Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Evaluate \(w_{x}, w_{y}\), and \(w_{z}\) at the point. $$ w=\sqrt{x^{2}+y^{2}+z^{2}} \quad(2,-1,2) $$

Step-by-Step Solution

Answer

\(w_{x} = \frac{2}{3}, w_{y} = -\frac{1}{3}, w_{z} = \frac{2}{3}\)

Step 1: Differentiate the function w with respect to x

Differentiate the function \( w = \sqrt{x^{2} + y^{2} + z^{2}} \) with respect to x. The derivative \( w_{x} = \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}} \)

Step 2: Differentiate the function w with respect to y

Differentiate the function \( w = \sqrt{x^{2} + y^{2} + z^{2}} \) with respect to y. The derivative \( w_{y} = \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}} \)

Step 3: Differentiate the function w with respect to z

Differentiate the function \( w = \sqrt{x^{2} + y^{2} + z^{2}} \) with respect to z. The derivative \( w_{z} = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} \)

Step 4: Substitute the given point into the equations

Substitute the values (2, -1, 2) into the equations found in Steps 1, 2, and 3. This yields \( w_{x} = \frac{2}{3}, w_{y} = -\frac{1}{3}, \) and \( w_{z} = \frac{2}{3} \)

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus; it involves functions of more than one variable. The exercise provided is a classic example of this, where we deal with a function, \( w \), that depends on three variables: \( x \), \( y \), and \( z \). Instead of finding the slope of a curve, which is done in single-variable calculus, multivariable calculus allows us to explore slopes in multiple dimensions—this is where partial derivatives come into play.

Partial differentiation, the process used in the exercise, involves taking the derivative of a multivariable function with respect to one variable while holding the others constant. This produces a gradient of sorts that shows how the function changes in different directions. Understanding the behavior of multivariable functions such as \( w \) is crucial for fields like physics, engineering, and economics, where multiple factors simultaneously influence outcomes.

Gradient Vector

The gradient vector is a fundamental concept in multivariable calculus. It is a vector of all the partial derivatives of a function. It points in the direction of the greatest rate of increase of the function and its magnitude gives the rate of increase in that direction. In the exercise, the partial derivatives \( w_x \), \( w_y \), and \( w_z \) can be combined to form the gradient vector for \( w \).

Here's a breakdown:

• \( w_x \) tells us how \( w \) changes as \( x \) increases while \( y \) and \( z \) are held fixed.
• \( w_y \) describes how \( w \) changes with respect to \( y \) when \( x \) and \( z \) remain constant.
• \( w_z \) indicates the rate of change of \( w \) with respect to \( z \), with \( x \) and \( y \) constant.

By evaluating these at a specific point, you gain a detailed understanding of the function's behavior at that location, informing applications such as optimizing multivariable functions.

Differentiation Techniques

Differentiation is a vast area in calculus, and multiple techniques can be applied depending on the function in question. The step-by-step solution reveals one such method: differentiation under the radical sign, which can be approached using the chain rule—an essential technique in calculus. This rule allows us to differentiate composite functions, such as \( w = \sqrt{x^{2} + y^{2} + z^{2}} \), by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

Applying the chain rule yields the derivatives with respect to each variable, considering the square root as the outer function and the squared terms as the inner functions. Other differentiation techniques include the product rule, quotient rule, and implicit differentiation—all tools that help solve a variety of problems in calculus.

Applied Mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. In our exercise, we encounter a real-world example of applied mathematics. Taking the gradient of a distance function has implications in physics, such as finding the direction of a force field, or in computer science for optimizing algorithms.

The power of applied mathematics lies in its ability to solve complex problems and improve practical situations. By mastering concepts such as partial differentiation and the gradient vector, students equip themselves with techniques that will aid in their pursuit of solving real-life challenges that require an understanding of multivariable systems and how they change under various conditions.