Question

Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) = 9 sin(xy), (0, 8)

Answer 1

Answer:

The magnitude is  $\Delta f(0,8)  =  72$

The  direction  is  $i$ i.e toward the x-axis

Step-by-step explanation:

From the question we are told that

   The function is  $f(x, y) = 9sin(xy)$

    The point considered is  $(0,8 )$

Generally the maximum rate of change of f at the given point and the direction is mathematically represented as

            $\Delta f(x,y) =  [\frac{\delta  f(x,y)}{\delta x } i + \frac{\delta  f(x,y)}{\delta y } j  ]$

             $\Delta f(x,y) =  [\frac{\delta  (9sin(xy))}{\delta x} i  + \frac{\delta  (9sin(xy))}{\delta y} i   ]$

            $\Delta f(x,y) = [9y cos (x,y) i +  9xcos (x,y) j]$

At  $(0,8 )$

            $\Delta  f (0,8) =  [9(8) cos(0* 8)i  + 9(8) sin(0* 8)j  ]$

            $\Delta  f (0,8) = 72 i$