Question

Find the derivative of the function at P0 in the
direction of u.

f(x, y) = -3x2 - 10y, P0(-8,
9), u = 3i - 4j

Answer #1

Calculate the directional derivative at point p in the direction a.
1) f (x, y) = (x ^ 2)*(y); p = (1,2); a ⃗ = 3i-4j
2) f (x, y, z) = (x ^ 3)*(y) - (y ^ 2)*(z ^ 2); p = (- 2,1,3); a ⃗ = i-2j + 2k

16.
a. Find the directional derivative of f (x, y) = xy at P0 = (1,
2) in the direction of v = 〈3, 4〉.
b. Find the equation of the tangent plane to the level surface
xy2 + y3z4 = 2 at the point (1, 1, 1).
c. Determine all critical points of the function f(x,y)=y3
+3x2y−6x2 −6y2 +2.

Find the directional derivative of the function at P in the
direction of v. f(x, y) = x3 − y3, P(8, 5), v = 2 2 (i + j)

find the derivative of the function f at the given
point P in the given direction u.
f(x,y,z)=zarctan(y/z)
P(3,-3,1)
u=4i-j+3k

. 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?

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)

Consider the function f(x, y) = 3e^(2y)cos x.
(a) Find the value of the direction derivative of f at the point
(1, 0) in the direction
of the point (2, 1).
(b) Find the direction of maximum increase of f from the point (π,
1).
Find the rate of that max increase.

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?

] 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.

Find the directional derivative of the function
f(x,y)=x^6+y^3/(x+y+6 ) at the point (2,-2) in the direction of the
vector < - 2 ,2>.
b) Also find the maximum rate of change of f at the given
point and the unit vector of the direction in which the maximum
occurs.

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