Answer:
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Question
Find a unit vector in the direction in which
f increases most rapidly at P and give the rate of change of f
in that direction; Find a unit vector in the direction in which f
decreases most rapidly at P and give the rate of change of f in
that direction.
f (x, y, z) = x²z e^y + xz²; P(1, ln 2, 2).
Explanation:
The function, z = f(x, y,z), increases most rapidly at (a, b,c) in the
direction of the gradient and decreases
most rapidly in the opposite direction
Given that
F=x²ze^y+xz² at P(1, In2, 2)
1. F increases most rapidly in the positive direction of ∇f
∇f= df/dx i + df/dy j +df/dz k
∇f=(2xze^y+z²)i + (x²ze^y) j + (x²e^y + 2xz)k
At the point P(1, In2, 2)
Then,
∇f= (2×1×2×e^In2+2²)i +(1²×2×e^In2)j +(1²e^In2+2×1×2)
∇f=12i + 4j + 6k
Then, unit vector
V= ∇f/|∇f|
Then, |∇f|= √ 12²+4²+6²
|∇f|= 14
Then,
Unit vector
V=(12i+4j+6k)/14
V=6/7 i + 2/7 j + 3/7 k
This is the increasing unit vector
The rate of change of f at point P is.
|∇f|= √ 12²+4²+6²
|∇f|= 14
2. F increases most rapidly in the positive direction of -∇f
∇f=- (df/dx i + df/dy j +df/dz k)
∇f=-(2xze^y+z²)i - (x²ze^y) j - (x²e^y + 2xz)k
At the point P(1, In2, 2)
Then,
∇f= -(2×1×2×e^In2+2²)i -(1²×2×e^In2)j -(1²e^In2+2×1×2)
∇f=-12i -4j - 6k
Then, unit vector
V= -∇f/|∇f|
Then, |∇f|= √ 12²+4²+6²
|∇f|= 14
Then,
Unit vector
V=-(12i+4j+6k)/14
V= - 6/7 i - 2/7 j - 3/7 k
This is the increasing unit vector
The rate of change of f at point P is.
|∇f|= √ 12²+4²+6²
|∇f|= 14
