Question

Consider the following. f(x, y, z) = xe5yz, P(1, 0, 2), u=1/3,-2/3,2/3. (a) Find the gradient of f. ∇f(x, y, z) = (b) Evaluate the gradient at the point P. ∇f(1, 0, 2) = (c) Find the rate of change of f at P in the direction of the vector u. Duf(1, 0, 2) =

Answer #1

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f(x, y, z) =
xe4yz, P(1, 0, 3),
u = <2/3, -1/3, 2/3>
(a) Find the gradient of f.
∇f(x, y, z) =
< , , >
(b) Evaluate the gradient at the point P.
∇f(1, 0, 3) = < , ,
>
(c) Find the rate of change of f at P in the
direction of the vector u.
Duf(1, 0, 3) =

Consider the following.
f(x, y) = x/y, P(4,
1), u =
3
5
i +
4
5
j
(a) Find the gradient of f.
(b) Evaluate the gradient at the point P.
(c) Find the rate of change of f at P in the
direction of the vector u.

In the following functions: a) Find the gradient of f. , b)
Evaluate
the gradient at point P. and
c) Find the rate of change of f in P, in the direction of
vector.
1- f(x. y) = 5xy^2 - 4x^3y, P( I , 2), u = ( 5/13, 12/13 )
2- f(x, y, z) = xe^2yz , P(3, 0, 2), u = (2/3, -2/3, 1/3)

Suppose ?(?,?)=??f(x,y)=xy, ?=(−4,−4)P=(−4,−4) and
?=3?+2?v=3i+2j.
A. Find the gradient of f.
∇?=∇f= ?+i+ ?j
Note: Your answers should be expressions of x and y; e.g. "3x -
4y"
B. Find the gradient of f at the point P.
(∇?)(?)=(∇f)(P)= ?+i+ ?j
Note: Your answers should be numbers
C. Find the directional derivative of f at P in the direction of
?v.
???=Duf=
D. Find the maximum rate of change of f at P.
E. Find the (unit) direction vector in which the maximum rate...

Let f(x, y) = x^2 ln(x^3 + y).
(a) Find the gradient of f.
(b) Find the direction in which the function decreases most
rapidly at the point P(2, 1). (Give the direction as a unit
vector.)
(c) Find the directions of zero change of f at the point P(2,
1). (Give both directions as a unit vector.)

For f(x,y,z) = sqrt(35-x^2-4y^2-2z) 1. Find the gradient of
f(x,y,z) 2. Evaluate delta f(x,y,z) 3. Find the unit vectors U+ and
U- , that give the direction of steepest ascent and the steepest
descent respectively.

Consider the function F(x, y, z) =x2/2−
y3/3 + z6/6 − 1.
(a) Find the gradient vector ∇F.
(b) Find a scalar equation and a vector parametric form for the
tangent plane to the surface F(x, y, z) = 0 at the point (1, −1,
1).
(c) Let x = s + t, y = st and z = et^2 . Use the multivariable
chain rule to find ∂F/∂s . Write your answer in terms of s and
t.

let
f(x,y) = 2xy+4y^2
a) find the rate of change f at the point P(3,2) in the
direction of u= [1,3]
b) in what direction does f have the maximum rate of change?
what is the maximum rate id change?

Consider the surface defined by z = f(x,y) = x+y^2+1.
a）Sketch axes that cover the region -2<=x<=2 and
-2<=y<=2.On the axes , draw and clearly label the contours
for the eights z=0 ,z=1,and z=2.
b)evaluate the gradients of f(x,y) at the point (x,y) = (0.-1),
and draw the gradient vector on the contour diagrqam .
c)compute the directional derivative at(x,y) = (0,-1) in the
direction V =<2,1>.

] Consider the function f : R 2 → R defined by f(x, y) = x ln(x
+ 2y). (a) Find the gradient of f(x, y) at the point P(e/3, e/3).
(b) Use the gradient to find the directional derivative of f at
P(e/3, e/3) in the direction of the vector ~u = h−4, 3i. (c) Find a
unit vector (based at P) pointing in the direction in which f
increases most rapidly at P.

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