Mathematics Applications & Interpretation Data Booklet PDF

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IB Mathematic Applications and Interpretations Data booklet....

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Diploma Programme

Mathematics: applications and interpretation 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

HL only

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

11

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 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

Prior learning – HL only

Solutions of a quadratic equation

The solutions of ax 2 + bx + c = 0 are x =

Mathematics: applications and interpretation formula booklet

−b ± b 2 − 4ac ,a ≠0 2a

2

Topic 1: Number and algebra – SL and HL

SL 1.2

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 =

SL 1.4

Compound interest

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

Percentage error

ε=

SL 1.3

SL 1.6

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 kn

vA − vE ×100% , where vE is the exact value and v A is vE

the approximate value of v

Mathematics: applications and interpretation formula booklet

3

Topic 1: Number and algebra – HL only

AHL 1.9

Laws of logarithms

log a xy = log a x + log a y x log a = log a x − log a y y

log a x m = m log a x for a, x, y > 0 AHL 1.11

The sum of an infinite geometric sequence

S∞ =

AHL 1.12

Complex numbers

z = a + bi

Discriminant

∆ = b 2 − 4ac

Modulus-argument (polar) and exponential (Euler) form

z = r (cos θ + isin θ ) = reiθ = r cis θ

AHL 1.13

AHL 1.14

AHL 1.15

u1 , r 0 1 + Ce −kx

Mathematics: applications and interpretation 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

c2 = a2 + b2 − 2 ab cos C ; cos C =

Area of a triangle

1 A = ab sin C 2

Length of an arc

l=

θ 360

2 2 2 a +b −c 2ab

× 2πr , where θ is the angle measured in degrees, r is

the radius

Area of a sector

A=

θ 360

× πr 2, where θ is the angle measured in degrees, r is

the radius

Mathematics: applications and interpretation formula booklet

6

Topic 3: Geometry and trigonometry – HL only

AHL 3.7

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 radians

AHL 3.8

Identities

2 2 cos θ + sin θ = 1

tan θ = AHL 3.9

Transformation matrices

sinθ cos θ

 cos 2 θ sin 2 θ   sin 2θ − cos 2θ  , reflection in the line y = (tan θ ) x   k 0   , horizontal stretch / stretch parallel to x-axis with a scale  0 1 factor of k 1 0    , vertical stretch / stretch parallel to y-axis with a scale 0 k  factor of k  k 0  0 k  , enlargement, with a scale factor of k, centre (0, 0)  

 cosθ  θ  sin

− sin θ   , anticlockwise/counter-clockwise rotation of cos θ  angle θ about the origin ( θ > 0 )  cosθ   − sin θ (θ > 0 )

sinθ   , clockwise rotation of angle θ about the origin cos θ 

Mathematics: applications and interpretation formula booklet

7

AHL 3.10

AHL 3.11

AHL 3.13

Magnitude of a vector

 v1  v = v + v + v , where v =  v2  v   3 2 1

2 2

2 3

Vector equation of a line

r = a + λb

Parametric form of the equation of a line

x = x 0 + λl, y = y 0 + λm, z = z 0 + λ n

Scalar product

 v1   w1      v ⋅ w = v1 w1 + v2 w2 + v3 w3 , where v =  v2  , w =  w2  v  w   3  3 v ⋅ w = v w cos θ , where θ is the angle between v and w v1w1 + v2 w2 + v3w3 v w

Angle between two vectors

cos θ =

Vector product

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

Mathematics: applications and interpretation formula booklet

8

Topic 4: Statistics and probability – SL and HL

SL 4.2

Interquartile range

IQR = Q3 − Q1

SL 4.3

k

SL 4.5

∑fx

i i

Mean, x , of a set of data

x=

i =1

, where n =

n

k

∑f

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)

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 4.6

P ( A ∩ B) P ( B)

X ~ B ( n , p) Mean

E ( X ) = np

Variance

Var ( X ) = np (1 − p)

Mathematics: applications and interpretation formula booklet

9

Topic 4: Statistics and probability – HL only

AHL 4.14

Linear transformation of a single random variable

E ( aX + b ) = aE ( X ) + b

Linear combinations of n independent random variables, X 1 , X 2 , ..., X n

E ( a1 X1 ± a2 X 2 ±... ± an X n ) = a1E ( X1 ) ± a2 E ( X 2 ) ± ... ± anE ( X n )

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

Var ( a1 X1 ± a2 X 2 ± ... ± an X n ) = a12 Var ( X1 ) + a2 2 Var ( X 2 ) + ...+ an2 Var ( X n )

Sample statistics Unbiased estimate of 2 population variance s n− 1

AHL 4.17

AHL 4.19

s 2n −1 =

n 2 sn n −1

Poisson distribution

X ~ Po( m) Mean

E(X ) = m

Variance

Var ( X ) = m

Transition matrices

T n s 0 = sn , where s0 is the initial state

Mathematics: applications and interpretation formula booklet

10

Topic 5: Calculus – SL and HL

SL 5.3 SL 5.5

Derivative of x

Integral of x

n

n ∫ x dx =

n

Area of region enclosed by a curve y = f ( x) and the

x-axis, where f ( x) > 0 SL 5.8

n n 1 f ( x) = x ⇒ f ′( x) = nx −

The trapezoidal rule

x+ + C , n ≠ −1 n +1 n 1

b

A = ∫ y dx a

1 h ( ( y0 + yn ) + 2( y1 + y2 + ...+ yn−1 )) , a 2 b −a where h = n

∫

b

y dx ≈

Topic 5: Calculus – HL only

AHL 5.9

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 tan x

f ( x) = tan x ⇒ f ′( x) =

Derivative of e

x

1 cos 2 x

f ( x) = e x ⇒ f ′( x) = e x 1 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 x d dx y= ⇒ = v v2 dx

Mathematics: applications and interpretation formula booklet

dy dy du = × dx du dx

dy dv du =u +v dx dx dx

11

AHL 5.11

Standard integrals

1

∫ x dx = ln

x +C

∫ sin x dx = − cos x + C ∫ cos x dx = sin x + C 1

∫ cos ∫e AHL 5.12

AHL 5.13

AHL 5.16

Area of region enclosed by a curve and x or y-axes

x

2

x

= tan x + C

d x = ex + C b

b

b

b

A = ∫a y d x or A = ∫a x dy

Volume of revolution about x or y-axes

2 2 V = ∫a πy dx or V = ∫a πx dy

Acceleration

a=

Distance travelled from t1 to t2

distance =

Displacement from t1 to t2

displacement =

Euler’s method

dv d 2s dv = 2 =v dt dt ds

∫

t2

t1

v (t ) dt

∫

t2

t1

v( t) d t

yn +1 = yn + h × f ( xn , yn ) ; x n+ 1 = x n + h , where h is a constant (step length)

Euler’s method for coupled systems

xn + 1 = xn + h × f 1( xn , yn , tn ) yn +1 = yn + h × f 2 ( xn , yn , tn ) tn +1 = tn + h where h is a constant (step length)

AHL 5.17

Exact solution for coupled linear differential equations

x = Ae λ1t p 1 + B e λ2t p 2

Mathematics: applications and interpretation formula booklet

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