Gradient fields on curves For the potential function φ and points A , B , C , and D on the level curve φ ( x, y ) = 0, complete the following steps. a. Find the gradient field F = ∇ φ . b. Evaluate F at the points A , B , C , and D . c. Plot the level curve φ ( x , y ) = 0 and the vectors F at the points A , B , C , and D . 46. φ ( x , y ) = 32 − x 4 − y 4 32 ; A (2, 2), B (−2, 2), C (−2, −2), and D (2, −2)
Gradient fields on curves For the potential function φ and points A , B , C , and D on the level curve φ ( x, y ) = 0, complete the following steps. a. Find the gradient field F = ∇ φ . b. Evaluate F at the points A , B , C , and D . c. Plot the level curve φ ( x , y ) = 0 and the vectors F at the points A , B , C , and D . 46. φ ( x , y ) = 32 − x 4 − y 4 32 ; A (2, 2), B (−2, 2), C (−2, −2), and D (2, −2)
Gradient fields on curves For the potential function φ and points A, B, C, and D on the level curve φ(x, y) = 0, complete the following steps.
a. Find the gradient field F = ∇φ.
b. Evaluate F at the points A, B, C, and D.
c. Plot the level curve φ(x, y) = 0 and the vectors F at the points A, B, C, and D.
46.
φ
(
x
,
y
)
=
32
−
x
4
−
y
4
32
; A(2, 2), B(−2, 2), C(−2, −2), and D(2, −2)
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
A. Find the gradient of f.
Vf
Note: Your answers should be expressions of x and y; e.g. "3x - 4y"
B. Find the gradient of f at the point P.
(Vƒ) (P) =
Note: Your answers should be numbers
Suppose f (x, y) = , P = (1, −1) and v = 2i – 2j.
=
C. Find the directional derivative of f at P in the direction of V.
Duf =
Note: Your answer should be a number
D. Find the maximum rate of change of f at P.
Note: Your answer should be a number
u=
E. Find the (unit) direction vector in which the maximum rate of change occurs at P.
Describe the two main geometric properties of the gradient V f.
Precalculus Enhanced with Graphing Utilities (7th Edition)
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