Question

The temperature on a surface can be described by the equation T(x, y, z) = 2xy + xy2 z + 12e 2z . Use this function to answer the following questions. Remember to clearly organize your work and indicate your final answer.

(a) Find the direction you should head to achieve the greatest rate of change in temperature at any point.

(b) Find the direction you should head to achieve the greatest rate of change in temperature if you are at P(3,2,3).

(c) Find the maximum rate of change in temperature at P(3,2,3)

(d) Find the rate of change in temperature as you head from P(3,2,3) to Q(0,6,3)

Answer #1

Consider the mountain known as Mount Wolf, whose surface can be
described by the parametrization r(u, v) = u, v, 7565 − 0.02u2 −
0.03v2 with u2 + v2 ≤ 10,000, where distance is measured in meters.
The air pressure P(x, y, z) in the neighborhood of Mount Wolf is
given by P(x, y, z) = 26e(−7x2 + 4y2 + 2z). Then the composition
Q(u, v) = (P ∘ r)(u, v) gives the pressure on the surface of the
mountain...

The temperature at a point (x, y, z) is given by T(x, y, z) =
400e−x2 − 5y2 − 9z2 where T is measured in °C and x, y, z in
meters.
(a) Find the rate of change of temperature at the point P(2, −1,
2) in the direction towards the point (3, −5, 6). °C/m
(b) In which direction does the temperature increase fastest at
P?
(c) Find the maximum rate of increase at P.

The temperature at a point (x, y, z) is given by T(x, y, z) =
100e^(−x^2 − 3y^2 − 9z^2) where T is measured in °C and x, y, z in
meters. (a) Find the rate of change of temperature at the point
P(2, −1, 2) in the direction towards the point (5, −2, 3). (b) In
which direction does the temperature increase fastest at P? (c)
Find the maximum rate of increase at P.

Consider the funtion T(x,y)=2xy-y^2(°C) which determines the
temperature for a metallic circular plate centered at the origin
and radius = 5, Explain why there's no direction in which the rate
of change of temperature in the point P(1,-1) is equals to 5°C.
consider units are cm.

The temperature T of a flat sheet, at the point (?x, y) ?, is
given by
T (?x, y) ? ? x ^ 3 - 2xy + ? 2y ^ 22. a) Determine the rate of
change of temperature in the direction of the vector v = (?2, −1)
?, from the point P (? − 1,2) ?. b) Calculate the highest rate of
temperature growth from P (? − 1,2) ?, as well as the direction in
which it...

Given the level surface S defined by f(x, y, z) = x −
y3 − 2z2 = 2 and the point P0(−4,
−2, 1).
Find the equation of the tangent plane to the surface S at the
point P0.
Find the derivative of f at P0in the direction of
r(t) =< 3, 6, −2 >
Find the direction and the value of the maximum rate of change
greatest increase of f at P0;
(d) Find the parametric equations of the...

please show me the steps
Suppose the temperature at (x, y, z) is given by T = xy +
sin(yz). In what direction should you go from the point (1, 1, 1)
to decrease the temperature as quickly as possible? What is the
rate of change of temperature in this direction?

The temperature on the surface of the sphere x^2+y^2+z^2=9 is
given by T(x,y,z)=2x+2y+z, What is the temperature at the hottest
point on the sphere? Enter your answer as a decimal.

The temperature in degrees Celsius on the surface of a metal
plate is
T (x, y) = 20 − 4x^(2) − y^(2) where x and y are measued in
centimeters.
In which direction from the point (2, −3) does the temperature
increase most rapidly?
What is this rate of increase in that direction?
c.∗ How much greater is the rate of increase from (2, −3) in the
direction of
the gradient, then the rate of change in the direction of...

The temperature at a point (x, y, z) is given by T(x, y, z) =
10e^(− 2x2 − y2 − 3z2). In which direction does the temperature
increase fastest at the point (2, 1, 3)? Express your answer as a
UNIT vector.

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