Mathematics 241 – Multivariable Calculus Make-up Exam 1 October 3, 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 = (1, 0, −1), Q = (3, 2, 0), and R = (2, −1, 6) in R^{3}. A) (10) Give a parametrization of the line segment from P to Q including the proper

range of t-values. B) (5) Is R on the line from part A? (Take the whole line here, not just the line segment.) B) (15) Find the equation of the plane containing R and perpendicular to the line from

part A. C) (10) What is the angle between v = R − Q and w = P − Q?

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

\frac{x · i}{x · j} = 1.

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{x^{3}+3x^{2}y}{x^{2}+4y^{2}} & if (x, y) ≠ (0, 0) \\ 0 & if (x, y) = (0, 0) \end{matrix}

A) (5) What would have to be true to say f is continuous at (0, 0)? B) (10) If we let (x, y) tend to (0, 0) along any curve y = mx, where m > 0, what is the

limiting value? Does your answer here answer the question of whether f is continuous at (0, 0)? Explain.

IV. All parts of this question refer to the function f : R^{2} → R defined by f(x, y) = sin (x^{2} + y^{2}). A) (10) Find the partial derivative \frac{∂f}{∂x} at (x, y) = (0, \sqrt{π}). B) (10) If we used cylindrical coordinates in R^{3} instead of rectangular coordinates, what

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

more than one connected piece!)

Extra Credit (10) Let S be the surface in R^{3} defined by 2x^{2} − 8y^{2} − 4z^{2} = 8. Describe the slices of S in planes of each of the forms x = a, y = b and z = c.