In what direction from (2, -1, 2) is the directional derivative of φ = 4xz³ - 3x²y²z minimum? And what is the magnitude of minimum change?
Understand the Problem
The question is asking for the direction from the point (2, -1, 2) where the directional derivative of the function φ = 4xz³ - 3x²y²z is minimized, along with the magnitude of that minimum change.
Answer
The direction is \( \frac{-80\hat{i} - 24\hat{j} - 60\hat{k}}{\sqrt{589}} \), and the magnitude is \( 4\sqrt{589} \).
Answer for screen readers
The direction from the point ( (2, -1, 2) ) where the directional derivative is minimized is:
$$ \frac{-80\hat{i} - 24\hat{j} - 60\hat{k}}{\sqrt{589}} $$
And the magnitude of that minimum change is:
$$ 4\sqrt{589} $$
Steps to Solve
- Calculate the gradient of the function
The gradient of the function $\phi = 4xz^3 - 3x^2y^2z$ gives the direction of the maximum increase.
We find the partial derivatives:
- $\frac{\partial \phi}{\partial x} = 4z^3 - 6xyz^2$
- $\frac{\partial \phi}{\partial y} = -6x^2zy$
- $\frac{\partial \phi}{\partial z} = 12xz^2 - 3x^2y$
- Evaluate the gradient at the point (2, -1, 2)
Substituting ( (x, y, z) = (2, -1, 2) ) into the partial derivatives we calculated:
-
$\frac{\partial \phi}{\partial x} = 4(2)(2^3) - 6(2)(-1)(2^2) = 32 + 48 = 80$
-
$\frac{\partial \phi}{\partial y} = -6(2^2)(2)(-1) = 24$
-
$\frac{\partial \phi}{\partial z} = 12(2)(2^2) - 3(2^2)(-1) = 48 + 12 = 60$
So, the gradient at the point is $\nabla \phi (2, -1, 2) = (80, 24, 60)$.
- Determine the direction of minimum change
The direction of minimum change occurs in the opposite direction to the gradient. Thus, we take the negative of the gradient:
$$ -\nabla \phi = (-80, -24, -60) $$
- Calculate the magnitude of the minimum change
The magnitude of the minimum change is calculated using the formula for the magnitude of a vector:
$$ |-\nabla \phi| = \sqrt{(-80)^2 + (-24)^2 + (-60)^2} $$
Calculating this gives:
$$ |-\nabla \phi| = \sqrt{6400 + 576 + 3600} = \sqrt{6400 + 576 + 3600} = \sqrt{10576} = 4\sqrt{589} $$
The direction from the point ( (2, -1, 2) ) where the directional derivative is minimized is:
$$ \frac{-80\hat{i} - 24\hat{j} - 60\hat{k}}{\sqrt{589}} $$
And the magnitude of that minimum change is:
$$ 4\sqrt{589} $$
More Information
The directional derivative indicates how the function changes as you move in a particular direction. The gradient always points in the direction of the steepest ascent, so the steepest descent occurs in the opposite direction. The magnitude provides a measure of how steep that change is.
Tips
- Failing to correctly compute partial derivatives can lead to incorrect gradient values.
- Not considering the negative gradient as the direction of minimum change.
- Miscalculating the magnitude of the gradient vector can yield incorrect results.
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