Problem 23 - Calculus with Concepts in Calculus

Question

From (1) it follows that the directional derivative of a function \(f\) at a point is smallest in the direction opposite to the gradient of \(f\) at that point. Thus we say that a function decreases most rapidly in the direction opposite the gradient.Find the direction in which the function decreases most rapidly at the given point. $$ f(x, y, z)=\frac{x-z}{y+z} ;(-1,1,3) $$

Step-by-Step Solution

Answer

The function decreases most rapidly in the direction \((-\frac{1}{2\sqrt{6}}, -\frac{1}{2\sqrt{6}}, \frac{1}{\sqrt{6}})\).

Step 1: Find the gradient of the function

To find the direction in which the function decreases most rapidly, we first need to determine the gradient of the function \( f(x, y, z) = \frac{x-z}{y+z} \). The gradient is a vector composed of the partial derivatives of the function with respect to each of its variables, \(x\), \(y\), and \(z\).

Step 2: Calculate the partial derivative with respect to x

The partial derivative of \( f \) with respect to \( x \) is calculated while treating \( y \) and \( z \) as constants. We have:\[ \frac{\partial f}{\partial x} = \frac{1}{y+z} \]

Step 3: Calculate the partial derivative with respect to y

The partial derivative of \( f \) with respect to \( y \) is calculated as follows:\[ \frac{\partial f}{\partial y} = -\frac{x-z}{(y+z)^2} \]

Step 4: Calculate the partial derivative with respect to z

The partial derivative of \( f \) with respect to \( z \) is:\[ \frac{\partial f}{\partial z} = -\frac{1}{y+z} + \frac{x-z}{(y+z)^2} \]

Step 5: Evaluate the gradient at the given point

Substitute \((x, y, z) = (-1, 1, 3)\) into each of the partial derivatives to find the components of the gradient at that point. 1. \( \frac{\partial f}{\partial x} = \frac{1}{1+3} = \frac{1}{4}\)2. \( \frac{\partial f}{\partial y} = -\frac{-1-3}{(1+3)^2} = \frac{4}{16} = \frac{1}{4}\)3. \( \frac{\partial f}{\partial z} = -\frac{1}{4} + \frac{-1-3}{16} = -\frac{1}{4} - \frac{4}{16} = -\frac{2}{4} = -\frac{1}{2} \)Thus, the gradient vector is \( abla f = \left( \frac{1}{4}, \frac{1}{4}, -\frac{1}{2} \right) \).

Step 6: Determine the opposite direction to the gradient

The direction in which the function decreases most rapidly is opposite to the gradient vector. Therefore, multiply each component of the gradient by -1: \( -abla f = \left( -\frac{1}{4}, -\frac{1}{4}, \frac{1}{2} \right) \).

Step 7: Normalize the direction vector

To find the unit vector in the direction of maximum decrease, divide the vector by its magnitude. Calculate the magnitude: \[ \sqrt{\left(-\frac{1}{4}\right)^2 + \left(-\frac{1}{4}\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{1}{16} + \frac{1}{16} + \frac{4}{16}} = \sqrt{\frac{6}{16}} = \sqrt{\frac{3}{8}} = \frac{\sqrt{3}}{2\sqrt{2}} \]The unit vector is:\[ \left( -\frac{1/4}{\frac{\sqrt{3}}{2\sqrt{2}}}, -\frac{1/4}{\frac{\sqrt{3}}{2\sqrt{2}}}, \frac{1/2}{\frac{\sqrt{3}}{2\sqrt{2}}} \right) = \left( -\frac{1}{2\sqrt{6}}, -\frac{1}{2\sqrt{6}}, \frac{1}{\sqrt{6}} \right) \]

Key Concepts

Gradient of a Function

The gradient of a function is a fundamental concept in vector calculus, playing a crucial role in understanding the directional rate of change of a function. For a multi-variable function \( f(x, y, z) \), the gradient, denoted as \( abla f \), is a vector that consists of all its first partial derivatives. This vector points in the direction of the steepest increase of the function.
The components of the gradient vector are the partial derivatives with respect to each variable. For the function \( f(x, y, z) = \frac{x-z}{y+z} \), the gradient is calculated as follows:

• \( \frac{\partial f}{\partial x} = \frac{1}{y+z} \)
• \( \frac{\partial f}{\partial y} = -\frac{x-z}{(y+z)^2} \)
• \( \frac{\partial f}{\partial z} = -\frac{1}{y+z} + \frac{x-z}{(y+z)^2} \)

The gradient helps us understand how rapidly the function changes and the direction of that change.
Once evaluated at a specific point, the gradient provides insightful information about the behavior of the function at that point.

Partial Derivatives

Partial derivatives represent how a function changes as one of its variables changes, while keeping all other variables constant. This concept is particularly useful for functions of several variables, such as \( f(x, y, z) = \frac{x-z}{y+z} \).
Calculating partial derivatives involves differentiating the function with respect to one variable at a time. For example, the partial derivative \( \frac{\partial f}{\partial x} \) shows the rate of change of \( f \) concerning \( x \), treating \( y \) and \( z \) as constants.

• \( \frac{\partial f}{\partial x} = \frac{1}{y+z} \) is found by treating \( y \) and \( z \) as constant.
• \( \frac{\partial f}{\partial y} = -\frac{x-z}{(y+z)^2} \) is derived by considering only \( y \) as a variable.
• \( \frac{\partial f}{\partial z} = -\frac{1}{y+z} + \frac{x-z}{(y+z)^2} \) focuses on the variable \( z \).

Understanding partial derivatives is essential for analyzing the behavior of functions and is the first step in computing the gradient.

Vector Calculus

Vector calculus is the branch of mathematics focusing on differentiation and integration of vector fields, usually in two or three-dimensional Euclidean space. It extends the ideas of single-variable calculus to functions with multiple variables.
One of the key outcomes of vector calculus is the gradient, which is a vector field representing the maximum rate of increase of a scalar function, like our function \( f(x, y, z) = \frac{x-z}{y+z} \).
Through vector calculus, we explore concepts like divergence and curl, which describe the behavior of vector fields. These tools are instrumental in physics and engineering for understanding phenomena like fluid flow and electromagnetism.
In our exercise, vector calculus is used to determine the gradient's magnitude and direction, ensuring we find the path of steepest ascent (or descent). The calculations reveal how multi-variable functions behave and interact, providing insights and tools to solve complex problems.

Function Optimization

Function optimization involves finding the maximum or minimum values of a function, a common problem in science and engineering. It often requires analyzing the function's gradient.
The gradient indicates the function's steepest ascent direction, while its opposite indicates the steepest descent. In our exercise, the goal is to find the direction in which the function \( f(x, y, z) = \frac{x-z}{y+z} \) decreases most rapidly at the point \((-1,1,3)\).
By evaluating the gradient at this point, we get \( abla f = \left( \frac{1}{4}, \frac{1}{4}, -\frac{1}{2} \right) \), which shows the direction of steepest increase. The opposite direction, \( -abla f = \left( -\frac{1}{4}, -\frac{1}{4}, \frac{1}{2} \right) \), represents the steepest decrease.
To optimize the function, we often also consider the magnitude of the directional vector, ensuring it is normalized to focus purely on direction rather than magnitude. Thus, function optimization leverages gradients to efficiently move towards the desired outcome, utilizing both magnitude and direction to evaluate potential solutions.