parametric functions x(t) = (x1(t), x2(t))
parametric functions x(t) = (x1(t), x2(t))
Question 1. For each of the following parametric functions x(t) = (x1(t), x2(t)) perform the following:
a) find the tangent vector to the curve x'(t) and evaluate it at t = 1;
b) use substitution to eliminate t and write down an implicit function g(x1, x2) = 0 representing the curve and find its gradient vector, ?g(x1, x2);
c) show that the vectors x'(1) and ?g(x1(1), x2(1)) are orthogonal;
d) sketch the curve x(t) and illustrate the condition in c).
i) x(t) = (1/t, t) for t in (0, 8).
ii) x(t) = (2t, 12 – 3t) for t in [0,4].
iii) x(t) = (2 + t, 4 – t2) for t in (-8,8).
Question 2. For each of the following functions,
a) determine whether f(x1, x2) is concave, strictly concave, convex or strictly convex using the test of the definiteness of the Hessian matrix of f(x1, x2).
b) find the stationary points of f(x1, x2), and determine (if possible) whether they are maximizers or minimizers of the function.
i) f(x1, x2) = x12 + 3×22 + 5 .
ii) f(x1, x2) = 10 – x12 – x22 + x1x2 .
iii) f(x1, x2) = x12 + x22 + 2×1
Question 3. Use the method of Lagrange multipliers to find the points that solve the following problems. Check the second order conditions to ensure your proposed solution solves the problem.
i) Minimize f(x1, x2) = 2×1 + 5×2 subject to x11/2×21/2 = 20.
ii) Maximize f(x1, x2) = 2 x11/2 + x2 subject to (1/3) x1 + x2 = 100.
iii) Maximize f(x1, x2) = 6x1x2 subject to x12 + x22 = 1.
ORDER THIS ESSAY HERE NOW AND GET A DISCOUNT !!!
You can place an order similar to this with us. You are assured of an authentic custom paper delivered within the given deadline besides our 24/7 customer support all through.
Latest completed orders:
# | topic title | discipline | academic level | pages | delivered |
---|---|---|---|---|---|
6
|
Writer's choice
|
Business
|
University
|
2
|
1 hour 32 min
|
7
|
Wise Approach to
|
Philosophy
|
College
|
2
|
2 hours 19 min
|
8
|
1980's and 1990
|
History
|
College
|
3
|
2 hours 20 min
|
9
|
pick the best topic
|
Finance
|
School
|
2
|
2 hours 27 min
|
10
|
finance for leisure
|
Finance
|
University
|
12
|
2 hours 36 min
|