CHEAT SHEET, Exams of Calculus

Download CHEAT SHEET and more Exams Calculus in PDF only on Docsity! Learn your uni course in one day. Check spoonfeedme.com for free video summaries, notes and cheat sheets by top students. CHEAT SHEET MATH1001 Dierential Calculus University of Sydney 1 Complex Numbers z = a+ ib Where: i 2 = 1 Modulus |z| = p a 2 + b 2 Conjugate If z = a+ ib then z = z ib Properties of Modulus and Conjugate |zw| = |z||w| z w |z| |w| |z + w|  |z|+ |w| |z w| |z| |w| z ± w = z ± w zw = zw Polar Form z = rcis✓ = r(cos ✓ + i sin ✓) Where r is the modulus ||z|| and ✓ the argument arg z. Polar Form Operations and Properties rcis(✓)⇥ pcis() = rpcis(✓ + ) rcis(✓) pcis() = r p cis(✓ ) arg(zw) = arg(z) + arg(w) arg ⇣ z w ⌘ = arg(z) arg(w) Euler’s Formula e ix = cosx+ i sinx Where e is Euler’s Number and x is real. De Moivre’s Theorem Given z = re i✓: z n = r n e in✓ Cos and Sin in Terms of Exponentials cosx = e ix + e ix 2 sinx = e ix e ix 2i 2 Binomial Theorem The coecients of the expansion of (a + b) n are given by the binomial theorem: f(x) = ✓ n x ◆ a x b nx For x = 0, 1, 2, 3..., n, where: ✓ n x ◆ = n! x!(n x)! Which are, alternatively, the entries in Pascal’s Triangle. 3 Cartesian Equation of a Plane ax+ by + cz = d 4 Rules for Dierentiation of a Sin- gle Variable Scalar Multiple d[kf(x)] dx = kf 0 (x) Power Function d[x n ] dx = nx n1 Addition Rule d[u(x) + v(x)] dx = u 0 (x) + v 0 (x) 1