution V about the x-axis from x = a to x = b

³

b

Mathematics SL formula booklet

ln x C , x ! 0

³ cos x dx x

d y du u du d x

du dv u dx dx v2

cos x C

A

Total distance travelled from t1 to t 2

v

dy dx

dv du v dx dx

³ sin x dx

³e Area under a curve between x = a and x = b

u

1 x

x n 1 C , n z 1 n 1

n ³ x dx

³ x dx

6.5

dy dx

Ÿ

Standard integrals

f c( x)

f ( x) Ÿ

g (u ) , u

y

1 cos 2 x

f c( x)

tan x Ÿ

f ( x)

a

a

y dx ʌy 2 Gx

distance

³

t2

t1

v(t ) dt

6

Mathematics SL formula booklet

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