Question

Given the function f(x, y, z) = (x^{2} + y2 +
z^{2} )^{−1/2}

a) what is the gradient at the point (12,0,16)?

b) what is the directional derivative of f in the direction of the vector u = (1,1,1) at the point (12,0,16)?

Answer #1

Calculate ∫ ∫S f(x,y,z)dS for the given surface and function.
x2+y2+z2=144, 6≤z≤12; f(x,y,z)=z2(x2+y2+z2)−1.

Find the directional derivative of the function
f(x,y,z)=ln(x2+y2−1)+y+6z at the point (1,1,0) in the direction of
the vector v→=i→−2j→+2k→

. For the function f(x, y) = xye^x−y , at the point (2, 2)
(a) find the gradient.
(b) find the directional derivative in the direction of the
vector 3i − j.
(c) in the direction of which unit vector is the rate of
increase maximum? What is the maximum rate of increase?
(d) in the direction of which unit vector(s) is the directional
derivative zero?

the function f(x; y) = xye^x-y, at the point (2; 2) (1)find the
gradient. (2) find the directional derivative in the direction of
the vector 3i - j. (3)find the direction of which unit vector is
the rate of increase maximum? What is the maxi- mum rate of
increase? (4)find the direction of which unit vector(s) is the
directional derivative zero?

Evaluate ∫∫Sf(x,y,z)dS , where f(x,y,z)=0.4sqrt(x2+y2+z2)) and S
is the hemisphere x2+y2+z2=36,z≥0

Consider the function f(x,y) = ( x2 +
z2)ln(y)
a)Find the gradient of f.
b) Find the rate of change of f at the point (2, 1, 1) in the
direction of ?⃗ = 〈−2, 4, −4〉

For f(x, y) = x2 + 4xy - y2 at the point P(2, 1), a) find the
unit vector u in the direction of steepest ascent; b) find the unit
vector u in the direction of steepest descent; and c) find a unit
vector u that points in the direction of no change in the function.
show all work please
a)
b)
c)

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)=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...

Find the linear approximation of the function f(x, y, z) = x2 +
y2 + z2 at (6, 2, 9) and use it to approximate the number 6.012 +
1.972 + 8.982 . (Round your answer to five decimal places.) f(6.01,
1.97, 8.98) ≈

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