Problem 32 - Calculus 3

Question

For the following exercises, find the vertical traces of the functions at the indicated values of \(x\) and \(y,\) and plot the traces. $$ z=\cos \sqrt{x^{2}+y^{2}} \quad x=1 $$

Step-by-Step Solution

Answer

Vertical trace: \(z = \cos \sqrt{1 + y^2}\). Plot for \(y\) values.

Step 1: Understanding the Problem

We need to find the vertical trace of the function \(z = \cos \sqrt{x^2 + y^2}\) at the point where \(x = 1\). This involves evaluating the function by substituting \(x = 1\) into the function.

Step 2: Substituting the Value of x

Substitute \(x = 1\) into the function so that we have the expression for \(z\) only in terms of \(y\). This gives: \[ z = \cos \sqrt{1^2 + y^2} = \cos \sqrt{1 + y^2}. \]

Step 3: Plotting the Vertical Trace

We plot the function \(z = \cos \sqrt{1 + y^2}\) against \(y\) to visualize the vertical trace. This is a graph that traces out the relationship between \(z\) and \(y\) when \(x = 1\). The plot resembles a damped wave as \(y\) moves away from zero.

Key Concepts

Multivariable Functions

In mathematics, multivariable functions are functions with more than one input variable. They extend the concept of functions from single-variable calculus into multiple dimensions, allowing for the modeling and visualization of more complex relationships.

A multivariable function is often written as \( z = f(x, y) \), where \( z \) depends on two variables, \( x \) and \( y \). These functions provide a framework for exploring phenomena where output depends on several inputs, making them vital in fields like physics, engineering, and economics.

Multivariable functions can be used to describe a surface in three-dimensional space. The function \( z = \cos \sqrt{x^2 + y^2} \) is an example, where \( z \) is determined by the inputs \( x \) and \( y \). Such a function creates a surface formed by all points \((x, y, z)\) that satisfy the equation.

• Understanding vertical traces is crucial when analyzing multivariable functions as it provides the cross-sections of the surface.
• Vertical traces help in breaking down a 3D graph into simpler 2D sections.

Analyzing these traces provides insights into the behavior of the function along specific directions.

Three-Dimensional Graphing

Three-dimensional graphing helps visualize multivariable functions by plotting them in a 3D space. It allows us to see how changes in the input variables \( x \) and \( y \) affect the output variable \( z \).

Plotting a function like \( z = \cos \sqrt{x^2 + y^2} \) involves creating a graph that shows a 3D surface. Each point on this surface corresponds to a combination of \( x \), \( y \), and \( z \).

• The x-axis and y-axis define the horizontal plane, representing all possible values of inputs.
• The z-axis represents the output or the height above or below the xy-plane.

Vertical traces highlight specific slices of the 3D surface, where one of the variables (either \( x \) or \( y \)) is held constant. By setting \( x = 1 \), the problem simplifies to a function of \( y \) alone, giving us the trace \( z = \cos \sqrt{1 + y^2} \).

This trace provides a 2D silhouette of the original 3D surface, making it easier to understand how the function behaves along the line \( x = 1 \). These simplified plots are useful in applications which necessitate understanding of the function's behavior under specific conditions.

Trigonometric Functions

Trigonometric functions are fundamental mathematical functions derived from angles and geometric measurements. They form the basis of studying periodic phenomena, such as waves and oscillations.

The cosine function, denoted \( \cos \theta \), is one of these basic functions. It describes the x-coordinate of a point on the unit circle as it moves counterclockwise from the positive x-axis by an angle \( \theta \). The cosine function is periodic with a cycle repeating every \( 2\pi \) radians.

• For the function \( z = \cos \sqrt{x^2 + y^2} \), the cosine component ensures the output \( z \) oscillates between -1 and 1.
• The presence of \( \sqrt{x^2 + y^2} \) inside the cosine modifies the input angle \( \theta \), making \( z \) less predictable but still limited to its periodic nature.

Understanding how trigonometric functions apply to multivariable contexts, such as the exercise at hand, involves recognizing these oscillating outputs and considering how they translate into three-dimensional surfaces. This enables us to predict and plot complex wave-like patterns, as seen in the vertical traces of the function when plotted against \( y \).