Question

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.

Answer #1

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.

The temperature at a point (x,y,z) is given by
T(x,y,z)=200e−x2−y2/4−z2/9, where T is measured in degrees celsius
and x,y, and z in meters. There are lots of places to make silly
errors in this problem; just try to keep track of what needs to be
a unit vector. A. Find the rate of change of the temperature at the
point (0, -1, 2) in the direction toward the point (-1, 4, 2). b)In
which direction (unit vector) does the temperature...

The temperature at a point (x,y,z) is given by
T(x,y,z)=200e−x2−y2/4−z2/9, where Tis measured in degrees celcius
and x,y, and z in meters. There are lots of places to make silly
errors in this problem; just try to keep track of what needs to be
a unit vector.
Find the rate of change of the temperature at the point (0, 1, -2)
in the direction toward the point (-1, -2, 5).
In which direction (unit vector) does the temperature increase the...

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.

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...

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...

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 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...

f(x, y, z) = x y2 z3 and consider the
point P(2, 1, 1).
(a) Find the directional derivative of f at P in the direction
of Q(0, −3, 5).
(b) In which direction does f increase fastest at P?
(c) What is the maximal rate of increase of f at P?

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...

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