Question

The rate of change of \( f(x, y, z) = x + x \cos z - y \sin z + y \) at \( P_0(2, -1, 0) \) in the direction from \( P_0(2, -1, 0) \) to \( P_1(0, 1, 2) \) is __________.

Hint:

The rate of change of a function in a given direction is the directional derivative, which is computed as the dot product of the gradient and the unit vector in that direction.

Answer 1

Correct Answer: 0

We are asked to find the rate of change of the function \( f(x, y, z) = x + x \cos z - y \sin z + y \) in the direction from \( P_0(2, -1, 0) \) to \( P_1(0, 1, 2) \). This is the directional derivative of \( f \) at \( P_0 \) in the direction of the vector from \( P_0 \) to \( P_1 \). Step 1: Compute the direction vector. The direction vector \( \vec{v} \) from \( P_0 \) to \( P_1 \) is: \[ \vec{v} = P_1 - P_0 = (0 - 2, 1 - (-1), 2 - 0) = (-2, 2, 2) \] Step 2: Normalize the direction vector. The magnitude of \( \vec{v} \) is: \[ |\vec{v}| = \sqrt{(-2)^2 + 2^2 + 2^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3} \] The unit vector \( \hat{v} \) in the direction of \( \vec{v} \) is: \[ \hat{v} = \frac{\vec{v}}{|\vec{v}|} = \frac{(-2, 2, 2)}{2\sqrt{3}} = \left( \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \] Step 3: Compute the gradient of \( f(x, y, z) \). The gradient of \( f(x, y, z) = x + x \cos z - y \sin z + y \) is: \[ \nabla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \] Computing each partial derivative: \[ \frac{\partial f}{\partial x} = 1 + \cos z, \quad \frac{\partial f}{\partial y} = -\sin z + 1, \quad \frac{\partial f}{\partial z} = -x \sin z - y \cos z \] Thus, the gradient is: \[ \nabla f(x, y, z) = (1 + \cos z, -\sin z + 1, -x \sin z - y \cos z) \] Step 4: Evaluate the gradient at \( P_0(2, -1, 0) \). At \( P_0(2, -1, 0) \), the gradient is: \[ \nabla f(2, -1, 0) = (1 + \cos 0, -\sin 0 + 1, -(2 \sin 0) - (-1 \cos 0)) = (1 + 1, 0 + 1, 0 + 1) = (2, 1, 1) \] Step 5: Compute the directional derivative. The directional derivative of \( f \) at \( P_0 \) in the direction of \( \hat{v} \) is the dot product: \[ D_{\hat{v}} f = \nabla f(2, -1, 0) \cdot \hat{v} = (2, 1, 1) \cdot \left( \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \] \[ D_{\hat{v}} f = 2 \times \frac{-1}{\sqrt{3}} + 1 \times \frac{1}{\sqrt{3}} + 1 \times \frac{1}{\sqrt{3}} = \frac{-2}{\sqrt{3}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{0}{\sqrt{3}} = 0 \] Final Answer: Thus, the rate of change of \( f(x, y, z) \) at \( P_0(2, -1, 0) \) in the direction from \( P_0 \) to \( P_1 \) is \( \boxed{0} \).