Calculus II, Spring 2016 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 10

1. Section 14.7: Problems 31, 35, 39, 61,

2. Section 14.7: Problems 44. Do this problem again using the gradient analysis (plotting ∇f near the critical point to determine whether the critical point is a local minimum, local maximum or neither).

3. Section 14.8: Problems 1, 23, 27, 33, 35.

4. Section 14.10: Problems 3, 7, 9, 12.

5. Continue on problem 12: Give a formula for (\frac{∂u}{∂x})_{y} where u = u(x, y, z, w) and f(x, y, z, w) = 0, g(x, y, z, w) = 0.

6. Follow up on example 5 of section 14.8: Let f(x, y, z) = x^{2} + y^{2} + z^{2} and compute \frac{df}{dx} at P_{1} = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 1 − \sqrt{2}), subject to the two constraints g_{1}(x, y, z) = x^{2} + y^{2} − 1 = 0 and g_{2}(x, y, z) = x + y + z − 1 = 0. That is, evaluate \frac{d}{dx}f(x, y(x), z(x)) at x = \frac{\sqrt{2}}{2}, where y = y(x) and z = z(x) are implicitly given by the constraints g_{1} = 0 and g_{2} = 0. Can you explain why you get \frac{df}{dx} = 0 there?