Problem 28 - Calculus for Biology and Medicine

Question

Given a function \(f(x, y)\) that is defined and differentiable on an open ball containing the point \(\left(x_{0}, y_{0}\right)\), show that the function \(f\) decreases most rapidly in the direction of \(-\nabla f\left(x_{0}, y_{0}\right)\).

Step-by-Step Solution

Answer

The function decreases most rapidly in the direction of \(-\nabla f(x_0, y_0)\), with the rate \(-\|\nabla f(x_0, y_0)\|\).

Step 1: Understanding the Gradient

The gradient of a function, denoted as \( abla f(x, y) \), is a vector that indicates the direction of the steepest ascent at any point \( (x, y) \). At the point \( (x_0, y_0) \), the gradient \( abla f(x_0, y_0) \) gives us this direction.

Step 2: Direction for Steepest Descent

The direction in which the function decreases most rapidly is in the opposite direction of the gradient. Therefore, the direction of steepest descent is given by \( -abla f(x_0, y_0) \).

Step 3: Directional Derivative

The rate of change of the function \( f \) in the direction of a unit vector \( \mathbf{u} \) is given by the directional derivative: \( D_\mathbf{u} f(x, y) = abla f(x, y) \cdot \mathbf{u} \). To find the rate of steepest decrease, we need to evaluate this for \( \mathbf{u} = -\frac{abla f(x_0, y_0)}{\|abla f(x_0, y_0)\|} \), which is the unit vector in the direction of \( -abla f(x_0, y_0) \).

Step 4: Calculating the Rate of Decrease

Compute the directional derivative in the direction of \( -abla f(x_0, y_0) \): \[ D_{-\mathbf{u}} f(x_0, y_0) = abla f(x_0, y_0) \cdot \left(-\frac{abla f(x_0, y_0)}{\|abla f(x_0, y_0)\|}\right) = -\|abla f(x_0, y_0)\| \]This result shows that the function decreases most rapidly in the direction of \( -abla f(x_0, y_0) \), with the rate of decrease being \(-\|abla f(x_0, y_0)\|\).

Step 5: Conclusion

The maximum rate of decrease occurs in the direction \( -abla f(x_0, y_0) \), confirming that this is indeed the direction of steepest descent for the function \( f \).

Key Concepts

Differentiable Functions

Differentiable functions are the main players in the world of calculus. Think of them as functions that have a smooth graph without any sharp turns or jumps. This smoothness allows us to talk about derivatives, which are rates of change of the function.

• If a function is differentiable at a point, it must also be continuous at that point.
• The derivative of a function at a certain point gives us the slope of the tangent line at that point.
• In multiple dimensions, we talk about the gradient, which generalizes the idea of a derivative to multidimensional spaces.

The gradient, denoted as \( abla f(x, y) \), provides a directional sense of change. For a function like \( f(x, y) \), being differentiable means you can find this gradient everywhere the function is defined and smooth. This gradient vector points in the direction of the steepest ascent at any given point on the surface described by \( f \). Hence, understanding differentiability sets the stage for deeper concepts like gradients and directional derivatives.

Directional Derivative

The directional derivative extends the idea of a derivative in a specified direction. It's like asking: how does my function change if I walk in this particular direction?

• Take a unit vector \( \mathbf{u} \) in the direction you want to measure the change.
• The directional derivative is computed as: \( D_\mathbf{u} f(x, y) = abla f(x, y) \cdot \mathbf{u} \).
• The dot product \( abla f(x, y) \cdot \mathbf{u} \) helps us project the gradient onto the direction of \( \mathbf{u} \).

This tells us the rate at which the function \( f \) is changing in the direction of \( \mathbf{u} \). If \( \mathbf{u} \) points in the same direction as the gradient, the directional derivative is maximized, suggesting the greatest positive change. Conversely, if \( \mathbf{u} \) points opposite to the gradient, the function decreases most rapidly, revealing the direction of steepest descent.

Steepest Descent

Steepest descent is a concept that's about finding the quickest way down a hill. In mathematical terms, for a function \( f \), it is the direction where \( f \) decreases at the maximum rate.

• This direction is precisely the opposite of where the gradient points, which is \( -abla f(x_0, y_0) \).
• When you compute \( -abla f(x_0, y_0) \), you're effectively flipping the gradient vector.
• The size of this vector, or its magnitude \( \| abla f(x_0, y_0) \| \), tells us how fast the function is decreasing.

By calculating the directional derivative along the vector \( -abla f(x_0, y_0) \), you find that the rate of change, or rate of descent, reaches its maximum negative value. This confirms you're on the path of steepest descent, giving you a powerful tool for optimization problems and gradient descent algorithms. It's like having a compass pointed straight down a steep slope, guiding you to the quickest way to decrease a function's value.