a.maximize and minimize −2xy on the ellipse x^2+4y^2=4
b.Determine whether or not the vector function is...
a.maximize and minimize −2xy on the ellipse x^2+4y^2=4
b.Determine whether or not the vector function is the gradient
∇f (x, y) of a function everywhere defined. If so, find all the
functions with that gradient. (x^2+3y^2)i+(2xy+e^x)j
Consider the vector field F(x, y) = <3 + 2xy, x2 − 3y 2>
(b) Evaluate integral (subscript c) F · dr, where C is the curve
(e^t sin t, e^t cost) for 0 ≤ t ≤ π.
Consider the following utility function U(X,Y) = X^1/4Y^3/4
Initially
PX = 2
PY = 4
I = 120
Suppose the price of X changes to PX = 3. Perform a decomposition
and fill in the table
X
Y
Substitution Effect
Income Effect
Total Effect
Consider the optimization problem of the objective function f(x,
y) = 3x 2 − 4y 2 + xy − 5 subject to x − 2y + 7 = 0. 1. Write down
the Lagrangian function and the first-order conditions. 1 mark 2.
Determine the stationary point. 2 marks 3. Does the stationary
point represent a maximum or a minimum? Justify your answer.
For the ellipse 6? 2 + 4? 2 = 36, find the eccentricity and
sketch the graph showing all main features including axis
intercepts, foci and directrices.
b) Using exclusively some part of your answer to part a),
determine the foci and directrices for the curve: (? + 2) 2 6 + (?
− 3) 2 9 =
The function f(x, y) = 10−x 2−4y 2+2x has one critical point.
Find that critical point and show that it is not a saddle point.
Indicate whether this critical point is a maximum or a minimum, and
find that maximum or minimum value.