Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Draw the graph and the corresponding contour plot. \(f(x, y)=\sin \left(x^{2}+y^{2}\right) /\left(x^{2}+y^{2}\right), f(0,0)=1 ;\) \(-2 \leq x \leq 2,-2 \leq y \leq 2\)

Step-by-Step Solution

Answer

Generate 3D and contour plots of \( f(x, y) \) for the range \([-2, 2] \times [-2, 2]\).

Step 1: Define the Function

The function given is \( f(x, y) = \frac{\sin(x^2 + y^2)}{x^2 + y^2} \) for all \( (x, y) eq (0, 0) \), and \( f(0, 0) = 1 \). This is due to the limit \( \lim_{(x, y) \to (0, 0)} f(x, y) = 1 \).

Step 2: Choose Range for x and y

The function should be evaluated over the range \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\) as provided in the problem.

Step 3: Prepare the Grid

Create a grid of values for \( x \) and \( y \) using values within the specified range. Commonly used increments might be 0.1 or smaller to create a smooth plot.

Step 4: Compute Function Values

Evaluate the function \( f(x, y) \) at each pair of \( (x, y) \) values on the grid. Be careful to treat \( f(0,0) = 1 \) separately.

Step 5: Plot the 3D Surface Plot

Use a plotting software (such as MATLAB or Python with Matplotlib) to create a 3D surface plot of \( f(x, y) \) using the grid and the computed function values.

Step 6: Create the Contour Plot

From the grid and function values, generate a contour plot to visualize lines of constant values of \( f(x, y) \) on the \( x, y \) plane.

Key Concepts

Contour Plots

Contour plots are a powerful tool in multivariable calculus that visualizes functions with two variables as curves on a two-dimensional plane. They represent lines of constant function value and offer an intuitive way to understand how a function behaves across a region.

For the function \( f(x, y) = \frac{\sin(x^2 + y^2)}{x^2 + y^2} \) given in the exercise, contour plots help us grasp regions where the function reaches identical values. In this context, these contours are useful for identifying key behaviors and trends in the function. In practical scenarios, think of contour plots like topographic maps marking identical elevation points.

To create a contour plot, a grid is established over a specified range, in our case \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\). At each grid point, the function value is computed, and lines that connect points having the same value are drawn. For students exploring the world of functions with multiple variables, contour plots offer a two-dimensional slice of three-dimensional insights.

3D Surface Plots

Moving beyond contour plots, 3D surface plots give us a more comprehensive view of a function in three dimensions. These plots use a grid in the \(x-y\) plane and elevate it by the function's value at any point to provide a textured surface.

In our function \( f(x, y) = \frac{\sin(x^2 + y^2)}{x^2 + y^2} \), the 3D surface plot illustrates how the function undulates across the plane. Peaks and valleys can be easily observed, offering more context than contours alone. The surface plot also distinctly showcases the behavior near \((0, 0)\), where the function reaches a value of 1.

Creating a 3D surface plot typically involves using specialized software like MATLAB or Python's Matplotlib library, which can knit together thousands of calculated function values into a visual terrain. This visualization can be rotated and viewed from different angles, providing a deeper understanding of how the function behaves globally.

Limits in Calculus

In calculus, limits define the value that a function approaches as the input approaches a certain point. They are an essential concept, especially when dealing with multivariable functions in cases where direct evaluation isn't possible.

For this function \( f(x, y) = \frac{\sin(x^2 + y^2)}{x^2 + y^2} \), at point \((x, y) = (0, 0)\), direct evaluation is undefined since it leads to a \( \frac{0}{0} \) indeterminate form. To solve this, the limit approach is used: \( \lim_{(x, y) \to (0, 0)} f(x, y) = 1 \). This tells us that, as \(x\) and \(y\) get closer to zero, the function value tends towards unity. Limits like these are commonly solved using techniques from calculus, such as L'Hôpital's rule or polar coordinate transformations.

Understanding limits in multivariable calculus provides clarity on how functions behave near boundaries or critical points, extending functionality beyond just numerical evaluations to insights into continuity and differentiability.

Sinusoidal Functions

Sinusoidal functions, such as the sine and cosine functions, are a cornerstone of trigonometry and multivariable calculus. They describe wave-like behaviors and periodic patterns in phenomena ranging from sound waves to harmonic oscillations.

In our example, the function \( f(x, y) = \frac{\sin(x^2 + y^2)}{x^2 + y^2} \) utilizes a sinusoidal component, \( \sin(x^2 + y^2) \), within its formulation. The sine function dictates the oscillatory nature of \( f(x, y) \) over its domain, contributing to the peaks and troughs seen in both contour and 3D surface plots.

Sinusoidal functions are important for modelling periodic phenomena and are often used in engineering, physics, and other applied sciences. They have properties like amplitude and frequency which influence the shape and scale of waves. As students learn to graph and analyse these functions, they uncover the rhythmic qualities and patterns essential in calculus-based applications.