For the shown system, Use the Routh criterion to determine the range of stability relative to the gain K. Note that K is always positive K> 0. Type a, b, c, d, e or f R(s) C(s) K(s? + 2s) (s2-4s + 8)(s +3) a) 1< K < 5 b) K > 5 c) 0 < K <4 d) K > 3 e) 3 < K < 8 ŋ not stable for any K Answer:

Problem #2 Calculate the Legendre transform (F1) of y = x². For your answer, give the new function F1 and its derivative dF1. (a(f(x)) Note that dy = C dx, C, = f(x), and dC, = (0) dx. dx

Find the gradient field F = Vo for the potential function o below. P(x,y.z) = In (3x +y° +z²) Vøxy.z) = (O Vo(x.y.z z)3D

Question 3 Consider the simple and close-packed hexagonal direct lattices. (a) Find out the general expression for the structure factor for each lattice type. In this context, what is the significance of a complex structure factor? (b) Without resorting to the structure factor, determine in which structure(s) X-ray Bragg reflections arise from the lattice planes (0001), (0002), and (1010). Justify your answer by the use of the above determined structure factor or otherwise and explain the meaning of the four indices describing the above cited lattice planes.