Calculus II, Spring 2009 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 09

Assigned April 23, 2009.

1. Section 13.6: Problem 57.

2. Section 13.7: Problems 17, 35, 36, 37, 42, 45. Try to use the gradient analysis (i.e. sketch the gradient vectors to help you find the answer) in problem 37.

3. Section 13.8: Problems 7, 8, 25, 27, 29, 31. (Use the method of Lagrange Multipliers only).

4. Sketch the level curves and gradient vectors of f(x, y) = x^{2} ± y^{2}, respectively.

5. Suppose that f_{x}(x, y) = 3x^{2} + 2x + 2y and f_{y}(x, y) = 2x + 2y. Does f has a local max, local min or a saddle point at (0, 0)? Hint: try the gradient analysis.

6. Find the equation of plane normal to the curve

\{ \begin{matrix}x^{2} + 2y^{2} + 3z^{2} = 6 \\ x + y + z = 3 \end{matrix}

at (1, 1, 1).

7. Suppose that the equation F (x, y, z) = 0 can define implicitly either x = f(y, z), y = g(z, x) or z = h(x, y) (F (x, y, z) = 2x + 3y − 4z − 1 is such an example). Show that the following identity holds:

f_{y} · g_{z} · h_{x} = −1.

This identity is sometimes written as x_{y} ·y_{z} ·z_{x} = −1, which might be a little misleading and hard to understand.

8. The following identity

∫_{a}^{b} \frac{d}{dy} f(x, y)dx = \frac{d}{dy} ∫_{a}^{b} f(x, y)dx  (1)

is valid provided the integrand is smooth enough. This is the case for most engineering applications including this homework problem. Use (1) and the Chain Rule to compute

\frac{d}{dy} ∫_{1}^{2} \frac{cos(xy)}{x}dx and \frac{d}{dy} ∫_{1+y^{2}}^{2+sin(y)} \frac{cos(xy)}{x}dx