MATHEMATICS METHODS SAMPLE FORMULA SHEET 2016 Copyright © School Curriculum and Standards Authority, 2014 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 by educational institutions, provided that it is not changed in any way and that the School Curriculum and Standards Authority is acknowledged as the copyright owner. Copying or communication for any other purpose can be done only within the terms of the Copyright Act or by permission of the Authority. Copying or communication of any third party copyright material contained in this document can be done only within the terms of the Copyright Act or by permission of the copyright owners. This document is valid for teaching and examining until 31 December 2016. Mathematics Methods Formula Sheet 2016 2014/10140 MATHEMATICs METHODs 2 SAMPLE Formula SHEET Measurement Circle: C = 2πr = πD, where C is the circumference, r is the radius and D is the diameter A = πr 2, where A is the area 1 Triangle: A = 2 bh, where b is the base and h is the perpendicular height Parallelogram: A = bh Trapezium: 1 A = 2 (a + b)h, where a and b are the lengths of the parallel sides Prism: V = Ah, where V is the volume and A is the area of the base Pyramid: V = 1 Ah 3 Cylinder: S = 2 πrh + 2 πr 2, where S is the total surface area V = πr 2h S = πrs + πr 2, where s is the slant height 1 V = 3 πr 2h Cone: Sphere: S = 4πr 2 V = Exponentials Index laws: 4 3 3 πr For a, b > 0 and m, n real, a m b m = (a b)m 1 a−m = a m (a m) n = a m n am an = a m+n am m−n an = a a0 = 1 m For a > 0 and m an integer and n a positive integer, a n = √ am = ( √ a) See next page n n m SAMPLE Formula SHEET 3 MATHEMATICs METHODs Logarithims For a, b, y, m and n positive real and k real: y = ax loga y = x logamn = logam + logan a = a1 loga a = 1 logex = ln x loga(mk) = k loga m 1= a0 Calculus Differentiation: Product rule: loga 1 = 0 1 dy dx n If f(x) = x then f '(x) = nx n–1 If f(x) = sin x then f ′(x) = cos x If f(x) = e x then f '(x) = e x If f(x) = cos x then f ′(x) = – sin x If f(x) = ln x then f ′(x) = x If f(x) = y then f '(x) = or If y = uv If y = f (x) g(x) then y' = f '(x) g(x) + f (x) g'(x) Quotient rule: f(x) Chain rule: dy du dv = v+u dx dx dx u or If y = v If y = g(x) then y' = then f '(x) g(x) – f(x) g'(x) (g(x))2 If y = f (g(x)) then dv dy du = dx v – u dx dx v2 or If y = f (u) and u = g(x) then y' = f '(g(x)) g'(x) then dy dy du = × dx du dx Powers: x n+1 n x dx = ∫ n +1 + c, n = −1 Exponentials: ∫ e xdx = e x + c Natural logarithm: 1 ∫ x dx = ln x + c and ∫ Trigonometry: ∫ sin x dx = −cos x + c and ∫ cos x dx = sin x + c Fundamental Theorem of Calculus: d x dx (∫a f(t) dt) = f (x) and ∫a Incremental formula: δy ~ Exponential growth and decay: If dy δx dx dy = ky, then y = Ae kt dt See next page f ′(x) f x dx = ln (f (x)) + c b f ' (x) d x = f (b) − f (a) MATHEMATICs METHODs 4 SAMPLE Formula SHEET Random variables, distributions, pobability and proportions Probability: For any event A and its complement A, and event B P(A) + P(A) = 1 P(A B) = P(A) + P(B) – P(A B) P(A B) = P(A) P(B|A) = P(B) P(A|B) In a Bernoulli trial: x is the sample proportion ˆp, Mean μ = p and standard deviation σ = √p(1–p) In a binomial distribution: n P (X = x) = ( x ) px (1–p)n–x Mean µ = np and standard deviation σ = √np(1– p) Expected value: If X is a discrete random variable, E(x) = ∑ipi xi, where xi are the possible values of X and pi = P(X = xi ) If X is a continuous random variable, ∞ E(x) = ∫–∞ xp(x)dx, where p(x) is the probability density function of X. Variance: If X is a discrete random variable, Var(x) = ∑i pi (xi – μ)2, where μ = E(X) is the expected value If X is a continuous random variable, ∞ Var(x) = ∫–∞(x – μ)2 p(x)dx. A confidence interval for the proportion, p, of a population is: ˆp(1 – ˆp) ˆp(1 – ˆp) ˆp + z , n n where ˆp is the sample mean, n is the sample size and z is the cut-off value on the standard normal distribution corresponding ˆp – z Margin of error: to the confidence level. E= z ˆp(1 – ˆp) is the half-width of the confidence interval n Note:Any additional formulas identified by the examination panel as necessary will be included in the body of the particular question. See next page