Binomial Theorem - Calculus Cheat Sheet

2 BinomialTheorem PolarForm z = rcisθ = r(cosθ + i sinθ) Wherer isthemodulus||z||andθ theargumentargz . ✓ PolarFormOperationsandProperties f (x) = Forx = 0, 1, 2, 3...,n,where: rcis(θ) × pcis(φ) = rpcis(θ + φ) z = a + ib Where: 2 i = −1 rcis(θ) r = cis(θ − φ) p pcis(φ) p |z| = a2 + b2 arg ⇣z ⌘ w =arg(z) − arg(w) z = a + ib then z = z − ib Wheree isEuler’sNumberandx isreal. z |z| |w| |zw+ w| ≤ |z| + |w| ScalarMultiple z n = rn einθ PropertiesofModulusandConjugate |zw| = |z||w| 4 RulesforDifferentiationofaSingleVariable DeMoivre’sTheorem Givenz = reiθ : d[kf (x)] = kf 0 (x) dx CosandSininTermsofExponentials cosx = ax bnx ax+ by+ cz= d eix = cosx + i sinx If x 3 CartesianEquationofaPlane Euler’sFormula Conjugate ◆ n ✓ ◆ n x!(nn! − x)! = x Whichare,alternatively,theentriesinPascal’sTriangle. arg(zw)=arg(z)+arg(w) Modulus Thecoecientsoftheexpansionof(a + b)n aregivenby thebinomialtheorem: eix + eix 2 sinx = eix − eix 2i PowerFunction d[xn ] = nxn1 dx |z − w| ≥ |z| − |w| AdditionRule z±w =z±w d[u(x) + v(x)] = u0 (x) + v 0 (x) dx zw= zw 1 ProductRule 6 Two-VariableOptimisation TotalDerivativeRule d[u(x)v(x)] = u(x)v 0 (x) + u0 (x)v (x) dx CriticalPoint When: ⇢ If: QuotientRule fx (a,b) = 0 ANDfy (a,b) = 0 u(x) If f (x) = then: v(x) Weusethetotalderivativerulegiven: Thenthereisacriticalpointat(a,b). f 0 (x) = v(x)u0 (x) − u(x)v 0 (x) [v(x)]2 dz ∂z dx ∂z dy = + dt ∂x dt ∂y dt Discriminant D(a,b) = fxx (a,b)fyy − [fxy (a,b)]2 Case1 D > 0 ANDfxx (a,b) > 0 then(a,b) isalocalminimum ChainRule If f (x) = u[v(x)]then: 0 0 0 f (x) = v (x)u (x) Case2 D > 0 ANDfxx (a,b) < 0 then(a,b) isalocalmaximum PartialDerivativeChainRule Whenwehavethecasewhere: ⇢ df df du = du dx dx AND: 7 LinearApproximations Tangenttosurfacez = f (x,y) at(a,b,f (a,b))is: Differentialiny z − f (a,b) = fx (a,b)(x − a) + fy (a,b)(y − b) ∂z ∂z ∂x ∂z ∂y = + ∂x ∂s ∂y ∂s ∂s Case4 D = 0 isinconclusive 5 TangentPlane = g(s,t) h(s,t) xy = Weusethechainrule: Case3 D < 0 then(a,b) isasaddlepoint Thatis: x = g(t) y = h ( t) ∂z ∂z ∂x ∂z ∂y = + ∂t ∂x ∂t ∂y ∂t DirectionalDerivatives dy= dy dx dx Forafunctionf (x,y) thedifferentialisgiven: ∂z ∂z dx+ dy ∂x ∂y 2 Du f = fx (x,y)u1 + fy (x,y)u2 a DifferentialofaFunctionof2Variables dz= Thedirectionalderivativeoff (x,y) inthedirectionofu is given: Where: uˆ = u1 ˆi + u2 ˆj GradientVector rf (x,y) = ∂f ˆi + ∂f ˆj ∂x ∂y 9HyperbolicTrigonometricFunctions 10Limits ImportantLimits RelationshiptoDirectionalDerivative lim sinx =1 x lim 1 1  cosx = x2 2 lim ex  1 =1 x x!0 Du f (x,y) = rf (x,y) ·uˆ y=0.5 x!0 8Logarithms x!0 BasicProperties L’Hopital’sRule loga 1 = 0 loga a = 1 x cosh(x) = x e +e 2 sinh(x) = x e e 2 loga xy=log a x +loga y loga tanhx = sinhx) coshx PropertiesofHyperbolicFunctions cosh2 x  sinh2 x = 1 ChangeofBase ax = bx logb a log b x loga x = logb a lim f (x) 0 ±1 = OR g(x) 0 ±1 lim f (x) f 0 (x) = lim x!a g(x) g 0 (x) x!a x!a loga xn = n log a x f 0 (x) = f (x)[lnf (x)]0 If Thenuse: x =loga x  log a y y LogarithmicDifferentiation x cosh(x + y) = coshx coshy + sinh x sinhy sinh(x  y) = sinh x coshy  sinhy coshx tanh(x + y) = tanhx + tanhy 1 + tanhx tanhy 3 11 TaylorSeries UsefulFunctions ex = 1 + x + cosx = 1  sinx = x  x2 2! + x3 + ...= Σ 3! x2 x4 x6 + ...= Σ +  6! 2! 4! x3 x5 x7 + ...= Σ +  7! 3! 5! 1 n=0 xn n! 1 nx n=0 (1) 1 n+1 n=0 (1) 2n 2n! x2n+1 (2n +1)! ErrorBound En = |f (x)  Pn (x)| < M Rn+1 (n +1) M isthemaximumvalueof|f (n+1) (x)| in|x  a| < R WithRemainderTerm f (x) = Pn (x) + En (x) With: Pn (x) = f (a) + (x  a)f 0 (a) + + (x  a)2 2! f 00 (a) + ... (x  a)n (n) f (a) n! And Z En (x) = x a (1)n (u  x)n (n+1) f (u) du n! 4