Question

Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) = 2 sin(xy), (0, 5)

Answer 1

Answer:

10

Step-by-step explanation:

Given the function f(x, y) = 2 sin(xy) at (0,5), the maximum rate of change of the function at that point will occur in the direction $\nabla f(0,5)$

$\nabla f(x, y) = \frac{\delta(2sin(xy)) }{\delta x} i + \frac{\delta(2sin(xy)) }{\delta y} j\\\\\nabla f(x, y) = 2ycos(xy)i + 2xcos(xy)j\\\\\nabla f(0, 5) = 2(5)cos(0*5)i + 2(0)cos(0*5)j\\\\\nabla f(0, 5) = 10cos0i+ 0j\\\\\nabla f(0, 5) = 10i+0j\\\\||\nabla f(0, 5)|| = \sqrt{10^2+0^2} \\\\||\nabla f(0, 5)|| = \sqrt{100} = 10$

Hence, the magnitude of the maximum rate of the function at the point (0, 5) is 10