Calculating Rate of ChangeAt the point (0,1,2) in which direction does the function f (...

szklanovqq

szklanovqq

Answered

2022-11-13

Calculating Rate of Change
At the point (0,1,2) in which direction does the function f ( x , y , z ) = x y 2 z increase most rapidly? What is the rate of change of f in this direction? At the point (1,1,0), what is the derivative of f in the direction of the vector 2 i ^ + 3 j ^ + 6 k ^ ?
I assumed that the rate of change is the same as the gradient of the function, namely f. Calculating this gave me:
f = ( x y 2 z ) x i ^ + ( x y 2 z ) y j ^ + ( x y 2 z ) z k ^
            = y 2 z   i ^ + 2 x z   j ^ + x y 2   k ^
Evaluating at point:
f ( 0 , 1 , 2 ) = 2   i ^
Hence, the function increases most rapidly in the x direction.
I am uncertain of how to approach solving the third part of the question, should I evaluate the rate of change at (1,1,0) and then find the difference between that and the vector 2 i ^ + 3 j ^ + 6 k ^ ?

Answer & Explanation

Miah Carlson

Miah Carlson

Expert

2022-11-14Added 17 answers

Hint: The directional derivative of f, in the direction of vector u , is just:
f u , or f u u
(there are different conventions, according to context).

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