Question

Show that the function f(z) =x^3-3xy^2+i((3x^2y-y^3) is differentiable

Answer #1

The real part of a f (z) complex function is given as
(x,y)=y^3-3x^2y. Show the harmonic function u(x,y) and find the
expressions v(x,y) and f(z). Calculate f'(1+2i) and write x+iy
algebraically.

Let f(x,y) = 3x^2y − 2y^2 − 3x^2 − 8y + 2.
(i) Find the stationary points of f.
(ii) For each stationary point P found in (i), determine whether
f has a local maximum, a local minimum, or a saddle point at P.
Answer:
(i) (0, −2), (2, 1), (−2, 1)
(ii) (0, −2) loc. max, (± 2, 1) saddle points

x^3y''' - x^2y'' - 3xy' =3x

find the total differential
(a) f(x,y)=x^2+3xy+2y
(b) f(x,y)=x-y/x+1

Find the maximum and minimum values of the function
f(x,y,z)=3x−y−3 subject to the constraints x^2+2z^2=324 and
x+y−z=−6 . Maximum value is , occuring at ( , , ). Minimum value is
, occuring at ( , , ).

F(x, y, z) =< 3xy^2 , xe^z , z^3 >, S is the solid bounded
by the cylinder y2 + z2 = 1 and the planes x
= −1 and x = 2 Find he surface area using surface integrals. DO NOT
USE Divergence Theorem. Answer: 9π/2

Find the maximum and minimum values of the function
f(x,y,z)=x+2y subject to the constraints y^2+z^2=100 and x+y+z=5. I
have: The maximum value is ____, occurring at (___, 5sqrt2,
-5sqrt2). The minimum value is ____, occurring at (___, -5sqrt2,
5sqrt2). The x-value of both of these is NOT 1. The maximum and
minimum are NOT 1+10sqrt2 and 1-10sqrt2, or my homework program is
wrong.

Consider the function g(x) = |3x + 4|.
(a) Is the function differentiable at x = 10? Find out using
ARCs. If it is not differentiable there, you do not have to do
anything else. If it is differentiable, write down the equation of
the tangent line thru (10, g(10)).
(b) Graph the function. Can you spot a point “a” such that the
tangent line through (a, f(a)) does not exist? If yes, show using
ARCS that g(x) is not...

Use
Gaussian Elimination to solve and show all steps:
1. (x+4y=6)
(1/2x+1/3y=1/2)
2. (x-2y+3z=7)
(-3x+y+2z=-5)
(2x+2y+z=3)

1. Let T(x, y, z) = (x + z, y − 2x, −z + 2y) and S(x, y, z) =
(2y − z, x − z, y + 3x). Use matrices to find the composition S ◦
T.
2. Find an equation of the tangent plane to the graph of x 2 − y
2 − 3z 2 = 5 at (6, 2, 3).
3. Find the critical points of f(x, y) = (x 2 + y 2 )e −y...

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