Partially differentiate the following multivariate functions for each of their individual choice variables: H (z, w)...
2. Consider the pde 0 <а < о, w(z,0) — 0, w(0, t) - t> 0, xwf = 0, = t Wr = (a) Use separation of variables to show that w(x, t) exp(k(t where k is a constant. (b) Show that the above solution does not satisfy both the initial and boundary conditions. (c) Use Laplace Transforms to solve the above pde.
2. Consider the pde 0
2. [20 points] A circuit with 4 inputs has to realize the following 3 functions z, w)-n (0, 1,3,4,9, 11) g (x, y,z, w)-2 (5, 8,9, 10, 11, 12, 13, 14, 15) In what follows the cost a circuit is defined as: Number of gates used + mumber of inputs to these es but not counting NOTs. So, assume that input variables are available in both complemented and un-complemented forms. (a) [10 points] Find simple SOP expressions using K-maps for...
Suppose that the covariates Xj,i for i 1, 2, , n and j 1, 2, , indicator variables for a single categorical variable in the manner covered in the course. Thus, suppose that for each individual i = 1,2,…,n we have that X1.i, X2.i,...,Xd,i this one is equal to the number 1. Let Bk be the (A , . . . , β 1), the minimizer of L (bi , b2, . . . ,勿of eq. (B. = Yn.(k), where...
Consider 2 firms with the following 2 different production functions (i.) y(L,K) = aL + bK (ii.) y(L,K) = L^0.5K^0.5 where y denotes the quantity produced and L and K are the amount of labor and capital, respectively. a. Assume K is fixed at 100. Do these production functions exhibit decreasing marginal products of labor? b. Assume K can be freely chosen. Do these production functions exhibit constant returns to scale? c. For each of the production functions, draw the...
Write each of the following functions in the form w = u(x,y) + iv(x,y) : h(z) = (3z + 2i)/(4z^2 + 5) Find limit: lim z→2+2i |z^2 − 4| =
12. Let g(x), h(y) and p(z) be functions and define f(x, y, z) = g(x)h(y)p(2). Let R= = {(x, y, z) E R3: a < x <b,c sy <d, eszsf} where a, b, c, d, e and f are constants. Prove the following result SS1, 5100,2)AV = L*()dx ["Mwdy ['Plzdz.
PrOBleM: SoLuTiONS To THE WAvE EQuATION a) By direct substitution determine which of the following functions satisfy the wave equation 1. g(z, t)-A cos(kr - wt) where A, k, w are positive constants 2. h(z,t)-Ae-(kz-wt)2 where A, k, ω are positive constants 3. p(x, t) A sinh(kx-wt) where A, k,w are positive constants 4. q(z, t) - Ae(atut) where A,a, w are positive constants 5. An arbitrary function: f(x, t) - f(kx -wt) where k and w are positive constants....
Consider these three moment generating functions, for X, Y and Z: (5 points each) m (t)=W-3 m, (t)=e + m,(t)=eW-7 a. What is the mean of X? b. What is the mean of Y? c. What is the mean of Z? d. What is the variance of X? e. What is the variance of Y? f. What is the variance of Z? Consider independent random variables X and Y with the following pmfs: y=1 (0.5 x=1 S(x)= {0.5 x =...
1. Find 8 different 2-level minimized circuits to realize each of the following functions. 1. F(W,X,Y,Z) = {m (2,4,6,7,12,14,15) 2. G(W,X,Y,Z) = (x + Y' + Z) (X' + Y + Z) W • Using algebraic techniques • Using network conversion
Mark which statements below are true, using the following Consider the diffusion problem u(0,t)=0, u(L,t)=50 where FER is a constant, forcing term Any attempt to solve this using separation of variables fails. This is because the PDE is not homogeneous. A more fruitful approach arises from splitting the solution into the sum of two u(z,t) = X(z)T(t) + us(z), where the subscript designates the function as the steady limit and does not represent a derlvative. BEWARE: MARKING A STATEMENT TRUE...