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ADEDEX424 Formula Sheet Arithmetic and Algebra Rules of Indices: xm × xn = xm+n . (xm )n = xmn . xm ÷ xn = xm−n . x0 = 1. x1 = x. 1 x−n = n . x (x × y)n = xn × y n . √ 1 n x = xn . Rules of Logarithms: Definition: loga x, is the number y such that x = ay . loga (xy) = loga x + loga y. m log ) = m loga x. a (x x = loga x − loga y. loga y loga 1 = 0. loga a = 1. logb x loga x = . logb a aloga x = x. loga (ax ) = x. The Binomial Theorem: n X n n−i i n x y. (x + y) = i i=0 n n−2 2 n n−1 n n x y + · · · + y n. x y+ (x + y) = x + 2 1 Binomial Coefficients: n! n n(n − 1)(n − 2) · · · (n − i + 1) = = . i i!(n − i)! i! n n . = n − i i n n n+1 . + = i i−1 i c UCD 2014/15 1 of 5 Lines and their Equations y2 − y1 . x2 − x1 The equation of a line is y = mx + c, where m is the slope and c is the y-intercept. p The length of a line segment between two points (x1 , y1 ) and (x2 , y2 ) is (x2− x1 )2 + (y2 − y1 )2 . x1 + x2 y1 + y2 The midpoint of the line seqment between two points (x1 , y1) and (x2 , y2) is . , 2 2 Given two points (x1 , y1 ) and (x2 , y2 ) on a line, then the slope of the line is m = Quadratic Equations √ b2 − 4ac . 2a b2 − 4ac b 2 . The turning point of the graph of y = ax + bx + c lies at − , − 2a 4a The equation ax2 + bx + c = 0 has solutions x = −b ± Trigonometry Figure 3: Definition of Trigonometric Functions. c UCD 2014/15 2 of 5 θ sin(θ) 0 π 6 π 4 0 1 2 1 √ 2 cos(θ) 1 tan(θ) 0 √ π 3 √ π 2 3 2 1 0 3 2 1 √ 2 1 2 1 √ 3 1 √ 3 ∗ Table 1: Values of sin(θ), cos(θ) and tan(θ) for important values of θ. sin(−θ) = − sin(θ) cos(−θ) = cos(θ) tan(−θ) = − tan(θ) cot(−θ) = − cot(θ) cosec(−θ) = − cosec(θ) sec(−θ) = sec(θ) Table 2: Parity Identities. sin π 2 − θ = cos(θ) tan π − θ = cot(θ) cosec 2 π 2 − θ = sec(θ) cos π 2 − θ = sin(θ) cot π − θ = tan(θ) sec 2 π 2 − θ = cosec(θ) Table 3: Co-function Identities. c UCD 2014/15 3 of 5 sin(A ± B) = sin(A) cos(B) ± cos(A) sin(B) cos(A ± B) = cos(A) cos(B) ∓ sin(A) sin(B) tan(A ± B) = tan(A) ± tan(B) 1 ∓ tan(A) tan(B) Table 4: Sum and Difference Formulae. sin2 (θ) = 1 − cos(2θ) 2 cos2 (θ) = 1 + cos(2θ) 2 tan2 (θ) = 1 − cos(2θ) 1 + cos(2θ) Table 5: Half Angle Formulae. b c a = = . sin(A) sin(B) sin(C) The Cosine Rule: a2 = b2 + c2 − 2bc cos(A). The Sine Rule: Differential Calculus f (x) f ′ (x) c 0 xn nxn−1 eax aeax 1 x a cos(ax) ln(ax) sin(ax) cos(ax) −a sin(ax) Comments Here c is any real number Here we must have ax > 0 Note the change of sign Table 6: Some Common Derivatives f (x + h) − f (x) . h→0 h ′ ′ ′ The sum rule for differentiation: (f + g) (x) = f (x) + g (x). The multiple rule for differentiation: (cf )′ (x) = cf ′ (x). Differentiation using first principles: f ′ (x) = lim c UCD 2014/15 4 of 5 Integral Calculus Z f (x) k xn 1 x eax sin(ax) cos(ax) f (x) dx kx + c 1 xn+1 + c n+1 ln(x) + c 1 ax e +c a 1 − cos(ax) + c a 1 sin(ax) + c a Comments Here k is any real number Here we must have n 6= −1 Here we must have x > 0 Note the change of sign Table 7: Some Common Integrals Evaluating definite integrals: Z b If F is an antiderivative of f then f (x) dx = [F (x)]ba = F (b) − F (a). Z b a Z b Z b The sum rule for integration: (f + g)(x) dx = f (x) dx + g(x) dx. a Z a Z a b b The multiple rule for integration: (kf )(x) dx = k f (x) dx. a a Statistics n 1 1P xi . (x1 + x2 + · · · + xn ) = n n i=1 Given a list numbers x1 , x2 , . . . , xn in ascending order, then their median is defined to be x n2 + x n2 +1 if n is even and x n+1 if n is odd. 2 2 n P (xi − x)2 . The variance is given by Var(x) = i=1 n p The standard deviation is given by σ = Var(x). The line of best fit y = mx + c is found using the formulae: n n n P P P n xi yi − xi yi i=1 i=1 i=1 m= and c = y − mx. n n 2 P 2 P n xi − xi The mean is given by x = i=1 c UCD 2014/15 i=1 5 of 5