Problem 281 - Calculus Volume 3

Question

For the following exercises, find the gradient. Find the gradient of \(f(x, y, z)=x y+y z+x z\) at point \(P(1,2,3)\).

Step-by-Step Solution

Answer

The gradient of the function at point \(P(1, 2, 3)\) is \((5, 4, 3)\).

Step 1: Identify the Function

First, we confirm the function provided: \(f(x, y, z) = xy + yz + xz\). We are tasked with finding the gradient at the point \(P(1, 2, 3)\).

Step 2: Understand the Gradient

The gradient is a vector consisting of partial derivatives of the function with respect to each variable \(x\), \(y\), and \(z\). It is denoted as \(abla f\) and given by \(abla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)\).

Step 3: Calculate Partial Derivative with Respect to x

The partial derivative of \(f\) with respect to \(x\) is found by treating \(y\) and \(z\) as constants: \( \frac{\partial f}{\partial x} = y + z \).

Step 4: Calculate Partial Derivative with Respect to y

The partial derivative of \(f\) with respect to \(y\) is found by treating \(x\) and \(z\) as constants: \( \frac{\partial f}{\partial y} = x + z \).

Step 5: Calculate Partial Derivative with Respect to z

The partial derivative of \(f\) with respect to \(z\) is found by treating \(x\) and \(y\) as constants: \( \frac{\partial f}{\partial z} = y + x \).

Step 6: Substitute the Point into the Gradient

Substitute \(x = 1\), \(y = 2\), and \(z = 3\) into each of the partial derivatives:- For \(\frac{\partial f}{\partial x}\), substitute to get \(2 + 3 = 5\).- For \(\frac{\partial f}{\partial y}\), substitute to get \(1 + 3 = 4\).- For \(\frac{\partial f}{\partial z}\), substitute to get \(2 + 1 = 3\).

Step 7: Form the Gradient Vector

The gradient vector at \(P(1, 2, 3)\) is \(abla f(1, 2, 3) = (5, 4, 3)\). This represents the direction and rate of the steepest ascent of the function at the point.

Key Concepts

Partial Derivatives

Partial derivatives are central in multivariable calculus as they help us understand how a function changes with respect to each of its variables independently. Essentially, by taking a partial derivative of a multivariable function, you're looking at the function's rate of change concerning one variable while keeping the others constant.

For example, with the function \(f(x, y, z) = xy + yz + xz\), to find the partial derivative with respect to \(x\), we treat \(y\) and \(z\) as constants, resulting in \(\frac{\partial f}{\partial x} = y + z\).

This process is repeated for the other variables, which allows us to see how changes in one variable, while others are held steady, affect the function's output. This is essential not only in calculus but also in fields like economics, engineering, and physics where variables frequently interact.

Gradient Vector

The gradient vector is a vital concept when dealing with functions of multiple variables. It represents the direction and rate of the steepest ascent of a function at a specific point. Essentially, this vector points towards the highest rate of increase of the function from that point.

In mathematical terms, the gradient vector, denoted by \(abla f\), is the collection of all partial derivatives of the function. For our function \(f(x, y, z) = xy + yz + xz\), the gradient is\[abla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right).\]

In practical terms, for functions describing real-world scenarios, this vector reveals how the function value changes as you move in different directions. Thus, understanding the gradient helps in optimization problems where you want to find maximum or minimum values.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions with more than one variable. Rather than dealing with lines and planes, it tackles surfaces and volumes, making it crucial for understanding phenomena in three-dimensional spaces and higher.

This branch of calculus encompasses several key operations, such as differentiation and integration, but applied to functions of two or more variables. Thus, concepts like partial derivatives and gradient vectors become extremely relevant here.

In fields such as engineering, physics, and computer graphics, multivariable calculus provides the tools to model and solve complex systems. With functions like \(f(x, y, z) = xy + yz + xz\), examining partial derivatives and gradients allows us to predict and optimize behavior in multidimensional spaces.

Calculus 3

Often referred to as Calculus 3, multivariable calculus is an extension of single-variable calculus presented in Calculus 1 and 2. This course typically involves exploring topics such as partial derivatives, multiple integrals, and vector calculus, focusing on more geometric and spatial reasoning.

One of the primary goals here is to understand how functions behave as they change in space. This includes finding the gradient vector of functions, like \(f(x, y, z) = xy + yz + xz\), to determine how these functions increase or decrease at given points in three-dimensional space.

By mastering these concepts, students gain powerful analytical skills to analyze and solve higher-level mathematics problems, which are pivotal in advanced science, engineering, and computer science disciplines.