Problem 21 - Calculus Early Transcendentals

Question

\(21-26\) Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$f(x, y)=y^{2} / x, \quad(2,4)$$

Step-by-Step Solution

Answer

The maximum rate of change is \(4\sqrt{2}\) in the direction of \((-1/\sqrt{2}, 1/\sqrt{2})\).

Step 1: Understanding the Concept of Gradient

The maximum rate of change of a function at a given point occurs in the direction of its gradient. The gradient of a function \(f(x, y)\) is given by the vector \(abla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)\).

Step 2: Compute the Partial Derivative with Respect to x

The function given is \(f(x, y) = \frac{y^{2}}{x}\). The partial derivative with respect to \(x\) is found by keeping \(y\) constant: \[ \frac{\partial f}{\partial x} = -\frac{y^2}{x^2} \]

Step 3: Compute the Partial Derivative with Respect to y

To find the partial derivative with respect to \(y\), keep \(x\) constant:\[ \frac{\partial f}{\partial y} = \frac{2y}{x} \]

Step 4: Evaluate the Gradient at the Point

Now, evaluate the gradient at the point \( (2, 4) \):\[ abla f (2, 4) = \left( -\frac{4^2}{2^2}, \frac{2(4)}{2} \right) = (-4, 4) \]

Step 5: Determine the Maximum Rate of Change

The magnitude of the gradient gives the maximum rate of change. Calculate the magnitude:\[ \|abla f (2, 4)\| = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]

Step 6: Determine the Direction of Maximum Rate of Change

The direction of the maximum rate of change is in the direction of the gradient vector \((-4, 4)\). The direction vector is \((-1, 1)\) after normalization \[ \text{Direction: } (-1/\sqrt{2}, 1/\sqrt{2}) \].

Key Concepts

Partial Derivatives

In calculus, a partial derivative represents how a function changes as one of its independent variables is varied. To find the partial derivative of a function with respect to

• a variable, treat all other variables as constants.
• For example, consider the function \( f(x, y) = \frac{y^2}{x} \).
• When calculating the partial derivative with respect to \( x \), the \( y \) term is treated as a constant.
• This results in the partial derivative \( \frac{\partial f}{\partial x} = -\frac{y^2}{x^2} \).
• Similarly, the partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = \frac{2y}{x} \), considering \( x \) constant.

Partial derivatives are crucial for exploring how each variable individually affects the function's overall change.
Each derivative provides insight into the function's behavior along a single direction.

Maximum Rate of Change

The maximum rate of change of a function at a given point is directed along its gradient. Imagine standing on a hilly landscape:

• The steepest ascent path from where you are is represented by the maximum rate of change.
• It always points in the direction of the gradient vector.
• For the function \( f(x, y) = \frac{y^2}{x} \), at the point \((2, 4)\), the gradient vector is \((-4, 4)\).

To find how fast the change occurs, you calculate the magnitude of the gradient. This tells you how steep or swift the change in the function is along that direction. At \((2, 4)\), the maximum rate of change is determined by the magnitude \( 4\sqrt{2} \).

Gradient Vector

The gradient vector is a powerful tool in multivariable calculus. It indicates both the direction and the rate of the steepest ascent for a function. If you visualize

• a mountain range, the gradient vector tells you which way is uphill and just how steep it is.
• Formally, it combines the function's partial derivatives into a vector.
• For example, \( abla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \).
• At the point \((2, 4)\), for our function, the gradient vector is calculated as \((-4, 4)\).

The crux of using the gradient is understanding that it immensely helps in determining optimization and is pivotal for real-life applications such as navigation and design.

Magnitude of Gradient

The magnitude of the gradient vector represents the maximum rate of change of a function at a specific point. Think of it as not just identifying the direction of change,

• but also evaluating how intense the change is.
• For any function \( f \), the gradient's magnitude at a given point is computed using:
• \[ \|abla f(x, y)\| = \sqrt{ \left( \frac{\partial f}{\partial x} \right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 } \].
• It helps quantify how rapidly the function value increases or decreases in the gradient's direction.

In our case, at the point \((2, 4)\), the magnitude is \( 4\sqrt{2} \). This gives a clear understanding that the function changes quite significantly in that particular direction.