1. Find Partials: Calculate $f_x$ and $f_y$ first.
2. Package the Vector: Write the result as $\nabla f = \langle f_x, f_y \rangle$.
3. Find Steepness: Calculate the magnitude $|\nabla f| = \sqrt{f_x^2 + f_y^2}$.
4. Direction of Descent: Multiply your gradient by -1 ($-\nabla f$).
Find the Gradient Vector $\nabla f$ for each function at the specified point.
$f(x, y) = x^2 + 3xy$ at point $(1, 2)$.
$f(x, y) = \sin(x) + e^y$ at point $(\pi, 0)$.
$f(x, y) = x^3 y^2$ at point $(2, 1)$.
Using the gradients from Part I, find the Steepest Slope at those points.
For Matrix 1 ($\langle 8, 3 \rangle$): What is the max slope?
For Matrix 2: What is the max slope?
If you are at the very top of a hill (a Maximum), what is the value of $f_x$? What is $f_y$? What is the magnitude of the Gradient vector? Why does the "Compass of Ascent" spin in circles when you reach the top?
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A set of contour lines (Level Curves) is shown below.
1. At point P, draw the Gradient arrow.
2. Remember: It must be exactly 90 degrees to the contour line!
You are standing on a surface defined by $f(x, y) = x^2 y$.
At point $(1, 10)$, find the direction you should walk to **Go Down** the fastest.
Objective: Explain the Gradient to a younger student using a compass and a toy mountain.
The Activity:
1. Show them a steep plastic hill.
2. Ask: "If you were a tiny ant, which way would you go to reach the top the fastest?"
3. Mark that line with a crayon.
4. "That line is the Gradient. It's the ant's 'Smart Compass'."
The Lesson: "God gives our heart a 'Smart Compass' called the Holy Spirit that always points toward the highest love."
Response: ___________________________________________________________