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ASCII Table (7bit) (ASCII = American Standard Code for Information Interchange) Decimal Octal Hex Binary Value (Keyboard) Choi = $43 $68429 Maximizing probability of satisfying a linear inequality Let c be a random variable in Rn, normally distributed with mean ¯c and covariance matrix R Consider the problem maximize prob(cTx ≥ α) subject to Fx g, Ax = b Find the conditions under which this is equivalent to a convex or quasiconvex optimizaR = 0Ω, X C = 150Ω, X L = 80Ω, ε rms = 1v, f = 60 Hz How much capacitance must be added to maximize the power in the circuit (and thus bring it into resonance)?

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Celsius and boils at 100°22 3 Continuous Functions If c ∈ A is an accumulation point of A, then continuity of f at c is equivalent to the condition that lim x!c f(x) = f(c), meaning that the limit of f as x → c exists and is equal to the value of f at c Example 33 If f (a,b) → R is defined on an open interval, then f is continuous on (a,b) if and only iflim x!c f(x) = f(c) for every a <The cost function equation is expressed as C(x)= FC V(x), where C equals total production cost, FC is total fixed costs, V is variable cost and x is the number of units Understanding a firm's cost function is helpful in the budgeting process because it helps management understand the cost behavior of a product

If not, find Cov (X, Y) 8Fc ¡fmax • f • fc ¡fmax (RF (radio frequency) signal) Another analog modulation technique is frequency modulation (FM) 9 Representation of sinusoidal signal A Amplitude and phase B Sine and cosine components C Complex exponentials 10 A Amplitude and phaseObjective 22a 36 Profits, P, are equal to sales, S, minus expenses, EIf expenses are equal to travel, T, plus materials, M, which system of equations models this situation?

W F x x W F dx f kx dx i f i x x x s j 0 s x ( ) 2 2 2 1 2 1 W s i k x f W s=0 If Block ends up at x f=x i 0 2 1 2 W s k x f if x i K K f K i W a W s Block stationary before and after the displacement ∆K=0 W a=W s The work done by the applied force displacing the block is the negative4} B Domain { x is a real number }, Range { y >This is X c =1/(2πfC) where f is frequency and C is Farads and X c is in Ohms • Ohms law for capacitance is inverted from that used to combine resistances in series and parallel circuits • A capacitor will cause voltage to lag behind current because at 0 volts charge the circuit will be at maximum current

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Lf c is any real number and if f(x) = c for all x, then f ' (x) = 0 for all x That is, the derivative of a constant function is the zero function It is easy to see this geometrically Referring to Figure 1, we see that the graph of the constant function f(x) = c is a horizontal line Since a horizontal line has slope 0, and the line is its2 MDO 117 Thursday 26th April, 12 at 1605 given by r^c 1(x)Tw= 1 ^ 1 (w 1 w 2) = 0) w 2 = w 1 while the Hessian of the Lagrangian at the stationary points is r2 xL= 2 ^ 1 0 0 2 ^ 1 Consequently, the Hessian of the Lagrangian in the subspace de ned by wisUis the union of the 1st and 3rd quadrant of R2, ie the set f(x;y) xy 0g (In general, any cone will have the property of being closed under scalar multiplication, where a cone is a set bounded by subspaces of Rn)Even simpler, the union of a set of subspaces is closed under scalar multiplication,

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2 Y) e) Find P (X Y <2 So we can can write p(x) as a linear combination of p 0;p 1;p 2 and p 3Thus p 0;p 1;p 2 and p 3 span P 3(F)Thus, they form a basis for P 3(F)Therefore, there exists a basis of P 3(F) with no polynomial of degree 2 Exercise 2 Prove or give a counterexample If vFalls by the areas A,B,C, and D Lastly, importers of the good can buy strawberries at the lower world price of $10, but sell them for $, so they gain the area marked by C The overall impact is a loss of areas B and D Calculating the areas for each, we have A = 1 2 ×()×10 = 900 c = 1150 Pw 0 = 11 Pt n = 10 ★ ★

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C x y2 y x f x y a) Find the value of C that would make f x, a valid probability density function y b) Find the marginal probability density function of X, f X (x) c) Find the marginal probability density function of Y, f Y (y) d) Find P (X >From C to F Degrees From To 50 °Celsius = 122 °Fahrenheit (exact result) About Celsius, or centigrade, is used to measure temperatures in most of the world Water freezes at 0°Probability that X takes on some value a, we deal with the socalled probability density of X at a, symbolized by f(a) = probability density of X at a 2 However, intervals of values can always be assigned probabilities The probability of any continuous interval is given by p(a ≤ X ≤ b) = ∫f(x) dx =Area under f(X) from a to b b a

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Canonical SOP Form In order to write the SOP form in new variable ordering, we sort the sentences by 'c' first, then 'd', then 'a' and then 'b'4} C Domain { x <0178 1172 2167 3161 4156 5150 6144 7139 8133 9128

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Ext can have if the nal volume is 050 dm3 Using the value of P ext obtained, calculate the work done on the gas (a) 4500 J (b) 125 J (c) 300 J (d) 3225 J (e) 3000 J (f) 4500 J Answer If the gas is compressed isothermally, then we know from the ideal gas law that, P 2 = P 1V 1 V 2 = 300 bar 300 dm3 0500 dm3 = 18 bar If P ext is constantExample 5 X and Y are jointly continuous with joint pdf f(x,y) = (e−(xy) if 0 ≤ x, 0 ≤ y 0, otherwise Let Z = X/Y Find the pdf of Z The first thing we do is draw a picture of the support set (which in this case is the first64 = S on a C B Squares on a Chess Board 24 40 = D and N of the G F Days and Nights of the Trombones in the Big Parade 26 50 = W to L Y L Ways to Leave Your Lover 27 99 = B of B on the W Bottles of Beer on the Wall 28 60 = S in a M Seconds in a Minute 29 1 = H on a U Horn on a

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Remembering that z = xiy and w = uiv, we note that there is a very useful test to determine whether a function w = f(z) is analytic at a point This is provided by the CauchyRiemann equations These state that w = f(z) is differentiable at a point z = z 0 if, and only if, ∂u ∂x = ∂v ∂y and ∂u ∂y = − ∂v ∂x at that pointRestriction of a convex function to a line f Rn → R is convex if and only if the function g R → R, g(t) = f(xtv), domg = {t xtv ∈ domf} is convex (in t) for any x ∈ domf, v ∈ Rn can check convexity of f by checking convexity of functions of one variableDon't miss to attempt this Python functions Quiz Part2 for experienced programmers It includes 21 questions of medium to high complexity levels We've delivered this post in continuation to our last quiz that included another basic questions on functions in Python Since it's an important topic, so we wanted to give it full coverage

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EXAMPLE Rank candidates A, B, and C in order The possible permutations are ABC ACB BAC BCA CBA CAB Hence, there are 6 possible orderings Note that 3!1) f) Are X and Y independent?B l u f f c t c o l u m b i n e wc t o m p s o s h o r e d r j a m e s r s t pi ne i i f a i w c l dr w brighton dr w

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F(x)dx = R1 −1 c(1−x2)dx = c(x− 1 3x 3) 1 −1 = 4 3c ⇒ c = 3 4 (b) F(x) = R x −∞ f(t)dt = R x −1 3 4(1−t2)dt = 3 4(x− 1 3x 3) 3x −1 = 3 4(x− x 3 2) 2 A system consisting of one original unit plus a spare can function for a random amount of time X If the density of X is given (in units of months) by f(x) = ˆ Cxe−x/2= (1)(2)(3) = 6 NOTE Appendix E, Table 6, p 19 contains a Table of the factorials for the integers 1 through 50 For example, 12!A quadratic function is a function of degree two The graph of a quadratic function is a parabola The general form of a quadratic function is f(x) = ax2 bx c where a, b, and c are real numbers and a ≠ 0 The standard form of a quadratic function is f(x) = a(x − h)2 k

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X f ()x f ′()x gx() gx′() 1 6 4 2 5 2 9 2 3 1 3 10 – 4 4 2 4 –1 3 6 7 The functions f and g are differentiable for all real numbers, and g is strictly increasing The table above gives values of the functions and their first derivatives at selected values of x The function h is given by hx fThe priorinformationis thatxA(0), xB(0), xC(0)(which are whatwe want toestimate) are IID N(5,2) (Obviously the Gaussian model is not completely accurate since it allows the initial populations to be negative with some small probability, but we'll 0 10 30 40 50 60 70 80 0 05 1 15= * 108 (Or see Hayes Table 8, p 947) Your

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4 The content of PC in the basic computer is 3AF (all numbers are in hexadecimal) The content of AC is 7EC3 The content of memory at address 3AF is 932E3 } { x >Celsius Fahrenheit is a scale commonly used to measure temperatures in the United States

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Todemonstrateconcavity let(w, x)and(w',x')betwocostminimizingpricefactorcombinationsand let w"=tw (1t)w'forany 0 ≤ t≤ 1 Concavity impliesthatC(w" y)≥ tC(w, y) (1t)C(w', y) Wecan prove this as follows We have that C(w" y) = w"
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Physics1214GeneralPhysicsII Midterm Solution 1 Three point charges are arranged on a line Charge q 3 = 500 nC and is located at the origin Charge q 2 = −300 nC and is at x = 400 cm Charge q 1 is at x = 0 cm What is q 1 (magnitude and sign) if the net force on qA Domain { x is a real number }, Range { y <Want X C = X L to minimize Z, so must decrease X C So we must add 155μF capacitance to maximize power XfCC C =Ω= =150 1/2 177π μF XX C CLnew new==Ω =80 332μF

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NOAANWSALERTSNY1261B8F37DE4RipCurrentStatement1261B9017C50NYOKXCFWOKX131e6a2d6cd7706baef4d9 wnwswebmaster@noaagov T ActualSince we want the coefficient of x3 to be equal to 6, we need 8y − 16 3 =6, or y = 17 4 =425 X With such y, the polynomial becomes P3(x)= 6x3 −x2 17x We can check whether this polynomial interpolates function f, that is, whether we got the0, we have that (c f)(x) = f(x)c is de ned for all x 2R, and furthermore, this is clearly positive Therefore, c f is in V (1) f g = g f To check that two functions are equal, we need to check that they are the same at all x 2R Thus, we need to

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Follows that f(x) f(b)(they may be equal) and f(y) f(c) (may be equal) Because b and c are at the deepest level of the tree we know that d(b) d(x)and d(c) d(y) Now switch the positions of xand bin the tree resultingin a different tree T0 and see how the cost changes Since T is optimum,–4 } { x(b) n(Ac), where A = {x xisanevennumberfrom1to10} (c) n(B), where B = {1,3,9} (d) n(?) Addition Rule for Sets Very Useful Formula If A and B are finite sets then n(AB)=n(A)n(B)n(A\B) 2 If n(B)=13,n(AB)=24,andn(A\B)=6,findn(A) 8 Fall 17, Maya Johnson 3 In a survey of 272 people, a pet food manufacturer found that 69 owned a dog but

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9 Fourier Transform Properties Solutions to Recommended Problems S91 The Fourier transform of x(t) is X(w) = x(t)e jw dt = fe t/2 u(t)e dt (S911) Since u(t)50/ eJ rIni _____ A) J1e 14 CD n CD (ID (ID cJ 0 0 I) cc J (N 0 CD LID CD IN \J'$_ U \J) ii Il 2221 Consider the differential equation dx/dt = kx — (a) If k 0, show that the only critical value c = 0 of x is stable (b) If k >A P S E E T M B P S E E T M C P S E E T M D P S E E T M 37 Tyrone wants to spend at most $10,000 on two televisions, R and S Each television must cost at least $3,000, and television R must cost at

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