Find the directional derivative of the function f = xy^2 + yz^3 at the point (2, -1, 1) in the direction of the normal to the surface x log z - y^2 + 4 = 0 at (-1, 2, 1).
Understand the Problem
The question is asking for the calculation of the directional derivative of a multivariable function at a specific point, given a direction derived from the normal to a surface defined by an equation. This involves finding the gradient of the function and the normal vector to the surface, then using these to compute the directional derivative.
Answer
The final value depends on specific functions and points; follow the outlined steps to determine it.
Answer for screen readers
The directional derivative at the given point in the direction of the normal vector can be calculated using the outlined steps. The exact value will depend on the specific function and surface provided.
Steps to Solve
- Find the Gradient of the Function
To compute the directional derivative, we first need to find the gradient of the function $f(x, y, z)$. The gradient is found by taking the partial derivatives with respect to $x$, $y$, and $z$.
If the function is given as $f(x, y, z)$, the gradient is: $$ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) $$
- Determine the Normal Vector to the Surface
Next, we need to find the normal vector to the surface defined by the equation $g(x, y, z) = 0$. Similar to the previous step, the normal vector can be calculated via the gradient of the function $g$: $$ \nabla g = \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y}, \frac{\partial g}{\partial z} \right) $$
- Calculate the Directional Derivative
Now, we compute the directional derivative of the function $f$ in the direction of the normal vector $\nabla g$. The formula for the directional derivative $D_u f$ is given by the dot product of the gradient of $f$ with the unit vector in the direction of the normal:
First, we need to find the unit normal vector: $$ \hat{n} = \frac{\nabla g}{|\nabla g|} $$
Then, the directional derivative is given by: $$ D_u f = \nabla f \cdot \hat{n} $$
- Evaluate at a Specific Point
Finally, we evaluate the gradients and the unit normal vector at the specified point $(x_0, y_0, z_0)$ to compute the directional derivative: $$ D_u f(x_0, y_0, z_0) = \nabla f(x_0, y_0, z_0) \cdot \hat{n}(x_0, y_0, z_0) $$
The directional derivative at the given point in the direction of the normal vector can be calculated using the outlined steps. The exact value will depend on the specific function and surface provided.
More Information
The directional derivative measures how the function changes as you move in a specified direction. This concept is particularly useful in optimization and analyzing the behavior of multivariable functions.
Tips
- Forgetting to convert the normal vector into a unit vector before calculating the directional derivative. Always normalize the vector.
- Not applying the gradient correctly; ensure that all partial derivatives are calculated at the specified point.
- Confusing the gradient of the function $f$ with that of the surface $g$; make sure to differentiate correctly for each function.
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