1. Find the Gradient ($\nabla f$): Calculate $\langle f_x, f_y \rangle$ and plug in your point.
2. Normalize your Direction: If the direction vector $\mathbf{v}$ is not length 1, divide it by its magnitude $|\mathbf{v}|$.
3. Dot Product: $D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}$.
4. Result Interpretation: Positive = Climbing. Negative = Falling. Zero = Level.
Turn each direction vector $\mathbf{v}$ into a Unit Vector $\mathbf{u}$.
Direction: $\mathbf{v} = \langle 3, 4 \rangle$
Direction: $\mathbf{v} = \langle 1, -1 \rangle$
Direction: Pointing exactly North ($\\langle 0, 10 \rangle$).
Find the directional derivative $D_{\mathbf{u}} f$ at the given point in the given direction.
$f(x, y) = x^2 + y^2$ at point $(1, 1)$ in direction $\mathbf{u} = \langle 0.6, 0.8 \rangle$.
$f(x, y) = 5x - 3y$ at $(0, 0)$ in direction $\mathbf{v} = \langle 4, 3 \rangle$.
(Hint: Normalize $\mathbf{v}$ first!)
In Problem 1, what is the Magnitude of the Gradient $|\nabla f|$? Compare it to your answer (2.8). Is your answer smaller or larger than the magnitude? Why can the Directional Derivative never be larger than the Gradient's length?
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The Orthogonal Walk: If $\nabla f = \langle 10, 0 \rangle$ (High place is straight East), what is your rate of change if you walk North ($\\mathbf{u} = \langle 0, 1 \rangle$)?
The temperature on a metal plate is $T(x,y) = e^x \cos y$.
You are at $(0, 0)$.
Task: Find the rate of change of temperature if you move toward the point $(3, 4)$.
Objective: Explain the Directional Derivative using a fan and a paper sail.
The Activity:
1. Point the fan directly at the sail. "Maximum Speed."
2. Turn the sail sideways. "Zero Speed."
3. Turn the sail 45 degrees. "Half Speed."
The Lesson: "How much power we get from God's wind depends on the 'Angle' of our heart. Alignment is everything!"
Response: ___________________________________________________________