Multivariable Calculus (Math 214)

1. Announcement

Mr. Hiroshi Tamura is now available to students in this course (division 2716 of AMAT 214) to answer questions about the subject matter of the course during the period 12:40-2:10 on Tuesdays and Fridays in the department's tutoring room ES 138.

2. Assignment due October 4, 1999

1. Let f be the function defined in the plane by

   f(x, y) = arctan ({y}/{x}) .

   1. What is the limit as t approaches 0 of f(2t, 3t) ?

   2. What is the limit as t approaches 0 of f(3t, 2t) ?

   3. What is the limit as (x, y) approaches the origin, while avoiding the line x = 0, i.e., the y-axis, of f(x, y) ?

   4. Find the gradient vector of f when x <> 0.

2. For the function of the previous exercise find the following:

   1. The gradient vector of f at the point P = (3, -4).

   2. A parametric representation of the line L in the plane that is perpendicular at P to the level set of f passing through the point P.

   3. An equation for the line L.

3. Use the notion of the ``cross product'' of two vectors to find a vector in R^{3} that is perpendicular to both of the vectors (1, -3, 2) and (-2, 1, 1).

4. Let g be the function that is defined in space by

   g(x, y, z) = x sin (yz) .

   Find the following:

   1. The gradient vector of g at the point Q = (1, pi/2, 1).

   2. A parametric representation of the line M in space that is perpendicular at the point Q to the surface S in space that is defined by the equation g(x, y, z) = 1.

   3. An equation of the plane in space that is tangent to the surface S at the point Q.

   4. Find two different planes in space that intersect in the line M.