Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MATH-4600

Due: Thursday September 21, 2023

Problem Set 3

Submissions are due in the LMS by the end of the day.

As a general hint, it will be extremely useful for you to employ a tool such as MAPLE, Mathematica, or even Matlab to help you visualize. They are all extremely powerful tools, and you have access via licenses through RPI ... take advantage! Note that my personal preferences is MAPLE, but others make other choices.

1. In PS1, #3 we found the intersection of the two surfaces S1 defined by 2x 2 + 3y 2 + 4z 3 = 9, and S2 defined by 2x 2 + 3y 2 + 4z 3 − 3x = 6. Call this curve C.

(a) Using a formulation with a single Lagrange multiplier, determine the minimum distance from the origin to C.

(b) Using a formulation with two Lagrange multipliers, determine the minimum distance from the origin to C.

2. Compute all extrema of the function f(x, y) = (x − 12 ) 2 − y 2 − x 3 4 within the region described by the curve x 2/16 + y 2 ≤ 1. Classify each point.

3. Let the function y(x) be defined implicitly from f(x, y) = (y + 1) ln(y) + x = 0.

(a) Verify that the point (x0, y0) = (−3 ln(2), 2) satisfies y(x0) = y0.

(b) Determine the 3-term Taylor polynomial of y(x) about the point x0.

(c) For which values of x does y(x) exist uniquely?

(d) (extra credit) Provide a plot of the 0 level curve of f(x, y).

4. Let x = u sin(v), and y = u cos(v), and assume f(u, v) is given. Determine fx and fy in terms of u, v, fu, and fv.

Suggested Problems form Colley

1. Section 4.1: 5, 7, 9, 11, 13, 15, 17, 19, 27, 29, 31, 41, 43

2. Section 4.2: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 29, 31, 33, 35, 37, 39, 47, 49, 51

3. Section 4.3: 1, 3, 5, 7, 9, 11, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39