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Math app&int Formula Booklet v1

Mathematics Formula Booklet

Subject

Mathematics: Applications and Interpretation SL

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Degree

IB

School

Grace Christian College Wodonga - Leneva

Academic year: 2023/2024

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Fritz Wolmarans

Grace Christian College Wodonga

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© International Baccalaureate Organization 201 9

Mathematics: applications and interpretation

formula booklet

For use during the course and in the examinations

First examinations 2021

Version 1.

Diploma Programme

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 9 HL only 10

Topic 5: Calculus

SL and HL 11 HL only 11

Topic 1: Number and algebra – SL and HL

SL 1. The nth 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 n n n n S = u + n − d S = u +u SL 1. The nth term of a geometric sequence 1 1 n 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. Compound interest 1 100 k n r 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. Exponents and logarithms a x= b ⇔ x = logab, where a > 0, b > 0, a≠ 1 SL 1 Percentage error A E E 100% v v v ε = − × , where vE is the exact value and vA is the approximate value of v

Topic 1: Number and algebra – HL only

AHL 1. Laws of logarithms log a xy = log a x +logay log a log a loga x x y y = − log a xm =m logax for a x y, , > 0 AHL 1. The sum of an infinite geometric sequence 1 1 u S r ∞ = − , r < 1 AHL 1. Complex numbers z = a +bi Discriminant ∆ = b 2 − 4 ac AHL 1. Modulus-argument (polar) and exponential (Euler) form z = r (cos θ + isin θ) = re i θ=rcisθ AHL 1 Determinant of a 2 × 2 matrix det a b ad bc c d   =   ⇒ = = −   A A A Inverse of a 2 × 2 matrix 1 1 , det a b d b ad bc c d c a =   ⇒ − =  −  ≠        −  A A A AHL 1. Power formula for a matrix M n =PD Pn − 1 , where P is the matrix of eigenvectors and D is the diagonal matrix of eigenvalues

Topic 3: Geometry and trigonometry – SL and HL

SL 3. Distance between two points ( x 1 , y 1 , z 1 )and ( x 2 , y 2 , z 2 ) 2 2 2 d = ( x 1 − x 2 ) + ( y 1 − y 2 ) + ( z 1 −z 2 ) Coordinates of the midpoint of a line segment with endpoints ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) 1 2 , 1 2 , 1 2 2 2 2  x + x y + y z +z      Volume of a right-pyramid 1 3 V = Ah, where A is the area of the base, h is the height Volume of a right cone 2 1 3 V = πr h , where r is the radius, h is the height Area of the curved surface of a cone A = πrl , where r is the radius, l is the slant height Volume of a sphere 3 4 3 V = πr , where r is the radius Surface area of a sphere A = 4 πr 2 , where r is the radius SL 3 Sine rule sin sin sin a b c A B C = = Cosine rule c 2 = a 2 + b 2 − 2 ab cosC; 2 2 2 cos 2 a b c C ab

− Area of a triangle 1 sin 2 A = ab C SL 3 Length of an arc 2 360 l r θ = × π , where θ is the angle measured in degrees, r is the radius Area of a sector 2 360 A r θ = × π , where θ is the angle measured in degrees, r is the radius

Topic 3: Geometry and trigonometry – HL only

AHL 3. Length of an arc l = r θ, where r is the radius, θ is the angle measured in radians Area of a sector 2 1 2 A = r θ, where r is the radius, θ is the angle measured in radians AHL 3. Identities cos 2 θ + sin 2 θ= 1 sin tan cos θ θ θ = AHL 3. Transformation matrices cos 2 sin 2 sin 2 cos 2 θ θ θ θ      −  , reflection in the line y =(tan θ)x 0 0 1  k      , horizontal stretch / stretch parallel to x-axis with a scale factor of k 1 0 0 k       , vertical stretch / stretch parallel to y-axis with a scale factor of k 0 0 k k       , enlargement, with a scale factor of k, centre (0, 0) cos sin sin cos θ θ θ θ  −      , anticlockwise/counter-clockwise rotation of angle θ about the origin ( θ > 0 ) cos sin sin cos θ θ θ θ      −  , clockwise rotation of angle θ about the origin ( θ > 0 )

Topic 4: Statistics and probability – SL and HL

SL 4. Interquartile range IQR = Q 3 −Q 1 SL 4. 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 Probability of an event A ( ) P ( ) ( ) n A A n U = Complementary events P ( A) + P ( A′) = 1 SL 4. 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 ( ) P ( ) P ( ) A B A B B ∩ = Independent events P ( A ∩ B) =P ( A) P ( B) SL 4. Expected value of a discrete random variable X

E ( X ) = ∑x P ( X =x)

SL 4. Binomial distribution X ~ B ( n , p) Mean E ( X )=np Variance Var ( X ) = np (1 −p)

Topic 4: Statistics and probability – HL only

AHL 4. Linear transformation of a single random variable

####### ( )

####### ( ) 2

E E ( ) Var Var ( ) aX b a X b aX b a X + = + + = Linear combinations of n independent random variables, X 1 , X 2 , ..., Xn

####### ( ) ( ) ( ) ( )

####### ( )

####### ( ) ( ) ( )

1 1 2 2 1 1 2 2 1 1 2 2 2 2 2 1 1 2 2 E ... E E ... E Var ... Var Var ... Var n n n n n n n n a X a X a X a X a X a X a X a X a X a X a X a X ± ± ± = ± ± ± ± ± ± = + + + Sample statistics Unbiased estimate of population variance s 2 n − 1 2 2 1 1 n n n s s n − = − AHL 4. Poisson distribution X ~ Po ( m) Mean E ( X )=m Variance Var ( X )=m AHL 4. Transition matrices T s n 0 =sn , where s 0 is the initial state

AHL 5. Standard integrals 1 d x ln x C x

∫ = +

∫ sin x d x = − cosx +C

∫ cos x d x = sinx +C

2 1 tan cos x C x

∫ = +

∫ e dx x = ex+C

AHL 5 Area of region enclosed by a curve and x or y-axes d b a

A = ∫ y xor d

b a

A = ∫ x y

Volume of revolution about x or y-axes b π 2 d a

V = ∫ y xor π 2 d

b a

V = ∫ x y

AHL 5 Acceleration 2 2 d d d d d d v s v a v t t s = = = Distance travelled from t 1 to t 2 distance 2 1 ( ) d t t

= ∫ v t t

Displacement from t 1 to t 2 displacement 2 1 ( ) d t t

= ∫ v t t

AHL 5. Euler’s method yn + 1 = yn + h ×f ( xn , yn); xn + 1 = xn + h, where h is a constant (step length) Euler’s method for coupled systems 1 1 1 2 1 ( , , ) ( , , ) n n n n n n n n n n n n x x h f x y t y y h f x y t t t h + + + = + × = + × = + where h is a constant (step length) AHL 5. Exact solution for coupled linear differential equations 1 2 e 1 e 2 x = A λt p +Bλtp

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