Multivariable Calculus (Math 214) Assignment

due September 27, 1999

1. Let A, B, and C be the points in Cartesian 3-space given by

   A = (2, -3, 5) , B = (0, 1, -1) , C = (1, -2, 4) .

   1. Find each of the three angles at the vertices of the triangle determined by the three given points.

   2. Verify explicitly that the lengths of the three sides of this triangle satisfy the classical ``Law of Cosines'' relative to the longest side as ``stand-in'' for the hypotenuse.

   3. Verify explicitly the classical law of sines for this triangle.

   4. Find the point where the three medians of this triangle meet.

2. Let C be the curve that is given by the equation

   x e^{y} - y e^{x} = 1 ,

   and let P be the point (1, 0).

   1. Find a vector that is normal (perpendicular) to C at P.

   2. Find an equation for the line tangent to C at P.

   3. What methods might be employed to sketch this curve?

3. Let C be the curve that is given parametrically by the formulas

   x = t - sin (t) , y = 1 - cos (t) ,

   and let P be the point where t = pi/4.

   1. Find a vector that is tangent to C at P.

   2. Find an equation for the line tangent to C at P.

   3. Find a parametric representation for the tangent line in the previous part.

4. For a given constant value of C consider the equation

   y^{2} - y + xy + x^{3} - 2 x = C .

   1. For given x (as well as the previously given C) under what circumstances is there a value y such that (x, y) is a solution?

   2. For given y under what circumstances is there a value x such that (x, y) is a solution?

   3. When C = 6, find an equation for each of the lines that is orthogonal to the curve given by this equation at the points of the curve where it crosses the y-axis.