Problem 299 - Calculus 3

Question

For the following exercises, find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$ f(x, y)=x e^{-y}, \quad(1,0) $$

Step-by-Step Solution

Answer

The maximum rate of change is \( \sqrt{2} \), and it occurs in the direction \( \left( \frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}} \right) \).

Step 1: Find the Gradient

The first step is to calculate the gradient of the function \( f(x, y) = x e^{-y} \). The gradient is a vector composed of partial derivatives with respect to \( x \) and \( y \). For the given function, \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \). This gives us:\[ \frac{\partial f}{\partial x} = e^{-y} \]\[ \frac{\partial f}{\partial y} = -x e^{-y} \]

Step 2: Evaluate the Gradient at the Given Point

Next, evaluate the gradient at the point \((1,0)\). Substitute \( x = 1 \) and \( y = 0 \) into the gradient:\[ abla f(1,0) = (e^{0}, -1 \cdot e^{0}) = (1, -1) \]

Step 3: Calculate the Maximum Rate of Change

The maximum rate of change of the function at a given point is the magnitude of the gradient vector at that point. Calculate the magnitude of the gradient vector \( (1, -1) \):\[ \| abla f(1,0) \| = \sqrt{1^2 + (-1)^2} = \sqrt{2} \]

Step 4: Determine the Direction of Maximum Rate of Change

The direction of the maximum rate of change is given by the direction of the gradient vector. Normalize the gradient vector \( (1, -1) \) to find the direction:\[ \text{Direction} = \left( \frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}} \right) \]

Key Concepts

Partial Derivatives

In multivariable calculus, partial derivatives are an essential concept. They help us understand how a function changes as each variable changes.

Think of the function as a surface. Each partial derivative provides the slope of this surface in the direction of one of the axes.

In this context, our function is given as \( f(x, y) = x e^{-y} \). We find the partial derivatives by keeping one variable constant and differentiating with respect to the other:

• Partial derivative with respect to \( x \): We treat \( y \) as a constant and differentiate \( f(x, y) \) with regard to \( x \), giving us \( \frac{\partial f}{\partial x} = e^{-y} \).
• Partial derivative with respect to \( y \): We treat \( x \) as a constant and differentiate \( f(x, y) \) with regard to \( y \), which results in \( \frac{\partial f}{\partial y} = -x e^{-y} \).

These derivatives help form the gradient vector, which is key to analyzing the behavior of our multivariable function.

Maximum Rate of Change

The maximum rate of change of a function at a given point is an intriguing concept. It tells us how quickly the function is changing at that point.

This rate is given by the magnitude of the gradient vector at that point. The bigger the magnitude, the faster the function is changing.

In our example, after finding the gradient vector \( (1, -1) \) at the point \((1, 0)\), we calculate its magnitude:

• The formula for the magnitude of vector \( (a, b) \) is \( \sqrt{a^2 + b^2} \).
• Applying this in our case, we get \( \sqrt{1^2 + (-1)^2} = \sqrt{2} \).

So, the maximum rate of change at the point \((1, 0)\) is \( \sqrt{2} \). This means the function is changing at a speed of \( \sqrt{2} \) units.

Gradient Vector

The gradient vector is a fundamental tool in multivariable calculus. It gives both the direction and the rate of steepest ascent of a function.

The gradient vector consists of all the first partial derivatives of the function. For our function \( f(x, y) = x e^{-y} \), the gradient \( abla f \) is given by:

• \( \frac{\partial f}{\partial x} = e^{-y} \)
• \( \frac{\partial f}{\partial y} = -x e^{-y} \)

Evaluated at point \((1, 0)\), the gradient vector is \( (1, -1) \).

This vector not only shows the intensity of change but also points in the direction where the function grows fastest. Thus, at any point, the gradient vector indicates how to move to increase the function's value most rapidly.

Directional Derivative

The directional derivative is an extension of the idea of partial derivatives. It tells us how a function changes as we move in a specified direction.

While the gradient vector gives the rate of maximum increase, the directional derivative allows us to measure the rate of change in any given direction.

The formula for the directional derivative of a function \( f(x, y) \) in the direction of a unit vector \( \mathbf{u} \) is:

• \( D_{\mathbf{u}}f = abla f \cdot \mathbf{u} \)

The dot product here reflects the projection of the gradient in the direction of \( \mathbf{u} \). For our exercise, the direction of the maximum rate of change is in the direction of the normalized gradient vector, which is \( \left( \frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}} \right) \).

This means if you move in this direction, the function changes at a rate dictated by the gradient vector's magnitude.