Question

f(x,y,z) = xe^{y+z}.

(a) Find the gradient of f, ∇f.

(b) Find the directional derivative of f at the point (2, 1, 2) in the direction of ? = 3? + 4?.

Answer #1

Find the gradient ∇f and the directional derivative at the point
P (1,−1,2) in the direction a = (2,−1,1) for the function f (x,y,z)
= x^3z − y(x^2) + z^2. In which direction is the directional
derivative at P decreasing most rapidly and what is its value?

find the directional derivative of f(x, y, z) = xy arctan (z) at
the point (1, 1, pi/4) in the direction <1, -1, 2>.

find the directional derivative of f(x,y) = x^2y^3 +2x^4y at the
point (3,-1) in the direction theta= 5pi/6
the gradient of f is f(x,y)=
the gradient of f (3,-1)=
the directional derivative is:

Find the directional derivative of the function f(x, y, z) = 4xy
+ xy3z − x z at the point P = (2, 0, −1) in the direction of the
vector v = 〈2, 9, −6〉.

Find the value of the directional
derivative of the function w = f ( x , y , z ) = 2 x y + 3 y z
- 4 x z
in the direction of the vector v =
< 1 , -1 , 1 > at the point P ( 1 , 1 , 1 ) .

a= 1, b =6 , c= 4
T(x,y,z) =(x^2 )y +(y^2 )z
Find the directional derivative of T at the point (a − 1, a, a +
1) along
the direction of the vector < a, b, c >.

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?

Suppose that f(x,y)=xy. Find the directional derivative of
f(x,y) in the directional 〈−6,3〉 and at the point (x,y)=(1,−4).
Answer exactly or round to 2 decimal places.

Find the gradient of the function and the max value of the
directional derivative at the given point.
f(x,y) = y2ex^(-2)y
+ x3y at the point (1,1)

Use the gradient to find the directional derivative of the
function at P in the direction of
PQ.
f(x, y) = 3x2 -
y2 + 4, P(7,
7), Q(2, 6)

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