We are tasked with finding the directional derivative of the function $f(u, v) = u^2 e^{-v}$ at the point $(3, 0)$ in the direction of the vector $\mathbf{v} = 3\mathbf{i} + 4\mathbf{j}$.

Step-by-Step Solution

1. Gradient of $f(u, v)$: The gradient of $f$, denoted as $\nabla f(u, v)$, is the vector of partial derivatives with respect to $u$ and $v$.

   $\frac{\partial f}{\partial u} = 2u e^{-v}$ $\frac{\partial f}{\partial v} = -u^2 e^{-v}$

   Therefore, the gradient is: $\nabla f(u, v) = \left( 2u e^{-v}, -u^2 e^{-v} \right)$

2. Gradient at the point $(3, 0)$: Substituting $u = 3$ and $v = 0$ into the gradient:

   $\nabla f(3, 0) = \left( 2(3) e^{0}, -(3)^2 e^{0} \right) = (6, -9)$

3. Unit vector in the direction of $\mathbf{v}$: The direction vector $\mathbf{v} = 3\mathbf{i} + 4\mathbf{j}$ needs to be converted into a unit vector by dividing it by its magnitude.

   The magnitude of $\mathbf{v}$ is: $|\mathbf{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

   So, the unit vector is: $\mathbf{u_v} = \left( \frac{3}{5}, \frac{4}{5} \right)$

4. Directional derivative: The directional derivative of $f$ in the direction of $\mathbf{v}$ at the point $(3, 0)$ is the dot product of $\nabla f(3, 0)$ and the unit vector $\mathbf{u_v}$:

   $D_{\mathbf{v}} f(3, 0) = \nabla f(3, 0) \cdot \mathbf{u_v}$

   $D_{\mathbf{v}} f(3, 0) = (6, -9) \cdot \left( \frac{3}{5}, \frac{4}{5} \right)$

   Compute the dot product: $D_{\mathbf{v}} f(3, 0) = 6 \times \frac{3}{5} + (-9) \times \frac{4}{5} = \frac{18}{5} - \frac{36}{5} = \frac{-18}{5}$

Thus, the directional derivative is: $D_{\mathbf{v}} f(3, 0) = \frac{-18}{5}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you compute the gradient of a multivariable function?
2. What is the significance of the directional derivative in geometry?
3. How do you find the unit vector of a given direction vector?
4. What is the relationship between the gradient and the directional derivative?
5. In what contexts are directional derivatives commonly used in physics or engineering?

Tip: Always remember to normalize the direction vector when computing the directional derivative to ensure it's unit length.