Reference no: EM13632456

Use the chain rule to find and where z = arcsin (2x - y) and x = .s2 + t2, y = 1 - 3 st. Let /,) = f(u(s, t), v(s, t)) where f, u, and v are differentiable and u(l,0) = 4, v(l, 0) = 7 us(1,0) = -1,vs(l,0) = - 2 ut(1,0) = 3,vt(1,0) = 5 fu4(4,7) =6, fv(4, 7) = 1. Find w8(l, 0) and wt(1,0). Find the directional derivative of the function f(x, y, z) = xsin yz at P = (1,1, pi /2) in the direction of the vector If g(x,y) = x2/9 + y2/4, find the gradient vector and use it to find the tangent line to the level curve g(x,y) = 1 at . Sketch the level curve, t he tangent line, and the gradient vector. Find the equation of t he tangent plane to the cone x2 +y2 = z2 at the point (x0, y0. Z0) (0,0,0) using the gradient method. Note: The equation of the cone must be in level surface form. Show that this tangent plane passes through the origin. Locale the critical points of f(x,y) = x3 +y3 + 3xy + 5 and classify them using the second derivative test. Find three positive numbers whose sum is 18 and the sum of whose squares is as small as possible.