MATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET

Transcription

1 MATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET 06 Copyright School Curriculum and Standards Authority, 06 This document apart from any third party copyright material contained in it may be freely copied, or communicated on an intranet, for non-commercial purposes in educational institutions, provided that it is not changed and that the School Curriculum and Standards Authority is acknowledged as the copyright owner, and that the Authority s moral rights are not infringed. Copying or communication for any other purpose can be done only within the terms of the Copyright Act 968 or with prior written permission of the School Curriculum and Standards Authority. Copying or communication of any third party copyright material can be done only within the terms of the Copyright Act 968 or with permission of the copyright owners. Any content in this document that has been derived from the Australian Curriculum may be used under the terms of the Creative Commons Attribution-NonCommercial 3.0 Australia licence. This document is valid for teaching and examining until 3 December /834[v] Mathematics Specialist Formula Sheet 06 Ref: 6-054

2 MATHEMATICS SPECIALIST Formula SHEET Index Vectors 3 Trigonometry 4 Functions 5 6 Complex numbers 6 Exponentials and logarithms 7 Mathematical reasoning 7 Measurement 8 Chance and data 8

3 Formula SHEET 3 MATHEMATICS SPECIALIST Vectors Magnitude: Dot product: (a, a, a ) = a + a 3 + a 3 a b = a b cosθ = a b + a b + a 3 b 3 Triangle inequality: Vector equation of a line in space: a + b a + b one point and the slope: two points A and B: r = a + λb r = a + λ(b a) x a Cartesian equations of a line in space: y a = z a = 3 b b Parametric form of vector equation of a line in space: x = a + λb...() y = a + λb...() z = a 3 + λb 3...(3) Vector equation of a plane in space: r n = c or r = a + λb + µc b 3 Cartesian equation of a plane: Vector cross product: ax + by + cz = d a b = a b sinθ n where a = the magnitude of vector a b = the magnitude of vector b θ is the angle between a and b and n is the unit vector perpendicular to vectors a and b a x a y b x b y Given a = and b = then a z b z a x a y a z b x b y b z a b = = a y b z a z b y a z b x a x b z a x b y a y b x

4 MATHEMATICS SPECIALIST 4 Formula SHEET Trigonometry In any triangle ABC: a b c sin A = sin B = sin C a = b + c bc cosa cosa = b + c a bc A = ab sin C In a circle of radius r, for an arc subtending angle θ (radians) at the centre: Length of arc = rθ Area of segment = r (θ sinθ) Area of sector = r θ Identities: cos θ + sin θ = + tan θ = sec θ cos θ = cos θ sin θ = cos θ = sin θ cos (θ + φ) = cosθ cosφ + sinθ sinφ sin (θ + φ) = sinθ cosφ + cosθ sinφ tan (θ + φ) = tanθ + tanφ + tanθ tanφ sinθ = sinθ cosθ tan θ = tanθ tan θ lim x 0 sin x = x lim x 0 cos x = 0 x Simple Harmonic Motion: If = k d x dt x then x = Asin(kt +α) or x = Acos(kt +β) and v = k (A x ), where A is the amplitude of the motion, α and β are phase angles, v is the velocity and x is the displacement.

5 Formula SHEET 5 MATHEMATICS SPECIALIST Functions Differentiation: If f(x) = y then f'(x) = dy If f(x) = e x then f'(x) = e x If f(x) = x n then f'(x) = nx n If f(x) = ln x then f'(x) = x If f(x) = sin x then f'(x) = cos x If f(x) = tan x then f'(x) = sec x = cos x If f(x) = cos x then f'(x) = sin x Product rule: If y = f (x) g(x) then y' = f'(x) g(x) + f(x) g'(x) or If y = uv dy du then = v + u dv Quotient rule: f(x) If y = g(x) then y' = f'(x) g(x) f(x) g'(x) (g(x)) or If y = u v dy then = du dv v u v Incremental formula: δy ~ dy δx or f (x + h) f (x) ~ f '(x)h Chain rule: If y = f(g(x)) then y' = f'(g(x)) g'(x) or If y = f (u) and u = g(x) dy dy du then = du Integration: Powers: x n = x n+ n + + c, n = Exponentials: e x = e x + c Logarithms: x = ln x + c Trigonometric: sin x = cos x + c cos x = sin x + c sec x = tan x + c Fundamental Theorem of Calculus: d x a f(t)dt = f (x) and b a f'(x) = f (b) f (a)

6 MATHEMATICS SPECIALIST 6 Formula SHEET Functions Quadratic function: If y = ax + bx + c and y = 0, then x = b ± b 4ac a for x C Absolute value function: x = x, for x 0 x, for x < 0 Complex numbers For z = a + ib, where i = b Argument: arg z = θ, where tan θ = a and π < θ π Modulus: mod z = z = a + ib = a + b = r Product: z = z arg (z ) = arg z + arg Quotient: z = z z arg z = arg z arg Polar form: For z = r cisθ, where r = z and θ = arg z: cis(θ + φ) = cis θ cis φ cis( θ) = cis θ z = r r cis (θ + φ) cis θ = cos θ + i sin θ cis(0) = z r z = cis (θ φ) r For complex conjugates: z = a + bi z = r cis θ z z = z z = a bi z = r cis ( θ) z z = = z z z + = z + z = z

7 Formula SHEET 7 MATHEMATICS SPECIALIST Exponentials and logarithms For a, b > 0 and m, n real: a m a n = a m+n a 0 = a m a n = a m n a n = a n (a m ) n = a mn (ab) m = a m b m For m an integer and n a positive integer: a n = n a a m n = n a m = ( n a) m For a, b, y, m and n positive real and k real: = a 0 log a = 0 y = a x log a y = x log a mn = log a m + log a n a = a log a a = log a m = log b m log b a (change of base) log a (m k ) = k log a m If dp dt = kp, then P = P 0 e kt Mathematical reasoning De Moivre's theorem: (cis θ) n = (cos θ + isin θ) n (cis θ) n = cos nθ + isin nθ z n = z n cis (nθ) q q θ + πk θ + πk z = z cos + isin for k an integer q q

8 MATHEMATICS SPECIALIST 8 Formula SHEET Measurement Circle: Triangle: Parallelogram: Trapezium: Prism: C = πr = πd, where C is the circumference, r is the radius and D is the diameter A = πr, where A is the area A = bh, where b is the base and h is the perpendicular height A = bh A = (a + b)h, where a and b are the lengths of the parallel sides V = Ah, where V is the volume and A is the area of the base Pyramid: V = Ah 3 Cylinder: Cone: S = πrh + πr, where S is the total surface area V = πr h S = πrs + πr, where s is the slant height V = 3 πr h Sphere: S = 4πr Chance and Data 4 V = 3 πr 3 A confidence interval for the mean of a population is: where μ is the population mean, X z σ n μ X + z σ n σ is the population standard deviation, X is the sample mean, n is the sample size, z is the cut off value on the standard normal distribution corresponding to the confidence level. Sample size: n = ( z σ d ) where d is the required value of the difference from the mean. Note: Any additional formulas identified by the examination panel as necessary will be included in the body of the particular question.