EXAMINATION 2

MATH 2331

The point value of each problem is indicated in parentheses. There are 30 points all together. Show your work or otherwise explain what you are doing. Write in complete sentences, especially when stating definitions and theorems. Begin each problem on a new page of your blue book. Put the problems in your blue book in the order given.

1. (3) For f(x, y) = 3x^{4} − x\sqrt{y} find

lim_{Δx→0} \frac{f(x + Δx, y) − f(x, y)}{Δx}

2. (3) If f(x, y) has continuous second order partial derivatives such that f_{x}(x, y) = xy^{3} and f_{y}(x, y) = ax^{2}y^{2} − 3y^{4}, what must the constant a be?

3. (4) Find an equation of the tangent plane to z = y^{2} − x^{2} at the point (−4, 5, 9).

4. (2) The attached Figure 1 shows a portion of the graph of z = f(x, y) and its tangent plane at a point (a, b, f(a, b)). The number A labels one of the quantities

f(a + Δx, b + Δy) − f(a, b) or f_{x}(a, b)Δx + f_{y}(a, b)Δy.

Which is it?

5. For the function f(x, y) = x^{2} − 4xy + 5y^{2} and the unit vector u = ⟨cos θ, sin θ⟩ find: (a) (2) The gradient of f at (x, y).

(b) (2) The directional derivative D_{u}f(x, y).

(c) (1) The second directional derivative D_{u}D_{u}f(x, y).

6. (2) Find an equation of the tangent line to the graph of 4x^{2} + 8xy + 5y^{2} = 1 at the point (1, −1).

7. (4) Find the critical points of the function

f(x, y) = −2 + 4x^{2} + 8xy + 5y^{2}

and use the second derivative test to classify them as local maxima, local minima or saddle points.

8. (3) Use the attached Figure 2 to estimate the maximum and minimum of f(x, y) = 2x^{3} + y^{3} + xy on the curve x^{2} + y^{2} = 1. Explain your reasoning.

9. (4) Use Lagrange multipliers to find the points on the surface z^{2} = xy + 1 that are closest to the origin.

Date: Friday, March 12, 1999.