Multivariable Calculus (Math 214) Assignment

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

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

Find the sum A + B.
Find the difference A - B.
Find the norms |A| and |B|.
Find the distance of A from B.
What is the angle at the origin in the triangle with vertices at A, B, and the origin?

2. Let f be the function that is defined by the formula

f(x, y) = y^{2} - y + x^{3} - 2 x^{2} + 1 .

Find the two partial derivatives of f.
For fixed a and b let g be the function given by the relation g(t) = f(ta, tb). Find the Taylor series of g.

3. Let

f(x, y, z) = x^{2} + y^{2} - z^{2} .

Find the three partial derivatives of f.
Describe the plane curve given by the equation f(x, y, 1) = 0.
Describe the plane curve given by the equation f(x, 0, z) = 0.
Describe the plane curve given by the equation f(x, 1, z) = 0.
What can be said about the set of all solutions of the equation f(x, y, z) = 0 ?

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