Let f(x, y) = x^2 ln(x^3 + y).
(a) Find the gradient of f.
(b) Find...
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.)
) Consider the function f(x,y)=−2x^2−y^2.
Find the the directional derivative of ff at the point (1,−3)(1,−3)...
) Consider the function f(x,y)=−2x^2−y^2.
Find the the directional derivative of ff at the point (1,−3)(1,−3)
in the direction given by the angle θ=π/2.
Find the unit vector which describes the direction in which ff
is increasing most rapidly at (1,−3).
A (–4, –1, 2), B (3, –2, –1) and C (–1, 3, –4),
AB= 7? −...
A (–4, –1, 2), B (3, –2, –1) and C (–1, 3, –4),
AB= 7? − ? − 3?
CB = 4? − 5? + 3?
AC = 3? + 5? - 2?
Question 7: Express the vector AC as the sum of two vectors: AC
= ? + ?, where ? is parallel to the vector CB and ? is
perpendicular to CB. Given that AC ∙ CB = −26 and that CB = √50,
determine the y-component of...
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...
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.
Find the distance from the point (1, 2, 3) to the
(a) XY
plane (b) Y...
Find the distance from the point (1, 2, 3) to the
(a) XY
plane (b) Y
axis
Find a unit vector that is perpendicular to
both
i + j and i -
k
Find the equation of the plane that contains the
point
(1, 2, -1) with normal
vector i – j.
f(x, y, z) =
xe4yz, P(1, 0, 3),
u = <2/3, -1/3, 2/3>
(a) Find the...
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) =
Find vector and parametric equations for:
a) the line that passes through the point P(9,-9,6) parallel...
Find vector and parametric equations for:
a) the line that passes through the point P(9,-9,6) parallel to
the vector u = <3,4,-2>
b) the line passing through the point P(6,-2,6) parallel to the
line x=2t, y = 2 - 3t, z = 3 +6t
c) the line passing through the point P(5, -2,1) parallel to the
line x = 3 - t, y = -2 +4t, z = 4 + 8t
A) Find a vector that measures 3 in the direction of the vector
2i + 3j...
A) Find a vector that measures 3 in the direction of the vector
2i + 3j - k
B) Given the vectors a = 2i + 3j - k and b = -2i + 3j + k. Find
2a 3 b and |a-b|
C) Find the vector that goes from point P (2, -1,4) to point Q
(3, -2,6)
Find the directional derivative of the function f (x, y) =
tan−1(xy) at the point (1, ...
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.
3. Consider the plane with a normal vector 〈2, 5, −1〉 which
contains the point (3,...
3. Consider the plane with a normal vector 〈2, 5, −1〉 which
contains the point (3, 5, −1), and the plane containing the points
(0, 2, 1), (−1, −1, 1), and (1, 2, −2). Determine whether the
planes are parallel, orthogonal, or neither.