The gradient points in the direction of the steepest increase of the function, and its magnitude represents the rate of that increase...give me numerical example

Let's consider a simple function \( f(x, y) = 3x^2 + 2y^2 \).

### Step 1: Compute the partial derivatives
First, we compute the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \).

- Partial derivative with respect to \( x \):
  \[
  \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(3x^2 + 2y^2) = 6x
  \]

- Partial derivative with respect to \( y \):
  \[
  \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(3x^2 + 2y^2) = 4y
  \]

### Step 2: Form the gradient vector
The gradient of \( f(x, y) \) is the vector formed by these partial derivatives:
\[
\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (6x, 4y)
\]

### Step 3: Evaluate the gradient at a specific point
Let's evaluate the gradient at the point \( (x, y) = (1, 2) \).

- \( \frac{\partial f}{\partial x} \) at \( (1, 2) \) is \( 6 \times 1 = 6 \).
- \( \frac{\partial f}{\partial y} \) at \( (1, 2) \) is \( 4 \times 2 = 8 \).

So, the gradient at \( (1, 2) \) is:
\[
\nabla f(1, 2) = (6, 8)
\]

### Step 4: Interpret the gradient
- **Direction**: The gradient vector \( (6, 8) \) points in the direction of the steepest increase of the function \( f(x, y) \) at the point \( (1, 2) \). This means that if you move in the direction of the vector \( (6, 8) \), the value of the function \( f(x, y) \) will increase the fastest.
  
- **Magnitude**: The magnitude of the gradient vector \( (6, 8) \) represents the rate of increase. The magnitude \( |\nabla f(1, 2)| \) is calculated as:
  \[
  |\nabla f(1, 2)| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
  \]
  This means that at the point \( (1, 2) \), if you move in the direction of the vector \( (6, 8) \), the function increases at a rate of 10 units per unit distance traveled in that direction.

### Do you want more details or have any questions?
### Here are 5 related questions:
1. How does the gradient change if you evaluate it at different points on the function?
2. What happens to the gradient at a local maximum or minimum?
3. How is the gradient used in gradient descent algorithms?
4. Can you explain the relationship between the gradient and the level curves of a function?
5. What is the significance of the gradient's direction in constrained optimization problems?

**Tip**: In optimization problems, you often move in the direction opposite to the gradient (negative gradient) when trying to minimize a function, as this direction points towards the steepest decrease.

Explore the concept of gradient in mathematics with a numerical example. Learn how the gradient points in the direction of the steepest increase of a function and its magnitude represents the rate of that increase. Includes step-by-step calculations and interpretations.

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