Definition of variance for continuous random variable with density function f (2 answers)\int_-\infty^\infty (t-\mu)^2 f(t)dt E(X^2) - E(X)^2Definition of standard deviationsqrt{variance} = sqrt{E(X^2)-E(X)^2}P(A or B)P(A) + P(B) - P(A and B)Events A and B are independentP(A and B) = P(A) x P(B)Events A and B are mutually exclusiveP(A and B) = 0, equiv. P(A or B) = P(A) + P(B)sin(2x)2sin(x)cos(x)cos(2x)2cos^2(x)-1 = cos^2(x) - sin^2(x) = 1-2sin^2(x)tan(2x)2tan(x)/(1-tan^2(x))sin(x+y)sin(x)cos(y)+sin(y)cos(x)cos(x+y)cos(x)cos(y) - sin(x)sin(y)sin(-x)-sin(x)cos(-x)cos(x)arcsin x (take derivate)1/sqrt{1-x^2} dx (take integral)arctan x (take derivate)1/1+x^2 dx (take integral)tan x (take derivative)sec^2 x (take integral)X is HausdorffFor any x, y \in X, there exist disjoint open neighborhoods of x and y.Indiscrete (trivial) topology on XTopology on X with empty set and XDiscrete topology on XTopology on X where every subset is openA topology is finer ifthe topology has more open setsInterior of Aunion of all open sets contained in AExterior of Aunion of all open sets that do not intersect ABoundary of Aall x in X such that every open set containing x intersects A and complement of A.Limit point of Aa point x in A such that every open set that contains x also contains at least one point of A other than xLower limit topology on RTopology on R generated by [a, b)Heine-Borel TheoremA subset of R^n (in standard topology)is compact if and only if it is closed and bounded.ConnectedThere do not exist non-empty open sets such that X = O_1 \cup O_2.A closed subset of compact space is compactIf a space is Hausdorff and compact, a compact subset is closedComplex logarithm Log(z)log|z|+i Arg(z), where Arg(z) \in (-\pi, \pi]cos(z) (for complex numbers)(e^{iz}+e^{-iz})/2sin(z) (for complex numbers)(e^{iz}-e^{-iz})/2definition cosh(x)(e^x + e^-x)/2definition sinh(x)(e^x - e^-x)/2Cauchy-Riemann equations. Say f(x+iy) = u(x, y) + iv(x, y)\partial u/\partial x = \partial v/ \partial y \partial v \partial x = -\partial u/\partial y f is differentiable IF AND ONLY IF the these equations hold (what are they called?)Definition of analytic at z_0f(z) is differentiable at every point in some open neighborhood of z_0If a complex function f(z) is analytic, what can you say about higher derivatives?All higher derivatives are analytic too! (When is this true?)Complex line integral \int_C f(z)dz (two answers)Need parametric parameterization z(t) for t going from a to b. Then it's \int_a^b f(z(t)) z'(t). Alternatively find F(z) such that F'(z) = f(z). Then \int_C f(z)dz = F(z(b)) - F(z(a)).Cauchy's TheoremIf f(z) is analytic on a simply connected open set then for every closed path we have \int_C f(z)dz = 0.Morera's TheoremIf f(z) is continuous on a simply connected open set O and \int_C f(z)dz = 0 for every closed curve C in O, then f(z) is analytic in OCauchy Integral FormulaIf f(z) is analytic at all points within and on a simple closed path C that surrounds z_0, then f(z_0) = 1/(2\pi i) \int_C f(z)/(z-z_0) dz f^(n)(z_0) = n!/(2\pi i) \int_C f(z)/(z-z_0)^{n+1} dzCauchy Derivative EstimatesIf f(z) is analytic on and within |z - z_0| = r and M = max |f(z)| on the circle, then |f^(n)(z_0)| \leq n!M/r^nLiouville's TheoremIf f(z) is a bounded entire function, then it is constantMaximum PrincipleIf f(z) is an analytic function on an open set O and there is a point z_0 in O such that |f(z)|\leq|f(z_0)| for all z, then f(z) is constant. If f(z) is an analytic function on a bounded set then |f(z)| must attain its maximum on the boundary of O.Radius of convergence for complex Taylor series1/\lim_{n --> \infty} 1/\sqrt[n]{|a_n|} (what is this limit in the context of a Taylor series?)Isolated singularityf(z) analytic at every point in some punctured disc around a singularityEssential singularityThere is no n such that (z-z_0)^n f(z) is analytic at z_0.Formula for Laurent coefficients a_{-n}1/2\pi i \int_C f(z)/(z-z_0)^{-n+1} dz (n \geq 0)Residue Formula (z_0 pole of order k)Res(z_0, f) = 1/(k-1)! \lim{z-->z_0} d^{k-1}/dz^{k-1} [(z-z_0)^k f(z)] In particular, for simple pole, Res(z_0, f) = \lim_{z\rightarrow z_0} (z-z_0) f(z)Residue Theorem (singularities z_i inside C)\int_C f(z)dz = 2\pi i \sum_{m=1}^n Res(z_m, f)Newton's MethodStart with approximation x_n of a root, set x_{n+1} = x_n - f(x_n)/f'(x_n) and iterateHarmonic conjugate of vA function u with u_x = v_y and u_y = -v_xLaplace's Equation (solutions are called what?)\partial^2 f/\partial x^2 + \partial^2 f/\partial y^2 = 0 solutions are called harmonicRational Root TheoremGiven p(x)=a_nx^2+a_{n-1}x^{n-1}+...+a_1x + a_0, the roots of p(x)=0 are of the form s/t where s|a_0 and t|a_n.hyperbolic cosh^2 and sinh^2 formulacosh^2-sinh^2 = 1sin^2(x) = ? (helpful to integrate)(1-cos 2x)/2arcsin(1)\pi/2arccos(1)0arctan(1)\pi/4Taylor polynomial approximationf(x) - P_n(x) = f^(n+1)(c)/(n+1)! x^{n+1} for some 0<c<xAlternating series test\sum_{n=1}^\infty (-1)^n a_n with a_n \geq 0 for all n will converge if a_n decrease monotonically with limit zero.Absolute convergence ruleEvery absolutely convergent series is convergentConvergence of a sequenceIf a sequence is monotonic and bounded, then it is convergentInverse function ruleIf f has nonzero derivative at x_0, then f^{-1} has a derivative at y_0 = f(x_0) which is f^{-1}'(y_0) = 1/f'(x_0)Complementary angle formulassin(\pi/2-\theta) = \cos\theta cos(\pi/2-\theta) = \sin\thetaRelation between arcsin and arccosarccos(x) = \pi/2 - arcsin(x)d(cot(x))-csc^2(x) dx (take integral)d(sec(x))sec x tan x dx (take integral)d(arcsin x)dx/\sqrt{1-x^2}d(arccos x)-dx/\sqrt{1-x^2}d(arctan x)dx/1+x^2Direction the gradient pointsDirection where derivative is maximized, quickest increase of scalar fieldLine integral of scalar field\int_C f ds = \int_a^b f(x(t), y(t)) (ds/dt) dt where ds/dt = \sqrt{x'(t)^2+y'(t)^2}Line integral of vector field\int_C F . dr = \int_a^b F(r(t)) . r'(t) dt = \int_C M dx + N dyFundamental Theorem of Calculus for Line Integralsint_C \Grad f . dr = f(B) - f(A)Polar coordinates volume formr dr d\thetaGreen's Theorem\int_C M dx + N dy = \int\int_R (\partial N/\partial x - \partial M/\partial y) dA \int_C F . dr = \int\int_S \Curl x F . n dSSurface of revolution: given f(x, y) = 0, how do you get the equation of revolution around x-axisreplace y with +/- \sqrt{y^2+z^2} so it's f(x, +/-\sqrt{y^2+z^2}) = 0homogeneous equationay''+by'+cy=0solving non-homogeneous equations (eg ay''+by'+cy=d(x).)y= y_h + y_phow to solve homogeneous equations1. form auxiliary polynomial 2. for each real root m, we have a term c.e^{mx} 3. for each pair of complex conjugate roots a +/- bi have e^{ax}(c_1 cos(bx) + c_2 sin(bx)) 4. for higher order roots have cx.e^{mx} (etc.)Max/min problem subject to constraint1. Solve for one variable through constraint, plug in, solve one-variable problem. OR 2. Lagrange multiplier: say asked to find extreme values of f(x, y) subject to constraint g(x, y) = c. Set \Grad f = \lambda \Grad g. This gives two equations; solve for \lambda in both and set equal. Substitute this relation back into the constraint.Definition of critical point (two variables)f_x = f_y = 0surface area of a sphere4 pi r^2volume of sphere4/3 pi r^3volume of solid of revolution around x-axis\pi \int_a^b y(x)^2 dxvolume of solid of revolution around y-axis\pi \int_a^b x(y)^2 dyintegral of odd function from -a to azeroarc length\int_a^b ds = \int_a^b \sqrt{dx^2+dy^2} = \int_a^b \sqrt{1+(dy/dx)^2} dx or \int_a^b \sqrt{dx^2+dy^2+dz^2} = \int_a^b \sqrt{x'(t)^2+y'(t)^2+z'(t)^2} dtLeibniz formula for differentiation under integral\int_a(x)^b(x) f(x, y) dy = f(x, b(x)) b'(x) - f(x, a(x)) a'(x) + \int_a(x)^b(x) \partial/\partial x f(x, y) dyinteresting function, good counter example possiblysin(1/x)standard deviation for n Bernoulli trials\sqrt{npq}, p=prob of success, q=1-p.tan(x/2)sin(x)/(1+cos(x))continuity on compact interval impliesuniform continuityarea of ellipse\pi a b