Question

A mountain are described with the following function f(x,y) = 3
– 3x^{2} + 3y^{2} - x^{4} – y^{4}
and are defined:

D_{f} = {(x,y) ∈ R^{2} | 3 – 3x^{2} +
3y^{2} - x^{4} – y^{4} ≥ 0}

Calculate (x,y,z)-coordinates where there is a flat surface. How can you se immediately that (0,0,3) are one of those points.

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Answer #1

Consider the function given by f(x,y) = 3x2 −6xy + 2y3 +
23.
(a) Find all critical points of f(x,y) and determine their
nature.
(b) What are the minimum and maximum values of f(x,y) on the
straight line segment given by 0 ≤ x ≤ 3, y = 2?

Use the Divergence Theorem to calculate the surface integral
S
F · dS;
that is, calculate the flux of F across
S.
F(x, y, z) = ey
tan(z)i + y
3 − x2
j + x sin(y)k,
S is the surface of the solid that lies above the
xy-plane and below the surface
z = 2 − x4 − y4,
−1 ≤ x ≤ 1,
−1 ≤ y ≤ 1.

Find the linearization of the function
f(x,y)=√(22−1x2−3y2 )at the point (-1,
2).
L(x,y)=_______
Use the linear approximation to estimate the value of
f(−1.1,2.1)=_________

Find the intervals where the function f (x) = ln(x) + 3x2 − x is
concave up or concave down. Include a sign chart indicated critical
points and test values.

The average value of a function f(x, y, z) over a solid region E
is defined to be fave = 1 V(E) E f(x, y, z) dV where V(E) is the
volume of E. For instance, if ρ is a density function, then ρave is
the average density of E. Find the average value of the function
f(x, y, z) = 5x2z + 5y2z over the region enclosed by the paraboloid
z = 4 − x2 − y2 and the...

Find all local extreme values of the function
f(x,y)=2x2−3y2−4xy−4x−16y+1.

Given the function f(x) = x3 - 3x2 - 9x + 10
Find the intervals where it is increasing and
decreasing and find the co-ordinates of the relative maximums &
minimums.
Find the intervals where it is concave up and down and
co-ordinates of any inflection points
Graph the f(x)

Calculate differentiability of f(x,y,z) = x^2 + y^2 + z^2
this function is defined in R^2

Consider the joint density function f (x, y) = 1 if 0<=
x<= 1; 0<=y<= 1. [0 elsewhere]
a) Obtain the probability density function of the v.a Z, where Z =
X^2.
b) Obtain the probability density function of v.a W, where W =
X*Y^2.
c) Obtain the joint density function of Z and W, that is, g (Z,
W)

Consider the function f(x,y) = (e^{2x})lny whose domain is
{(x,y): y>0}. What is the equation of the plane
tangent to the surface z=f(x,y) at (x,y) = (3,5)?

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