ASSIGNMENT 3 1. [4 marks] Find the unit tangent vector T and the principal unit normal vector N for the given value of t: x = cosh t, y = sinh t, z = t; t = ln 2. 2. [3 marks] Find the curvature at...

ASSIGNMENT 3 1. [4 marks] Find the unit tangent vector T and the principal unit normal vector N for the given value of t: x = cosh t, y = sinh t, z = t; t = ln 2. 2. [3 marks] Find the curvature at the indicated point: x = e t , y = e -t , z = t; t = 0. 3. [3 marks] Show that for a plane curve described by y = f(x) the curvature ?(x) is ?(x) = |d 2 y/dx2 | [1 + (dy/dx) 2 ] 3/2 [Hint: Let x be the parameter so that r(x) = xi + yj = xi + f(x)j.] 4. [3 marks] Let f(x, y) = x + (xy) 1/3 . Find (a) f(t, t2 ), (b) f(x, x2 ), (c) f(2y 2 , 4y). 5. [2 marks] Find g(u(x, y), v(x, y)) if g(x, y) = y sin(x 2 y), u(x, y) = x 2 y 3 , v(x, y) = pxy. 6. [3 marks] Using the inverse function theorem, prove that the function y = x cos x has an inverse function g that is defined in some neighbourhood of 0 and such that g(0) = 0. Compute the derivative of g at 0. 7. [2 marks] Determine the level surfaces passing through the point (1, 1, 1). (a) f(x, y, z) = 3x - y + 2z (b) f(x, y, z) = z - x 2 - y 2 .