1. Find the are length of the curve F(t) = (cos(t),sin(t),t) from the point (1,0,0) to the point (1,0,2%). (25 pts.) 2. Let F(t) = (t,2sin(t),2cos(t)). (a) Find N. the unit normal vector at the point where t = a (15 pts.) (b) Find the curvature of i(1) at at (x,0,-2).(15 pts.) 3. For this problem, (a) Write equations for two grid curves, a and C2. where G is the grid curve passing through F(1,2) with e = 2 fixed. and C2 is the grid curve passing through r(1,2) with u = 1 fixed. (4 pts.) (b) Find the derivatives of these functions. (4 pts.) (i(u)= (2(0)= (c) Find a plane parallel to both (i(1) and (2(2). which passes through F(1.2). (12 pts) 4. Find xy' lim if it exists. If it does not exist, show that it does not exist. (25 pts.) 5. (a) If =emitation tantn, where + a², = 1. show that gu aru = u. ar? arz art In other words, show that u is equal to it's own Laplacian. (15 pts.) (b) Show that u = In(V22+47). satisfies Laplace's equation, uzz + uyy = 0. (10 pts.) 6. (5 pts each) Answer each of the following questions either true or false (not T or F). You need not justify your answers. (a) The curve given by F(t) = + -4,In(t) - 1), t> 0 is a line. (b) fxy = of (c) The osculating circle of a curve C at a point has the same tangent vector, normal vector, and curvature as C at that point. (d) If Sa(2,y) and fy(T,y) both exist, the f(x,y) is differentiable. (e) If f(x,y) is continuous at (a,b), then f(x,y) f(a,b) as (x.y) (a.b) along any line through (a,b). BONUS: This problem will only be graded if you have answered all other questions on this test. Prove that for any 6 people on Facebook, there are either 3 people who are all friends with each other or 3 people, none of whom are friends with each other. (NOTE: This is a much harder problem than the last bonus was, don't feel bad if you try hard but don't get it) (15 pts.) The above problem is an example of a result in an area of mathematics called Ramsey theory. It studies the conditions under which certain types of order must appear. In other words, it looks at when you are guaranteed to have a certain amount of order, even if everything is trying to be completely chaotic.