Math 174  Exam II  3/5/99  D. Gottlieb

Name

1. a) Give an example of a set in the plane which is open AND

b) Give an example of a set which is closed in the plane.

2. Do one of the two questions.

a) A function f(x, y) is continuous at the point (x_{0}, y_{0}) if what three conditions are

satisfied? OR

b) Consider the function \{ \begin{matrix}f(x, y) = \frac{x^{4}}{x^{4} + y^{2}} \\ f(0, 0) = 0. \end{matrix} for x and y not both zero. Is this

function continuous at (0, 0)?

3. a) Suppose f(x, y) and all partial derivatives are defined in an open neighborhood of a point (a, b). What does Euler’s theorem say about f_{xy}?

OR

b) Let f(x, y) = xy^{3}e^{x} cos y + y^{4} ln(x). What number does f_{xx}(f_{xy} − f_{yx}) equal when evaluated at (π/2, 1)?

4. a) Suppose f(x, y, z) is a differentiable function so that f_{x}(a, b, c) = 2, f_{y}(a, b, c) = −1 and f_{z}(a, b, c) = 0. If the values of b and c are changed by dy = 1 and dz = .1, what change dx of a will result in df = 0?

OR

b) Let f(x, y, z) = (e^{x+y})z + \sqrt{x^{2} + y^{2}} ln z. Given dy = .1, dz = .1 and df = 0 at the point (3, 2, 1), what value must dx have?

5. Suppose f(x, y, z) = x^{2} + y^{2} + z^{2} and g(x, y, z) = x + y + z. Sketch the level surfaces f(x, y, z) = f(1, −1, 1) and g(x, y, z) = g(1, −1, 1).

6. Find the gradient vectors of f(x, y, z) = x^{2} + y^{2} + z^{2} and g(x, y, z) = x + y + z at the point (1, −1, 1) and find a vector perpendicular to both of them.

7. a) Which function f(x, y, z) = x^{2} + y^{2} + z^{2}, or g(x, y, z) = x + y + z has the fastest rate of increase at (1, −1, 1).

AND

b) What is the tangent vector to the curve of intersection at (1, −1, 1)?

8. At what point is the maximum value of g(x, y, z) = x + y + z subject to the constraint f(x, y, z) = x^{2} + y^{2} + z^{2} = 3 taken on?

9. Find the equation of the tangent plane to the sphere x^{2} + y^{2} + z^{2} = \frac{1}{3} at the point

(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}).

10. Use Taylor’s formula for f(x, y) = e^{x} cos y to find a quadratic approximation near the origin.