MATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET

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1 MATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET /2643 Mathematics Specialist Fmula Sheet 2018

2 MATHEMATICS SPECIALIST 2 FORMULA SHEET Inex Differentiation an integration 3 Applications of calculus Functions Statistical inference 4 Mensuration Vects in 3D 5 Complex numbers 6 Trigonometry 7

3 FORMULA SHEET 3 MATHEMATICS SPECIALIST Differentiation an integration x (xn ) = nx n 1 x n x = x n+1 n +1 + c, n = 1 x (eax ) = ae ax e ax x = 1 a e ax + c x (1n x) = 1 x 1 x x = ln x + c f (x) (1n f (x)) = x f (x) f (x) f(x) x = 1n f (x) + c (sin f (x)) = f (x) cos ( f (x)) x sin (cos f (x)) = f (x) sin ( f (x)) x cos x (tan f (x)) = f (x) f (x) sec2 ( f (x)) = cos 2 f (x) (ax) x = 1 a (ax) x = 1 a sec 2 (ax) x = 1 a cos (ax) + c sin (ax) + c tan (ax) + c If y = uv If y = f (x) g(x) Prouct rule v (uv) = u x x + u x v y' = f'(x) g(x) + f (x) g'(x) If y = u v If y = f (x) g(x) Quotient rule x u v = v u x u v x v 2 y' = f'(x) g(x) f(x) g'(x) (g(x)) 2 If y = f(u) an u = g(x) If y = f(g(x)) Chain rule y x = y u u x y' = f'(g(x)) g'(x) x Funamental theem f(t)t = f (x) x a an b a f'(x)x = f (b) f (a)

4 MATHEMATICS SPECIALIST 4 FORMULA SHEET Applications of calculus Growth an ecay Exponential equation P t = kp P = P 0 ekt Logistic equation Volumes of solis of revolution About the x-axis About the y-axis P t = rp(k P) P = kp 0 P 0 + (k P 0 )e rkt b V = [ f(x)] 2 x a V = [ f(y)] 2 y c Simple harmonic motion If 2 x = k t 2 2 x x = A sin (kt + α) x = A cos (kt + β) where A is the amplitue, α an β are phase angles, v is the velocity an x is the isplacement v 2 = k 2 (A 2 x 2 ) Perio: T = 2 k Frequency: f = 1 T Increments fmula δy y x δx Acceleration v t v v x x 1 2 v2 Functions Quaratic function Absolute value function If f(x) = ax 2 + bx + c an f(x) = 0, x = b + b2 4ac 2a x = x, f x 0 x, f x < 0 Statistical inference Confience interval f the mean of the population Sample size X z s n μ X + z s n n = ( z s ) 2

5 FORMULA SHEET 5 MATHEMATICS SPECIALIST Mensuration Parallelogram A = bh Triangle A = 1 2 bh A = 1 2 ab sin C Trapezium A = 1 2 (a + b)h Circle Prism A = πr 2 an C = 2πr = π V = Ah, where A is the area of the cross section Pyrami V = 1 3 Ah, where A is the area of the cross section Cyliner V = πr 2 h S = 2πrh + 2πr 2 Cone V = 1 3 πr2 h S = πrs + πr 2, where s is the slant height Sphere V = 4 3 πr3 S = 4πr 2 Vects in 3D Magnitue (a 1, a 2, a ) = a a a 3 Dot prouct a b = a b cosθ = a 1 b 1 + a 2 b 2 + a 3 b 3 Cross prouct a b = a 1 a 2 a 3 b 1 b 2 b 3 = a 2 b 3 a 3 b 2 a 3 b 1 a 1 b 3 a 1 b 2 a 2 b 1 Equation of a line One point an irection r = a + λu Two points A an B r = a + λ(b a) Equation of a plane r = a + λu 1 + µu 2 r n = a n r = r Equation of a sphere (x a) 2 + (y b) 2 + (z c) 2 = r 2 Cartesian equation of a line Cartesian equation of a plane Parametric equation of a line x a 1 u 1 = y a 2 u 2 = z a 3 u 3 ax + by + cz = x = a 1 + λu (1) y = a 2 + λu (2) z = a 3 + λu (3)

6 MATHEMATICS SPECIALIST 6 FORMULA SHEET Complex numbers Cartesian fm z = a + bi z = a bi Mo (z) = z = a 2 + b 2 = r Arg(z) = θ, tan θ = b a, π < θ π z = 2 z = 2 arg ( ) = arg ( ) + arg ( ) arg = arg ( ) arg ( ) z z = z 2 z 1 = 1 z = z z 2 + = + = Polar fm z = a + bi = r(cos θ + i sin θ) = r cis θ z = r cis ( θ) = r 1 r 2 cis (θ 1 + θ 2 ) = r 1 r 2 cis (θ 1 θ 2 ) cis(θ 1 + θ 2 ) = cis θ 1 cis θ 2 cis( θ) = 1 cis θ De Moivres theem z n = z n cis (nθ) (cis θ) n = cos nθ + i sin nθ 1 z q = r 1 q cos θ + 2πk q + isin θ + 2πk q, f k an integer

7 FORMULA SHEET 7 MATHEMATICS SPECIALIST Trigonometry Ientities a b c sin A = sin B = sin C a 2 = b 2 + c 2 2bc cosa cosa = b2 + c 2 a 2 2bc cos 2 x + sin 2 x = 1 Length of arc = rθ Area of segment = 1 2 r 2 (θ sinθ) Area of sect = 1 2 r2 θ 1 + tan 2 x = sec 2 x cos (x + y) = cos x cos y + sin x sin y sin (x + y) = sin x cos y + cos x sin y cos 2x = cos 2 x sin 2 x = 2 cos 2 x 1 = 1 2 sin 2 x sin 2x = 2sin x cos x tan (x + y) = tan x + tany 1 + tan x tan y tan 2x = 2 tan x 1 tan 2 x cos A cos B = 1 2 (cos(a B) + cos(a + B)) sin A cos B = 1 2 (sin(a + B) + sin(a B)) sin A sin B = 1 2 (cos(a B) cos(a + B)) cos A sin B = 1 2 (sin(a + B) sin(a B)) Note: Any aitional fmulas ientifie by the examination panel as necessary will be inclue in the boy of the particular question.

8 Copyright School Curriculum an Stanars Authity, 2016 This ocument apart from any thir party copyright material containe in it may be freely copie, communicate on an intranet, f non-commercial purposes in eucational institutions, provie that it is not change an that the School Curriculum an Stanars Authity is acknowlege as the copyright owner, an that the Authity s mal rights are not infringe. Copying communication f any other purpose can be one only within the terms of the Copyright Act 1968 with pri written permission of the School Curriculum an Stanars Authity. Copying communication of any thir party copyright material can be one only within the terms of the Copyright Act 1968 with permission of the copyright owners. Any content in this ocument that has been erive from the Australian Curriculum may be use uner the terms of the Creative Commons Attribution 4.0 International (CC BY) licence. This ocument is vali f teaching an examining until 31 December Publishe by the School Curriculum an Stanars Authity of Western Australia 303 Sevenoaks Street CANNINGTON WA 6107

MATHEMATICS: SPECIALIST UNITS 3C AND 3D FORMULA SHEET 2015

MATHEMATICS: SPECIALIST UNITS 3C AND 3D FORMULA SHEET 05 Copyright School Curriculum and Standards Authority, 05 This document apart from any third party copyright material contained in it may be freely