Question

Suppose f(x,y)=sqrt(tan(x)+y) and u is the unit vector in the
direction of 〈2,−1〉. Then,

(a) ∇f(x,y)=∇f(x,y)=

(b) ∇f(0.4,9)=∇f(0.4,9)=

(c) fu(0.4,9)=Duf(0.4,9)=

Answer #1

Given f(x,y) = 2y/sqrt(x) , find a unit vector u =
(u1,u2) such that Duf(1,3) = 0 and
u1 > 0

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, 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) =

1. Let f(x, y) = 2x + xy^2 , x, y ∈ R.
(a) Find the directional derivative Duf of f at the point (1, 2)
in the direction of the vector →v = 3→i + 4→j .
(b) Find the maximum directional derivative of f and a unit
vector corresponding to the maximum directional derivative at the
point (1, 2).
(c) Find the minimum directional derivative and a unit vector in
the direction of maximal decrease at the point...

Solve the following
a)y=tan^-1 (sqrt((x+1)/(x+2)
b)y=ln(sin^-1(x))
c)d/dx[sec^-1(x)]=1/(x(sqrt(x^2 -1)
d)Find Y' tan^-1(x^2 y)=x+xy^2

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.

Find the directional derivative of the function f (x, y) =
tan−1(xy) at the point (1, 3) in the direction of the unit vector
parallel to the vector v = 4i + j.

Calculate the derivative of f(x,y) = tan^-1(x/y) in the
direction (sqrt3,1)

In regards to the function f(x,y) = xye^y+2x. Find Duf(1,0) in
the direction of the vector <3,4>. And determine the greatest
slope along the surface that occurs at the point (1,0).

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)

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