Question

Find the maximum rate of change of f at the given point and the direction in which it occurs.

f(p, q, r) = arctan(pqr), (4, 1, 1)

Answer 1

To find the maximum rate of change of f at the point (4, 1, 1) and the direction in which it occurs, we need to find the gradient vector of f at that point. The gradient vector represents the direction of maximum change of a function.

Step-by-step explanation:

To find the maximum rate of change of f at the point (4, 1, 1) and the direction in which it occurs, we need to find the gradient vector of f at that point. The gradient vector represents the direction of maximum change of a function.

To find the gradient vector, we first find the partial derivatives of f with respect to each variable, p, q, and r. Then we evaluate those partial derivatives at the point (4, 1, 1) to get the components of the gradient vector.

In this case, we have:

∂f/∂p = qrcos⁡(pqr) = (1)(1)(1)cos⁡(4(1)(1))

∂f/∂q = prcos⁡(pqr) = (4)(1)cos⁡(4(1)(1))

∂f/∂r = pcos⁡(pqr) = (4)cos⁡(4(1)(1))

Evaluated at the point (4, 1, 1), the gradient vector is (cos⁡(4), 4cos⁡(4), 4cos⁡(4)). The maximum rate of change of f at that point occurs in the direction of the gradient vector, which is (cos⁡(4), 4cos⁡(4), 4cos⁡(4)).