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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 1.0

© International Baccalaureate Organization 2019

Contents

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 HL only

9 10

Topic 5: Calculus SL and HL

11

HL only

12

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 = (bh) , where b is the base, h is the height 2

Area of a trapezoid

1 A = ( a + b) h , where a and b are the parallel sides, h is the height 2

Area of a circle

A = πr 2 , where r is the radius

Circumference of a circle

C = 2πr , 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 = πr 2h , 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 = 2πrh , where r is the radius, h is the height

Distance between two points ( x1 , y1 ) and ( x2 , y2 )

d = ( x1 − x2 ) 2 + ( y1 − y2 ) 2

Coordinates of the midpoint of a line segment with endpoints ( x1 , y1 ) and ( x2 , y2 )

 x1 + x2 y1 + y2  ,   2   2

Mathematics: analysis and approaches formula booklet

2

Topic 1: Number and algebra – SL and HL

SL 1.2

SL 1.3

The nth term of an arithmetic sequence

u n = u1 + (n − 1) d

The sum of n terms of an arithmetic sequence

Sn =

The nth term of a geometric sequence

u n = u1 r n − 1

The sum of n terms of a finite geometric sequence

Sn =

n n (2u 1 + (n − 1) d ) ; S n = (u 1 + u n ) 2 2

u1 (rn − 1) u1 (1− rn ) , r ≠1 = 1− r r −1

SL 1.4

Compound 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 x log a = log a x − log a y y

kn

r   FV = PV ×  1 +  , where FV is the future value,  100k  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

log a x m = m log a x log a x =

logb x logb a

SL 1.8

The sum of an infinite geometric sequence

S∞ =

SL 1.9

Binomial theorem

( a + b) n = a n + n C a n− 1b +… + n C a n− rb r +… + b n 1 r

u1 , r 0, a ≠ 1

Topic 2: Functions – HL only

AHL 2.12

Sum and product of the roots of polynomial equations of the form n

∑a x r =0

r

r

−an −1 ; product is Sum is an

( −1)

n

a0

an

=0

Mathematics: analysis and approaches formula booklet

5

Topic 3: Geometry and trigonometry – SL and HL

SL 3.1

Distance between two points ( x1 , y1 , z1 ) and

d = ( x1 − x2 ) 2 + ( y1 − y2 ) 2 + ( z1 − z 2 ) 2

( x2 , y2 , z2 ) Coordinates of the midpoint of a line segment with endpoints ( x1 , y1 , z1 )

 x1 + x2 y1 + y2 z1 + z2  , ,   2 2   2

and ( x2 , y2 , z2 )

V=

Volume of a right cone

1 V = πr 2h , where r is the radius, h is the height 3

Area of the curved surface of a cone

SL 3.2

SL 3.4

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

Volume of a right-pyramid

A = πrl , where r is the radius, l is the slant height

Volume of a sphere

4 V = πr 3 , where r is the radius 3

Surface area of a sphere

A = 4πr 2 , where r is the radius

Sine rule

a b c = = sin A sin B sin C

Cosine rule

c 2 = a 2 + b 2 − 2 ab cos C ; cos C =

Area of a triangle

A=

Length of an arc

l = rθ , where r is the radius, θ is the angle measured in radians

Area of a sector

1 A = r 2 θ , where r is the radius, θ is the angle measured in 2

a2 + b 2 − c 2 2ab

1 ab sin C 2

radians

Mathematics: analysis and approaches formula booklet

6

sinθ cosθ

SL 3.5

Identity for tanθ

tan θ =

SL 3.6

Pythagorean identity

cos2 θ + sin 2 θ = 1

Double angle identities

sin 2θ = 2sin θ cos θ

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

Topic 3: Geometry and trigonometry – HL only

AHL 3.9

Reciprocal trigonometric identities

secθ =

1 cosθ

cosecθ = Pythagorean identities

1 sinθ

1 + tan 2 θ = sec 2 θ 1 + cot2 θ = cosec2 θ

AHL 3.10

Compound angle identities

sin ( A ± B ) = sin A cos B ± cos A sin B cos( A ± B ) = cos A cos B ∓ sin A sin B tan ( A ± B) =

Double angle identity for tan AHL 3.12

Magnitude of a vector

tan 2θ =

tan A ± tan B 1 ∓ tan A tan B

2 tan θ 1 − tan 2 θ

 v1  v = v12 + v22 + v32 , where v =  v2  v   3

Mathematics: analysis and approaches formula booklet

7

AHL 3.13

Scalar product

 v1   w1      v ⋅ w = v1 w1 + v2 w2 + v3 w3 , where v =  v2  , w =  w 2  v  w   3  3 v ⋅ w = v w cos θ , where θ is the angle between v and w

AHL 3.14

AHL 3.16

v1w1 +v 2w2 +v 3w 3 v w

Angle between two vectors

cos θ =

Vector equation of a line

r = a + λb

Parametric form of the equation of a line

x = x0 + λl , y = y0 + λm, z = z0 + λn

Cartesian equations of a line

x − x0 y − y0 z − z 0 = = l m n

Vector product

 v2 w3 − v 3w2   v1   w1      v × w =  v3 w1 − v1 w3  , where v =  v2  , w = w2  v w −v w  v  w   1 2 2 1  3  3 v × w = v w sin θ , 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 + µc

Equation of a plane (using the normal vector)

r ⋅n = a ⋅n

Cartesian equation of a plane

ax + by + cz = d

Mathematics: analysis and approaches formula booklet

8

Topic 4: Statistics and probability – SL and HL

SL 4.2

Interquartile range

SL 4.3

k

SL 4.6

∑fx

i i

Mean, x , of a set of data

SL 4.5

IQR = Q3 − Q1

x=

i =1

, where n =

n

i

i =1

n ( A) n (U )

Probability of an event A

P ( A) =

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 ) P ( A ∩ B) P ( B)

Conditional probability

P ( A B) =

Independent events

P ( A ∩ B) = P ( A) P ( B)

SL 4.7

Expected value of a discrete random variable X

E ( X ) = ∑ x P ( X = x)

SL 4.8

Binomial distribution

SL 4.12

k

∑f

X ~ B ( n , p) Mean

E ( X ) = np

Variance

Var ( X ) = np (1 − p)

Standardized normal variable

z=

Mathematics: analysis and approaches formula booklet

x− µ

σ

9

Topic 4: Statistics and probability – HL only

AHL 4.13

Bayes’ theorem

P (B | A ) =

P ( B) P ( A | B) P (B ) P (A | B ) + P (B ′) P ( A | B ′ )

P (B i | A ) = AHL 4.14

P(Bi ) P( A | Bi ) P(B1 ) P( A | B1 ) + P(B2 )P( A | B2 ) + P(B3 ) P(A | B3 )

k

Variance σ

∑ f (x i

2

σ2=

−µ)

i

Standard deviation σ

Linear transformation of a single random variable

Expected value of a continuous random variable X

σ=

=

n k

i =1

i

i

− µ)

∑fx

2

i i

i= 1

∑ f (x

k

2

i= 1

n

−µ 2

2

n

E ( aX + b ) = aE ( X ) + b Var ( aX + b ) = a 2 Var ( X )

E ( X ) = µ=

∫

∞

−∞

x f ( x ) dx

Variance

Var ( X ) = E ( X − µ )2 = E ( X 2 ) − [E (X ) ]

Variance of a discrete random variable X

Var ( X ) = ∑ ( x − µ )2 P ( X = x) = ∑ x2 P ( X = x) − µ2

Variance of a continuous random variable X

Var ( X ) = ∫ ( x − µ) 2 f ( x) dx = ∫ x2 f ( x) d x − µ2

2

Mathematics: analysis and approaches formula booklet

∞

∞

−∞

−∞

10

Topic 5: Calculus – SL and HL

SL 5.3 SL 5.5

Derivative of x

n

Integral of xn

Area between a curve y = f ( x) and the x-axis, where f ( x) > 0 SL 5.6

SL 5.9

n n 1 f ( x) = x ⇒ f ′( x) = nx −

∫x

n

dx =

xn + 1 + C , n ≠ −1 n+ 1

b

A = ∫ y dx a

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) =

Chain rule

y = g (u ) , where u = f ( x) ⇒

Product rule

y = uv ⇒

Quotient rule

du dv v −u u dy y= ⇒ = dx 2 dx v v dx

Acceleration

a=

Distance travelled from t1 to t 2

distance =

Displacement from t1 to t 2

displacement =

Mathematics: analysis and approaches formula booklet

1 x dy dy du = × d x du dx

dy dv du =u +v dx dx dx

dv d 2s = dt dt 2

∫

t2

t1

v (t ) dt

∫

t2

t1

v( t)d t

11

SL 5.10

Standard integrals

1

∫ x dx = ln

x +C

∫ sin x dx = − cos x + C ∫ cos x dx = sin x + C ∫e SL 5.11

Area of region enclosed by a curve and x-axis

x

x dx = e + C

b

A = ∫ y dx a

Topic 5: Calculus – HL only

AHL 5.12

Derivative of f ( x) from first principles

AHL 5.15

Standard derivatives

y = f ( x) ⇒

dy  f ( x + h ) − f (x )  = f ′( x ) = lim   h →0 dx h  

tan x

f ( x) = tan x ⇒ f ′( x) = sec 2 x

sec x

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

cosec x

f ( x) = cosec x ⇒ f ′ ( x) = −cosec x cot x

cot x

f ( x) = cot x ⇒ f ′( x) = −cosec 2 x

ax

f ( x) = a x ⇒ f ′( x) = a x(ln a)

loga x

f ( x) = log a x ⇒ f ′( x) =

arcsin x

f ( x) = arcsin x ⇒ f ′( x) =

arccos x

f ( x) = arccos x ⇒ f ′( x) = −

arctan x

f ( x) = arctan x ⇒ f ′( x) =

Mathematics: analysis and approaches formula booklet

1 xln a 1 1− x

2

1 1 −x2

1 1 + x2

12

AHL 5.15

Standard integrals

∫a ∫a

AHL 5.17

AHL 5.18

Integration by parts

Area of region enclosed by a curve and y-axis

1 x a +C ln a

1 1  x d x = arctan   + C 2 a +x a

x d x = arcsin   + C ,  a a −x 2

2

dv

du

∫ u d x dx = uv − ∫ v d x d x

x...