Use the Chain Rule to find the indicated partial derivatives. N
= p + q p + r , p = u + vw, q = v + uw, r = w + uv; ∂N ∂u , ∂N ∂v ,
∂N ∂w when u = 4, v = 2, w = 8
Given a function φ(z) with z = x+iy let
U(x, y) = ½ [φ(x+iy) + φ(x-iy)] and V(x, y) = i/2
[φ(x+iy) –φ(x-iy)]
A) For φ(z) = z2 find U and V and their induced
vector fields E =▼U and
F =▼V also show that ▼2U =
▼2V = 0
B) Repeat for f(z) = z3
C) For f(z) = ln z we get U(x, y) = ½ ln
(x2+y2) and V(x, y) = arctan (y/x) Find ▼U
(electrostatic...
Consider the following utility function U(x,y)=x^r+
y^r,0<r<1 Find the income effect and substitution effect? Does the substitution effect increase or decrease when
r increases?
Given function f(x,y,z)=x^(2)+2*y^(2)+z^(2), subject to two
constraints x+y+z=6 and x-2*y+z=0. find the extreme value of
f(x,y,z) and determine whether it is maximum of minimum.
2.48. Is it true that {ax + by + cz|x, y, z ∈ Z} = {n · gcd(a,
b, c)|n ∈ Z}?
2.49. What are all the integer values of e for which the
Diophantine equation 18x + 14y + 63z = e has an integer solution.
Find a solution for each such e.
2.50. For integers a, b and k > 0, is it true that a | b iff
a^k | b^k ?
Suppose a consumer's utility function is given by U ( X , Y ) =
X 1 2 Y 1 2. The price of X is PX=8 and the price of Y
is PY=5. The consumer has M=80 to spend.
You may find that it helps to draw a graph to organize the
information in this question. You may draw in the blank area on the
front page of the assignment, but this graph will not be
graded.
a) (2...
Let z=e^(x) tan y.
a. Compute the first-order partial derivatives of z.
b. Compute the second-order partial derivatives of z.
c.∗ Convert z = f(x,y) into polar coordinates and then compute
the first- order partial derivatives fr and fθ by directly
differentiating the com- posite function, and then using the Chain
Rule.