Math 222 - 2 First Midquarter Exam Name_________________ Treibergs Wednesday, January 24, 1996 This is a closed book exam. No books, papers or calculators. 1.) Let h( t ) be a space curve in R^3 with constant speed. Show that the velocity vector and acceleration vector of h( t ) are perpendicular. 2.) Let f( x, y, z ) = [ x^3 z + y^2, y exp( y + z^2 ] Find ∂f / ∂y. Find the affine function which best approximates f at the point (x,y,z) = (1,0,2). u^2 3.) Let g(u, v) = ( 1 + 4uv, 3, --------- ) u^2 + v^2 Find lim g(u, v) if it exists. (u,v)->(0,0) 4.) Find a normal vector and the equation of the tangent plane to the surface x^3 + y^3 + z^3 + 4xyz = 60 at the point (x,y,z) = (1,2,3). 5.) Let / x^3 + y^3 | -----------, if (x,y) ≠ (0,0); k(x,y) = < x^2 + y^2 | \ 0, if (x,y) = (0,0). a. Let u = (a,b). Show that there is a directional D_u k at (0,0). b. Find a candidate for the affine function a(x,y) (constant plus linear) which would best approximate k at (0,0). c. Show that k is not differentiable at (0,0).