Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 12910. Find the tangent plane and normal line to the following surfaces at the points shown (seeProblem 9):a) z = x2 + y2 at (I, 1,2),. ,b) z = Jw at (:, 3, f),c) z = at(2, 1,2),d) z = log(x2 + y2) at (i, $, 0).11. For each of the following curves (represented by intersecting surfaces), find the equationsof the tangent line at the point indicated, verifying that the point is on the curve:a) 2x + y - z = 6, x + 2y + 22 = 7 at (3, 1, 1):b) x2 + y2 _f z2 = 9, x2 + y2 - 8z2 = 0 at (2,2, 1);c) x2+y2= l,x+y+z=Oat(l,O,-1);d) x2 + y2 + z2 = 9, x2 + 2y2 + 3z2 = 9 at (3,0,0).Why does the procedure break down in (d)? Show that solution is impossible.12. Show that the curveis tangent to the surfaceat the polnt (1, 1, 1).2 2x2-y +z =1,xyz - x2 - 6 y = -6xy+xz=213. Show that the equation of the plane normal to the curveF(x, y, z) = 0, G(x, y, z) = 0at the point (xl, yl, zl) can be written in the vector formdr . grad F x grad G = 0,and write out the equation in rcctangular coordinates. Use the results to find the normalplanes to the curves of Problem 1 I(b) and (c) at the points given.14. Determine a plane normal to the curve x = t2, y = t, z = 2t and passing through thepoint (1.0, 0).15. Find the gradient vectors of the following functions:a) F = x2 + y2 + z2 b) F =2x2+y2Plot a level surface of each function and verify that the gradient vector is always normalto the level surface.16. Three equations of form x = f (u, v), y = g(u, v), z = h(u, v) can be considered asparametric equations of a surface, for elimination of u and v leads in general to a singleequation F(x, y, z) = 0.a) Show that the vector3 "is normal to the surface at the point (XI, yl, zl), wherexl = f (uI, vl), yl = g(ur, VI),zl = h(u1, VI) and the derivatives are evaluated at this point.