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- 1. Compute E yz dV , where E is the region above z 0, below z y, and inside x 2 y 2 4What is the absolute extrema of Q(y,z) = y2z2 on a region with vertices at the points (0,0), (0,4) and (4,0)?(2) Solve the initial boundary value problem in the region x ∈ R, t > 0.ut + 2xtux = e^(−t), u(x, 0) = f (x).
- Find the critical point of ƒ(x, y) = xy + 2x - ln x2y in the open first quadrant (x >0, y>0) and show that ƒ takes on a minimum there.Solve the initial boundary value problem in the region x in R t greater than 0. u_t + 2tu_x = e^(2t) u(x,0) = f(x).It posits only ∫ ∫ f(x,y)dA in the region bounded by; the x-axis, y - 2 = 0,y + x - 5 = 0 and x = y.
- Approximate the critical points of g(x) = x cos−1 x and estimate the maximum value of g.Show that if φ is continuously differentiable in a given region V and on itsboundary S, then∫S φ dS =∫V ∇φ dVIf we rotate the region under the curve y = 8 between x = 2 and x = 3 about the x-axis, what answer should the Shell Method give us?
- apply the separation of variables method, to find nontrivialsolutions to the mixed boundary value problem associated with the heatequation { ut = uxx, 0 < x < 6, t > 0,u(0, t) = 0, ux(6, t) = 0, t > 0Find the absolute maximum and absolute minimum values of the function f(x,y)=x2+y3 on the region bounded by x2+3y=4 and the x-axis.Let f(x) = 1/[π(1 + x2)], −∞< x < ∞ be the pdf of the Cauchy random variable X. Showthat E (X) does not exist. (hint: split into two integrals (−∞,0) and (0,∞))