Mathematics: analysis and approaches formula booklet

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Diploma Programme

Mathematics: analysis and approaches formula booklet

For use during the course and in the examinations First examinations 2021

Version 1.1

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Prior learning

SL and HL 2

Topic 1: Number and algebra

SL and HL 3

HL only 4

Topic 2: Functions

SL and HL 5

HL only 5

Topic 3: Geometry and trigonometry

SL and HL 6

HL only 7

Topic 4: Statistics and probability

SL and HL 9

HL only 10

Topic 5: Calculus

SL and HL 11

HL only 12

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Prior learning – SL and HL

Area of a parallelogram A bh = , where b is the base, h is the height

Area of a triangle 1 ( )

A = 2 bh , where b is the base, h is the height

Area of a trapezoid 1 ( )

A = 2 a b h + , where a and b are the parallel sides, h is the height Area of a circle A = π r

²

, where r is the radius

Circumference of a circle C = π 2 r , where r is the radius

Volume of a cuboid V lwh = , where l is the length, w is the width, h is the height Volume of a cylinder V = π r h

²

, where r is the radius, h is the height

Volume of a prism V Ah = , where A is the area of cross-section, h is the height Area of the curved surface of

a cylinder A = π 2 rh , where r is the radius, h is the height Distance between two

points ( , ) x y

₁ ₁

and ( , ) x y

₂ ₂

d = ( x x

¹

−

²

)

²

+ ( y y

¹

−

²

)

²

Coordinates of the midpoint of

a line segment with endpoints

1 1

( , ) x y and ( , ) x y

₂ ₂

1 2

,

1 2

2 2

x x y + + y

 

 

 

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Topic 1: Number and algebra – SL and HL

SL 1.2 The n th term of an

arithmetic sequence u

_n

= + u

¹

( n − 1) d The sum of n terms of an

arithmetic sequence ( 2

1

( 1) ; ) (

1

)

2 2

n

S = u + n − d S = u u +

SL 1.3 The n th term of a geometric sequence

1 n1

u

n

= u r

⁻

The sum of n terms of a

finite geometric sequence

¹

( 1)

¹

(1 )

1 1

n n

n

u r u r

S r r

− −

= =

− − , r ≠ 1

SL 1.4 Compound interest 1

100 r

k n

FV PV

k

 

= × +     ^{, where} FV is the future value, PV is the present value, n is the number of years, k is the number of compounding periods per year, r% is the nominal annual rate of interest

SL 1.5 Exponents and logarithms a

^x

= ⇔ b x = log

_a

b , where a > 0, b > 0, a ≠ 1 SL 1.7 Exponents and logarithms log

_a

xy = log

_a

x + log

_a

y

log

_a

x log

_a

x log

_a

y

y = −

log

_a

x

^m

= m log

_a

x log log

log

^b

a

b

x x

= a SL 1.8 The sum of an infinite

geometric sequence

¹

, 1

1 S u r

∞

= r <

−

SL 1.9 Binomial theorem ( a b + )

ⁿ

= a

ⁿ

+ n C 1 a b

ⁿ⁻1

+ +  n C r a b

^{n r r}⁻

+ +  b

ⁿ

C !

!( )!

n n

r = r n r

−

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Topic 1: Number and algebra – HL only

AHL 1.10 Combinations C !

!( )!

n n

r = r n r

−

Permutations P !

( )!

n n

r = n r

− AHL 1.12 Complex numbers z a b = + i AHL 1.13 Modulus-argument (polar)

and exponential (Euler) form

(cos isin ) e

i

cis z r = θ + θ = r

^θ

= r θ

AHL 1.14 De Moivre’s theorem [ ^r ^(cos ^θ ⁺ ^{isin )} ^θ ]

ⁿ

⁼ ^r

ⁿ

^(cos ⁿ ^θ ⁺ ^{isin )} ⁿ ^θ ⁼ ^r

^{n n}

^e

ⁱ^θ

⁼ ^r

ⁿ

^cis ⁿ ^θ

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Topic 2: Functions – SL and HL

SL 2.1 Equations of a straight line y mx c = + ; ax by d + + = 0 ; y y m x x −

1

= ( −

1

)

Gradient formula

² ¹

2 1

= −

− y y

m x x

SL 2.6 Axis of symmetry of the graph of a quadratic function

( )

2

2 f x ax bx c x b

= + + ⇒ axis of symmetry is = − a SL 2.7 Solutions of a quadratic

equation Discriminant

2

0

2

4 , 0

2 b b ac

ax bx c x a

a

− ± −

+ + = ⇒ = ≠

2

4 b ac

∆ = − SL 2.9 Exponential and

logarithmic functions

e

ln

x x a

a = ; log

_a

a

^x

= = x a

^log^a^x

where a x , > 0, a ≠ 1

Topic 2: Functions – HL only

AHL 2.12 Sum and product of the roots of polynomial equations of the form

0 n

0

r r

r

a x

=

∑ =

Sum is

ⁿ ¹

n

a a

⁻

− ; product is ( ) 1

ⁿ ₀

n

a a

−

(7)

Topic 3: Geometry and trigonometry – SL and HL

SL 3.1 Distance between two points ( , , ) x y z

₁ ₁ ₁

and

2 2 2

( , , ) x y z

2 2 2

1 2 1 2 1 2

( ) ( ) ( )

= − + − + −

d x x y y z z

Coordinates of the

midpoint of a line segment with endpoints ( , , ) x y z

₁ ₁ ₁

and ( , , ) x y z

₂ ₂ ₂

1 2

, ,

1 2 1 2

2 2 2

+ + +

 

 

 

x x y y z z

Volume of a right-pyramid 1

V = 3 Ah , where A is the area of the base, h is the height

Volume of a right cone 1

²

V = π 3 r h , where r is the radius, h is the height Area of the curved surface

of a cone A = π rl , where r is the radius, l is the slant height

Volume of a sphere 4

³

V = π 3 r , where r is the radius

Surface area of a sphere A = 4π r

²

, where r is the radius SL 3.2 Sine rule

sin sin sin

a b c

A = B = C

Cosine rule c

²

= a

²

+ b

²

− 2 cos ab C ; cos

² ² ²

2 a b c

C ab

+ −

=

Area of a triangle 1 sin

A = 2 ab C

SL 3.4 Length of an arc l r = θ , where r is the radius, θ is the angle measured in radians

(8)

SL 3.5 Identity for tan θ _tan ^sin cos θ θ

= θ

SL 3.6 Pythagorean identity cos

²

θ + sin

²

θ = 1

Double angle identities sin 2 θ = 2sin cos θ θ

2 2 2 2

cos2 θ = cos θ − sin θ = 2cos θ − = − 1 1 2sin θ

Topic 3: Geometry and trigonometry – HL only

AHL 3.9 Reciprocal trigonometric identities

sec 1 θ cos

= θ cosec 1

θ sin

= θ

Pythagorean identities

² ²

2 2

1 tan sec 1 cot cosec

θ θ

+ =

AHL 3.10 Compound angle identities sin ( A B ± ) sin cos = A B ± cos sin A B cos( A B ± ) cos cos = A B  sin sin A B

tan tan tan ( )

1 tan tan

A B

A B A B

± = ±

 Double angle identity

for tan tan 2 2tan

²

1 tan θ θ

= θ

−

AHL 3.12 Magnitude of a vector v = v

₁²

+ v

₂²

+ v

₃²

, where

1 2

v v v

   

=  

   

v

(9)

AHL 3.13 Scalar product v w ⋅ = v w v w v w

_{1 1}

+

_{2 2}

+

_{3 3}

, where

1 2 3

v v v

   

=  

    v ,

1 2 3

w w w

 

 

=  

 

  w

cos θ

v w ⋅ = v w , where θ is the angle between v and w

Angle between two vectors

1 1 2 2 3 3

cos θ = v w v w ⁺ ⁺ v w v w

AHL 3.14 Vector equation of a line r a = + λ b Parametric form of the

equation of a line x x =

⁰

+ λ l y y , , =

⁰

+ λ m z z =

⁰

+ λ n Cartesian equations of a

line x x

⁰

y y

⁰

z z

⁰

l m n

− − −

= =

AHL 3.16 Vector product

2 3 3 2

3 1 1 3

1 2 2 1

v w v w v w v w v w v w

 − 

 

× =  − 

 − 

 

v w , where

1 2 3

v v v

   

=  

    v ,

1 2 3

w w w

 

 

=  

 

  w sin θ

× =

v w v w , where θ is the angle between v and w Area of a parallelogram A = × v w where v and w form two adjacent sides of a

parallelogram AHL 3.17 Vector equation of a plane r a = + λ b + c µ

Equation of a plane

(using the normal vector) r n a n ⋅ = ⋅ Cartesian equation of a

plane ax by cz d + + =

(10)

Topic 4: Statistics and probability – SL and HL

SL 4.2 Interquartile range IQR Q =

3

− Q

1

SL 4.3

Mean, x , of a set of data

¹

k i i i

f x x = ∑

⁼

n

, where

1 k i i

n f

=

= ∑

SL 4.5 Probability of an event A P( ) ( ) ( ) A n A

= n U Complementary events P( ) P( ) 1 A + A′ =

SL 4.6 Combined events P( A B ∪ ) P( ) P( ) P( = A + B − A B ∩ ) Mutually exclusive events P( A B ∪ ) P( ) P( ) = A + B

Conditional probability P( ) P( ) P( ) A B A B

B

= ∩

Independent events P( A B ∩ ) P( ) P( ) = A B SL 4.7 Expected value of a

discrete random variable X E( ) X = ∑ x P( X x = ) SL 4.8 Binomial distribution

~ B ( , )

X n p

Mean E( ) X = np

Variance Var( ) X = np (1 − p ) SL 4.12 Standardized normal

variable z x µ

σ

= −

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Topic 4: Statistics and probability – HL only

AHL 4.13 Bayes’ theorem P( | ) P( ) P( | )

P( ) P( | ) P( ) P( | ) B A B

B A = B A B B A B

+ ′ ′

1 1 2 2 3 3

P( )P( | ) P( | )

P( )P( | ) P( )P( | ) P( )P( | )

ⁱ ⁱ

i

B A B

B A = B A B B A B B A B

+ +

AHL 4.14

Variance σ

²

( )

² ²

2 1 1 2

k k

i i i i

i i

f x f x

n n

µ

σ

⁼ ⁼

µ

−

= ∑ = ∑ −

Standard deviation σ ( )

²

1 k

i i i

f x n

µ

σ

⁼

−

= ∑

Linear transformation of a

single random variable ( )

( )

²

E E( )

Var Var( )

aX b a X b

aX b a X

+ = +

+ = Expected value of a

continuous random variable X

E( ) X µ

^∞

x f x x ( )d

= = ∫

−∞

Variance ^{Var( ) E(} ^X ⁼ ^X ⁻ ^µ ⁾

²

⁼ ^{E( ) E( )} ^X

²

⁻ [ ^X ]

²

Variance of a discrete random variable X

2 2 2

Var( ) X = ∑ ( x − µ ) P( X x = ) = ∑ x P( X x = ) − µ

Variance of a continuous

random variable X Var( ) X

^∞

( x µ ) ( )d

²

f x x

^∞

x f x x

²

( )d µ

²

−∞ −∞

= ∫ − = ∫ −

(12)

Topic 5: Calculus – SL and HL

SL 5.3 Derivative of x

ⁿ

f x ( ) = x

ⁿ

⇒ f x ′ ( ) = nx

ⁿ⁻¹

SL 5.5 Integral of x

ⁿ

d

¹

, 1

1

n

x

n

x x C n

n

=

+

+ ≠ −

∫ + Area between a curve

y f x = ( ) and the x -axis,

where f x > ( ) 0 A = ∫

_a^b

y x d

SL 5.6 Derivative of sin x f x ( ) sin = x ⇒ f x ′ ( ) cos = x Derivative of cos x f x ( ) cos = x ⇒ f x ′ ( ) = − sin x Derivative of e

^x

f x ( ) e =

^x

⇒ f x ′ ( ) e =

^x

Derivative of ln x f x ( ) ln x f x ( ) 1

′ x

= ⇒ =

Chain rule y g u = ( ) , where ( ) d d d

d d d

y y u

u f x

x u x

= ⇒ = ×

Product rule d d d

d d d

y v u

y uv u v

x x x

= ⇒ = +

Quotient rule

₂

d d

d d d

d

u v

v u

u y x x

y v x v

= ⇒ = −

SL 5.9 Acceleration d d

²₂

d d

v s

a = t = t

Distance travelled from

t

1

to t

₂

^distance

2 1

( ) d

t

v t t

= ∫

Displacement from

t

1

to t

₂

displacement

²

1

( )d

t t

v t t

= ∫

(13)

SL 5.10 Standard integrals 1 d ln x x C

x = +

∫

sin d x x = − cos x C +

∫

cos d x x = sin x C +

∫

e d

^x

x = e

^x

+ C

∫

SL 5.11 Area of region enclosed

by a curve and x -axis A = ∫

_a^b

y x d

Topic 5: Calculus – HL only

AHL 5.12 Derivative of f x ( ) from

first principles

0

d ( ) ( )

( ) ( ) lim

d

^h

y f x h f x

y f x f x

x

^→

h

+ −

 

= ⇒ = ′ =    

AHL 5.15 Standard derivatives

tan x f x ( ) tan = x ⇒ f x ′ ( ) sec =

²

Mathematics: analysis and approaches formula booklet

Diploma Programme