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X z a b c y x z a b c y If y and z are adjacent, then both of them must have two neighbors among a,b,c to have degree equal to 3 By Pigeonhole principle, y and z must have a common neighbour By symmetry, we may assume that b is their common neighbour At this point, x and b have degree 3, vertices y,z have degree 2 and a and c have degree, respectively, and set Y 2 = W 1, Y 3 = αY 2 W 2, and Y 1 = βY 2 γY 3 W 3 Determine the variancecovariance matrix of Y 1, Y 2, and Y 3 (c) Determine the values of α, β, σ2 1, σ22, and σ2 3 in order that the variancecovariance matrices in (a) and (b) coincide Third Practice Second Midterm Exam 23Y t ∈ R p is the observed output • wt ∈ R n is called process noise or state noise • vt ∈ R p is called measurement noise w x y v z−1 A C The Kalman filter 8–8

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40 'ã ƒwƒAƒXƒ^ƒCƒ‹ ƒƒ"ƒY-Y m x m a m s a ' ' Strategy Find final velocity in y direction and use it in 2 2 1, tan y xy x v v v v v T §¨¸ ©¹ v v a t yf y y 0 ts 146 0 ( 98 / )(146 )m s s2 143 /ms 22 (115 / ) ( 143 / )22 184 / v v v xy m s m s ms tan tan11 143 / 115 / 513 y x v ms v m s T T §§¨¸ ©¹ ©¹ v m s (184 / , 513 )Consumption Function w/ respect to Y C = 100 2/3 (Y – 90) = 40 2/3 Y Savings Function w/ respect to Y S = 100 1/3(Y – 90) = 130 1/3 Y (d) 600 = 35(T – Y) 50C Net Taxes = 02Y => 50C = 600 35(Y – T) => C =12 07(Y – T) Autonomous Consumption Level a = 12 MPC b = 07 MPS MPS = 1MPC = 03

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A)w B)x C)y D)z E)x and y 19) )The normal boiling point of the substance with the phase diagram shown above is _____ °C A)10 B) C)30 D)40 E)50 ) 21)The phase diagram of a substance is shown above The area labeled _____ indicates the gasThis equation cannot be written in the form (1) Remarks on "Linear" Intuitively, a second order differential equation is linear if y00 appears in the equation with exponent 1 only, and if either or both of y and y0 appear in the equation, then they do so with exponent 1 onlyFor example, suppose we have a line Ax By C = 0 Replacing x and y with x/w and y/w yields A(x/w) B(y/w) C = 0 Multiplying by w changes it to Ax By Cw = 0 Let the given equation be a second degree polynomial Ax 2 2Bxy Cy 2 2Dx 2Ey F = 0 After replacing x and y with x/w and y/w and multiplying the result with w 2, we have

We designate (3, 5) as (x 2, y 2) and (4, 2) as (x 1, y 1) Substituting into Equation (1) yields Note that we get the same result if we subsitute 4 and 2 for x 2 and y 2 and 3 and 5 for x 1 and y 1 Lines with various slopes are shown in Figure 78 belowY Ax x x x F M Vx Vx x Mxx Mxx = = æöç÷ çç ÷÷ èø÷ =⋅ = =⋅ P52 Write the equations for shear and moment between points D and E Select the origin at D 5ʹ 3ʹ 10ʹ 6ʹ 8 kips AE D C B w = 3 kips/ft x P52 3 kips/ft 8 kips A yE D B C 3 kips/ft D E y x 10x V x M xX is an Erlang (n,λ) random variable with parameter λ = 1/3 and expected value EX = 15 (a) What is the value of the parameter n?

X° y° d° B C D O Example 1 Find the value of each of the pronumerals in the diagram O is the centre of the circle and ∠AOB = 100 A z° x° y° 100° B O Solution Theorem 1 gives that z = y = 50 The value of x can be found by observing either of the following Reflex angle AOB is 260 Therefore x = 130 (Theorem 1) or y x = 180Solve the equation for different variables stepbystep \square!B x = 34, y = 17 3 d x = 17 3, y = 34 ____ 15 A piece of art is in the shape of an equilateral triangle with sides of 13 in Find the area of the piece of art

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(c) y00 xy2y0 −y3 = exy is a nonlinear equation;Problem 24 The strain (plane strain) in a given point of a body is described by the 2x2 matrix below Find components of the strain tensor H x''y in a new coordinate system rotated by the angle Consider four cases = 45o, 45o, 60o and 60o ª0005º H xy « » ¬005 0 ¼ Problem 24 Solution Transformation equations for plane strain, from old coordinate system xy to new coordinate(c) What is VarX?

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41 Chapter 4 Discretetime Fourier Transform (DTFT) 41 DTFT and its Inverse Forward DTFT The DTFT is a transformation that maps Discretetime (DT) signal xn into a complex valued function of the real variable w, namely −= ∑ ∈ℜ ∞ =−∞Depends on future values of the input, eg for t = 3 we have y(−3) = x(−1) Invertibility The system is invertible by applying the function w(t) = y(3t) Linearity The system is linear because if y1(t) = x1(t/3), and (31) y2(t) = x2(t/3) (32) and x(t) = αx1(t) βx2(t), then the output y(t) corresponding toT h e c i t y of n e w y o r k d e p a r t m e n t o f f i n a n c e o f f i c e o f t a x p o l i c y a n n u a l r e p o r t o n t a x e x p e n d i t u r e s fiscal year 13 bill de

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Solution for Given that x, y, z >1 and w be a positive number such that log, w = 24, log, w = 40 and logrvz W = 12 Find log, w %3DY= x(1 x) (1=2;1=4) The region Dis both vertically simple and horizontally simple, but the bounds for yin terms of x are simpler than the bounds for xin terms of y, so when we use Fubini's theorem to evaluate the integral, we take the yintegral on the inside and the xintegral on the outside This gives us ZZ D f(x;y)dA= Z x=1=2 x=0 Z y=x(1Let A={w, x, y, z} and B={a, b} Use the setroster notation to write each of the following sets, and indicate the number of elements that are in each set a

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C U(x, y) = x2/3 y1/3 Since the indifference curves are bowed towards the origin, they do obey the assumption of diminishing MRS y x 8 8 1 1 512 512 Economics 3070 d U(x, y) =min(2X, 3Y) This is an example of perfect complements The MRS is undefined at the vertex where 2X=3Y But lets graph the indifference curve, remember they L shaped(c) If the ant is to return directly to the starting point, how far and in whatWe take the third term in the first parenthesis and multiply it to every term in the second parenthesis, to obtain c x c y c z cx cy cz c x c y c z The final product is equal to the sum of all of these terms, which is a x a y a z b x b y b z c x c y c z

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Find the rate of change of total revenue, cost, and profit with respect to time Assume that R(x) and C(x) are in dollars {eq}R(x) = 45x 05x^2, \ \ C(x) = 4x 5, \ \ when \ x = 40 \ and \ dx(b) What is the PDF of X?Simple and best practice solution for Y=40x equation Check how easy it is, and learn it for the future Our solution is simple, and easy to understand, so don`t hesitate to use it as a solution of your homework If it's not what You are looking for type in the equation solver your own equation and let us solve it

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Ex) X and Y have joint PDF f(x,y) = c x y2 if 0 < x < y < 1 = 0 elsewhere • Find c First, draw the region where f > 0 (not 0 y x 1 1 cxy dxdy y ³ ³ 0 2 1 0 1 cxy dxdy y x ³ ³ 0 2 1 cxy dydx x 2³ ³ 1 1 0Solution for xy=40 equation Simplifying x 1y = 40 Solving x 1y = 40 Solving for variable 'x' Move all terms containing x to the left, all other terms to the right Add 'y' to each side of the equation x 1y y = 40 y Combine like terms 1y y =Angle x measures 42° Find the measure of angle y A) 42° B) 48° C) 138° D) 148° Explanation Since both angles make up the sum of the straight line, they are supplementary, or 180°, so 180° 42° = 138° 4) The angles of a triangle are 2x, 3x, and 4x degrees Find the value of x A) B) 30 C) 40 D) 50 Explanation

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Solution Fix x= (˘ j), y= ( j) and z= ( j) in l1Usual triangle inequality on real numbers yields j˘ j jj j˘ j jj j j jj sup j2N j˘ j jj sup j2N j j jj = d(x;z) d(z;y) Taking supremum over j2N on both sides gives the desired inequalityOverdetermined System for a Line Fit (2) Writing out the αx β = y equation for all of the known points (x i,y i), i =1,,mgives the overdetermined system 2 6 6 4 x1 1 x2 1 x m 1 3 7 7 5 » α β – = 2 6 6 4 y1 y2 y m 3 7 7 5 or Ac = y where A = 2 6 6 4 x1 1 x2 1 x m 1 3 7 7 5 c = α β – y = 2 6 6 4 y1 y2 y m 3 7 7 5 Note We cannot solve Ac = y with Gaussian elimination Unless the60º North of East Let the positive x direction be East and the positive y direction be North (a) What are the x and y components of d 1,d 2 and d 3?

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(1) 5 minutes 00 second X X Y Z Outcrop B W Z Outcrop C Z Z Z Z W W Z X 10 9 8 7 6 5 4 3 2 1 0 Muscovite mica Fluorite Olivine Graphite Mohs' Hardness Scale 23 Landscapes characterized by gentle slopes and(b) What are the x and the y components, the magnitude and the direction of the ant's net displacement?411 Recognize a function of two variables and identify its domain and range 412 Sketch a graph of a function of two variables 413 Sketch several traces or level curves of a function of two variables 414 Recognize a function of three or more variables and identify its level surfaces Our first step is to explain what a function of

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P yare quadratic surds and if a p x= p y,thena= 0 and x= y 22 If p x;40 seconds after the earthquake How long after the first Pwave arrives will the first Swave arrive?Math 461 Introduction to Probability AJ Hildebrand Variance, covariance, correlation, momentgenerating functions In the Ross text, this is covered in Sections 74 and 77

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X;Y 2X B Y B stX BY B = 5) 8 >< > MU X B MU Y B = p X B Y B = 5) 8 < Y B X B = 1 2 2X2 = 5) 8 >< > X B = p 10 2 Y B = p 10 Notice that on the graph in (v) this allocation is marked as point D the intersection of the B's indi erence curve that goes through the endowment and the contract line viii Assume that A still knows B'sP Ay By L x y A B Figure 108 Problem 2 M V Ay = P/2 x Figure 109 Problem 2 For 0 x L/2 Hence, EI d2y dx2 = M = Px 2 EI dy dx = Px2 4 C1 integrating with respect to x EIy= Px3 12 C1x C2 integrating again with respect to x Use boundary condition y = 0 at x = 0A 8˚C b 18˚C c 40˚C d 50˚C 15 10 05 010 10 30 40 50 60 70 P,atm T,°C W X Z y 0 4 Region "d" on the heating curve shown (Temperature versus heat, "q") corresponds to a a pure gas increasing in temperature b a liquid increasing in temperature c a solid increasing in temperature d a solid melting e a liquid boiling a

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P yare quadratic surds and if a p x= b p ythen a= band x= y 23 If a;m;nare positivereal numbersanda6=1,thenlog a mn=log a mlog a n 24 If a;m;nare positive real numbers, a6=1,thenlog a m n =log a m−log a n 25 If aand mare positive real numbers, a6=1thenlog a mnYc(t) = c ej!t = c xc(t) where c = Aej') The efiect of a linear dynamical system on a complexvalued sinusoidal can be characterized in terms of a multiplication with a complex number;X y A 1 A 2 3 Symmetric Areas An area is symmetric with respect to a line when every point on one side is mirrored on the other The line divides the area into equal parts and the centroid will be on that axis An area can be symmetric to a center point

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Xt1 = Axt wt, yt = Cxt vt • xt ∈ R n is the state;Page 2 c) d) wx x′y BD B′D′ A′B 312) Simplify the following Boolean functions in products of sums a) F(w, x, y, z) = ∑ (0, 2, 5, 6, 7, 8, 10)Yc(t) = c xc(t) If we can write c = Aej', then A is the amplitude and ' is the phase of the output relative to the amplitude and phase of the input 32

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FOL Semantics (6) Consider a world with objects A, B, and C We'll look at a logical languge with constant symbols X, Y, and Z, function symbols f and g, and predicate symbols p, q, and rGet stepbystep solutions from expert tutors as fast as 1530 minutes Your first 5 questions are on us!Problem 343 Solution From Appendix A, an Erlang random variable X with parameters λ > 0 a postive real number and n a positive integerhas PDF fX (x) = ˆ

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Without the information that fXY(x;y) = 0 for (x;y) outside of A, we could plot the full surface, but the particle is only found in the given triangle A, so the joint probability density function is shown on the right This gives a volume under the surface that is above the region Aequal to 1 x y y) x y y) Not a pdf A pdf 18Writing Algebraic Expressions Just like any language, math has a way to communicate ideas An algebraic expression is a compact way of describing mathematical objects using a combination of numbers, variables (letters), and arithmetic operations namely addition, subtraction, multiplication, and division In other words, the three main components of algebraic expressions are numbers,

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δ()x = 016x Rex 1/7 Cf x = 0664 Rex Cf x = 0027 Rex 1/7 CD = 1328 ReL C D = 0031 ReL 1/7 for turbulent flow over entire plate, 0 Œ L, ie assumes turbulent flow in the laminar region where Cf x = τw 1 2 ρU∞ 2 local drag coefficient based on local wall shear stress (laminar or turbulent flow region) and CD = total drag coefficient

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