Let F(x, y, z)=2xy^2 +2yz^2 +2x^2z.
(a) Find the directional derivative of F at the point (−2, 1,
−1) in the direction parallel to the line x = 3 + 4t, y = 2 − t, z
= 3t.
Answer: 58/√26
(b) Determine symmetric equations for the normal line to the
level surface F(x, y, z)= −2 at the point (−1, 2, 1).
Answer: x = −1 + 4t, y = 2 − 6t, z = 1 + 10t
The directional derivative of a function F(x,y) at a point P
(1,-3) in the direction
of the unit vector - 3/5 i + 4/5 j is equal to - 4 while its
directional derivative
at P in the direction of the unit vector (1/root5) i + (2/root5) j
is equal to 0.
Find the directional derivative of F(x,y ) at P in the direction
from P(1,-3)
towards the point Q(5,0).
Calculate the directional derivative of
f(x,y,z)=x(y^2)+y((1-z)^(1/2)) at the point P(1,−2,0) in the
direction of the vector v = 5i+2j−k. (a) Calculate the directional
derivative of f at the point P in the direction of v. (b) Find the
unit vector that points in the same direction as the max rate of
change for f at the point P.
Find the directional derivative of f(x,y)=arctan(xy) at the point (-2,5) in the direction of maximum decrease. What is the Domain and Range of f(x,y)=arctan(xy)?
1. (a) Find the directional derivative of f at (2,1) in the
direction from (2,1) to (6,-2). Show your work clearly.
(b) Find a unit vector u such that the directional derivative of
f in the direction of u at (2,1) is zero. Show all work required to
justify your answer.
A) Find the directional derivative of the function at the given
point in the direction of vector v. f(x, y) = 5 + 6x√y, (5, 4), v =
<8, -6>
Duf(5, 4) =
B) Find the directional derivative,
Duf, of the function at the given
point in the direction of vector v.
f(x, y)
=ln(x2+y2), (4, 5),
v = <-5, 4>
Duf(4, 5) =
C) Find the maximum rate of change of f at the given
point and the direction...
1. For z=x^2-xy^2 find:
a. the gradient for z;
b. the directional derivative in direction of <3,-1>;
c.the approximation for Δz when taking a Δs =0.05 step from
(1,2) in direction of <3,-1>;
d. the approximation to the maximum Δz possible when taking a Δs
=0.05 step from (1,2).