Mathematics 241 – Multivariable Calculus Exam 1 September 28, 2007

Directions

Do all work in the blue exam booklet. Do not place anything you want considered for credit on this sheet. There are 100 possible points.

I. All parts of this question refer to the three points P = (0, 0, 1), Q = (1, 2, −4), and R = (1, −1, 0) in R^{3}. A) (15) Give a parametrization of the line segment from P to Q including the proper

range of t-values. Which t-value gives the point on the line segment equidistant from P and Q in your parametrization?

B) (15) Find the equation of the plane containing P, Q, R. C) (10) What is the angle between v = Q − P and w = R − P?

II. (15) Let x = (x, y, z) be the position vector of a point in R^{3}, let k = (0, 0, 1), and consider the equation

\frac{x · k}{∥x∥} = \frac{1}{\sqrt{2}}.

Rewrite the equation using spherical coordinates, and describe the set of all points (x, y, z) satisfying the equation.

III. Consider the function

f(x, y) = \{ \begin{matrix}\frac{xy−y^{2}}{x^{2}+4y^{2}} & if (x, y) ≠ (0, 0) \\ 0 & if (x, y) = (0, 0) \end{matrix}

A) (10) If we let (x, y) tend to (0, 0) along the line y = mx what is the limiting value? B) (5) Is f continuous at (0, 0)? Why or why not?

IV. All parts of this question refer to the function f : R^{2} → R defined by f(x, y) = cos(\sqrt{x^{2} + y^{2}}). A) (10) If we used cylindrical coordinates in R^{3} instead of rectangular coordinates, what

would be the equation of the graph of f? B) (10) Describe or sketch the level curve of f for the value c = 1. (Careful – there is

more than one connected piece!) C) (10) Find the partial derivative \frac{∂f}{∂x} at (x, y) = (\frac{π}{2} , 0).

Extra Credit (10) Let S be the surface in R^{3} defined by \frac{x^{2}}{9} + \frac{y^{2}}{16} − z^{2} = 1. Describe the slices of S in planes of each of the forms x = a, y = b and z = c.