Problem 43 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

Plot a 3D surface graph and a 2D contour plot using the grid from \(-2\) to \(2\) for both \(x\) and \(y\).

Step 1: Understand the Function

The function we need to work with is \( f(x, y) = \sin \sqrt{2x^2 + y^2} \). This is a two-variable function involving trigonometric and square root components. It oscillates because of the sine function.

Step 2: Define the Grid for Input Variables

Since \( x \) and \( y \) both range from -2 to 2, we will create a grid over these values. Typically, this can be achieved using a mesh grid in software tools like Python with libraries such as NumPy.

Step 3: Compute the Function Values

For each combination of \( x \) and \( y \) on our grid, compute \( 2x^2 + y^2 \), take the square root of the result, and then take the sine. This gives us the values of \( f(x, y) \) across the grid.

Step 4: Draw the Graph

Plot the values of \( f(x, y) \) over the grid as a 3D surface. This graph will show how \( f(x, y) \) changes over the range of \( x \) and \( y \). Use software like Matplotlib which provides functions like `plot_surface` to create 3D plots.

Step 5: Draw the Contour Plot

Plot lines of constant \( f(x, y) \) values over the grid to form a contour plot. This can be done using contour plotting functions available in plotting libraries. It reduces the 3D plot into a 2-dimensional graph where different colors represent different ranges of function values.

Key Concepts

Trigonometric Functions

Trigonometric functions are widely used in calculus to describe periodic phenomena. One such function, the sine function, is characterized by its wave-like oscillations. In our exercise, the function given is \( f(x, y) = \sin \sqrt{2x^2 + y^2} \). Here, the sine function is applied to the square root of a combination of \( x \) and \( y \), resulting in a complex oscillatory behavior.

• Sine functions oscillate between -1 and 1.
• This oscillation causes any output to consistently rise and fall in a regular pattern.
• In multivariable functions like this one, the sine function creates a surface that goes up and down, forming ripples or waves.

When studying such functions, it's crucial to understand how the trigonometric component contributes to the behavior of the graph. This understanding provides insight into the potential shapes and patterns that can be expected in both the 3D and contour plots.

3D Plotting

3D plotting is a technique that allows us to visualize functions of two variables in three dimensions. For the function \( f(x, y) = \sin \sqrt{2x^2 + y^2} \), the 3D plot will visualize how the function value (z-axis) changes with varying \( x \) and \( y \) inputs.

• The x-axis and y-axis represent the range of input variables as defined from -2 to 2.
• The z-axis displays the corresponding sine values, resulting in a surface that reflects the function's oscillatory nature.
• Tools like Matplotlib's `plot_surface` function are typically used to generate these plots.

Through 3D plotting, we observe the dynamic nature of surfaces formed by multivariable functions and are able to visually understand their complexities. It renders the intricate waves and peaks of the sine function in a comprehensible manner.

Contour Plots

Contour plots provide a two-dimensional representation of a three-dimensional surface by using lines to connect points at the same function value. In the context of our function, the contour plot helps simplify visual complexity by encapsulating the oscillations of \( f(x, y) \) in a 2D framework.

• Each contour line represents a constant function value.
• Different colors can indicate higher or lower function values.
• These plots are useful for finding gradients and analyzing function behavior at a glance.

Utilizing contour plots allows us to identify key features, such as peaks, valleys, and slopes, without the need to interpret a full 3D graph. This is particularly helpful when analyzing large data sets or when we need a quick and clear depiction of function trends.

Function Analysis

Function analysis involves examining the characteristics and behavior of a function to understand its properties and potential applications. For the function \( f(x, y) = \sin \sqrt{2x^2 + y^2} \), conducting a function analysis includes:

• Identifying the domain and range of the function, as provided with \( x \) and \( y \) values ranging from -2 to 2.
• Understanding the periodic nature induced by the sine function.
• Anticipating how changes in \( x \) and \( y \) can affect the resulting output, particularly regarding the oscillation patterns.
• Checking symmetry and possible maxima or minima to uncover critical points.

Effective analysis of such functions helps predict their behaviors in real-world scenarios and allows for improved problem-solving strategies in multivariable calculus. It's about diving deeper into how various factors interact within the function to shape its output dynamics.