Calculating Rate of ChangeAt 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 ^ ?

Question

Calculating Rate of ChangeAt 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      ^      ?

Miah Carlson · Accepted Answer

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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