/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Plot the following surfaces: (... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 8

Plot the following surfaces: (a) \(z=\sin x \sin y\) for \(-3 \pi \leq x \leq 3 \pi\) and \(-3 \pi \leq y \leq 3 \pi\), (b) \(z=\left(x^{2}+y^{2}\right) \cos \left(x^{2}+y^{2}\right)\) for \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\)

Short Answer

Expert verified
Plot surface (a) using meshgrid and \(z = \sin x \sin y\) over \(-3\pi \leq x,y \leq 3\pi\); plot surface (b) with \(z = (x^2 + y^2)\cos(x^2 + y^2)\) over \(-1 \leq x,y \leq 1\).

Step by step solution

01

Understanding Surface (a)

We need to plot the surface given by the equation \(z = \sin x \sin y\). This surface is a product of two sine functions in terms of \(x\) and \(y\), forming a wave-like pattern.
02

Defining the Domain for (a)

The domain for this function is defined as \(-3\pi \leq x \leq 3\pi\) and \(-3\pi \leq y \leq 3\pi\). This means that both \(x\) and \(y\) range from \(-3\pi\) to \(3\pi\).
03

Plotting the Surface (a)

In a suitable software (like MATLAB or Python's Matplotlib), create a meshgrid covering the specified domain. Calculate \(z\) for each point in the mesh grid, using the formula \(z = \sin x \sin y\). Use a 3D plot function to visualize the surface.
04

Understanding Surface (b)

For surface (b), the equation is \(z = (x^2 + y^2) \cos (x^2 + y^2)\). This involves both quadratic and trigonometric operations, creating a unique pattern.
05

Defining the Domain for (b)

The specified domain is \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\). This means both \(x\) and \(y\) range from \(-1\) to \(1\).
06

Plotting the Surface (b)

Again, use a meshgrid for the specified domain in a computational tool. Calculate \(z\) using the formula \(z = (x^2 + y^2) \cos (x^2 + y^2)\) for each point in the mesh. Use a 3D plot function to display the surface.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Plots
Surface plots allow us to visualize how a particular function behaves across a specified domain in 3D space. They are essential in understanding multivariable functions by providing a visual representation of how two variables interact to produce a third outcome, known as the 'z' value. In MATLAB, creating a surface plot often involves using meshgrids to create a grid of 'x' and 'y' values over a given range. Once these grids are defined, the function's output for each point on the grid can be computed, and MATLAB's built-in 3D plotting tools can render the surface.
  • They are useful in fields such as engineering and physics where understanding the topology of surfaces is crucial.
  • They visually convey peaks, valleys, and contours of the function in the domain.
This makes interpreting complex mathematical functions significantly more tangible.
Trigonometric Functions
Trigonometric functions are a cornerstone in mathematics, often used to describe the properties of waves, oscillations, and periodic phenomena. In this context, the function \( z = \sin x \sin y \) represents a product of two sine functions, which creates a series of waves that oscillate both horizontally and vertically. This pattern generates a ripple effect akin to waves on a pond.
  • The periodic nature of sine functions results in repeated patterns, which is useful for studying repetitive or cyclical systems.
  • They are essential in modeling real-world systems, such as sound waves, light waves, and alternating current electricity.
Understanding how these functions influence surface plots can better equip a learner in predicting and analyzing the behavior of various physical systems.
Domain Definition
Defining the domain of a function is a fundamental step in mathematical modeling and analysis. In surface plotting, the domain specifies the range of input values鈥攖he 'x' and 'y' values鈥攖hat the function will evaluate.
  • For the function \( z = \sin x \sin y \), the domain is defined as \(-3\pi \leq x \leq 3\pi\) and \(-3\pi \leq y \leq 3\pi\). This large range captures several oscillations of the sine wave, allowing for a detailed visualization.
  • For the function \( z = (x^2 + y^2) \cos (x^2 + y^2) \), the domain is \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\), focusing on a smaller area where subtle changes and local variations are of interest.
A clear understanding of the domain helps tailor the visual analysis and ensures the plot effectively communicates the function鈥檚 behavior over the desired range.
3D Visualization
3D visualization is a powerful tool that enhances the interpretation of complex data by offering a spatial perspective. It provides insights into the depth and detailed structure of mathematical functions.
  • By presenting data in three dimensions, users can observe not just the outcomes of \( z \) values, but also visually track how changes in \( x \) and \( y \) affect these outcomes.
  • In MATLAB, this involves using functions like plot3D, mesh, or surf, which help in rendering a detailed and interactive surface plot.
3D visualization is not only beneficial for academic endeavors but is also crucial in professional fields where spatial analysis is needed, such as in computer graphics, geographical mapping, and even in the medical imaging domain, providing three-dimensional views of anatomical structures.

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Most popular questions from this chapter

Evaluate the following limits: (a) \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\) (b) \(\lim _{x \rightarrow-\pi} \frac{1+\cos x}{x+\pi}\) (c) \(\lim _{x \rightarrow \infty} x^{2} e^{-x}\) (d) \(\lim _{x \rightarrow 1^{-}} \frac{1}{x-1}\) (e) \(\lim _{x \rightarrow n+} \sin \left(\frac{1}{x}\right)\).

Find the Taylor polynomial of the indicated degree \(n\) at the indicated point \(c\) for the following functions: (a) \(f(x)=e^{x}, n=6, c=0\) (b) \(f(x)=\sin x, n=4\) and \(6, c=0\) (c) \(f(x)=\sin x, n=5, c=2\) (d) \(f(x)=\tan x, n=6, c=0\) (e) \(f(x)=\ln x, n=4, c=1\)

| Genotype of Parental
Offspring | Genotype | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | | AA-AA | AA-Aa | AA-aa | Aa-Aa | Aa-aa | aa-aa | | AA | 1 | 1//2 | 0 | 1//4 | 0 | 0 | | Aa | 0 | 1//2 | 1 | 1//2 | 1//2 | 0 | (a) Write these equations as a single matrix equation \(X_{n+1}=M X_{n}, n \geq 0\). Explain carefully what the entries of the column matrix \(X_{n}\) are, and what the coefficients of the square matrix \(M\) are. (b) Apply the matrix equation recursively to express \(X_{n}\) in terms of \(X_{0}\) and powers of \(M\). (c) Next use MATLAB to compute the eigenvalues and eigenvectors of \(M\). (d) From Problem 12 you know that \(M U=U R\), where \(R\) is the diagonal matrix of eigenvalues of \(M\). Solve that equation for \(M\). Can you see what \(R_{\infty}=\lim _{n \rightarrow \infty} R^{n}\) is? Use that and your above expression of \(M\) in terms of \(R\) to compute \(M_{\infty}=\lim _{n \rightarrow \infty} M^{n}\). (e) Describe the eventual population distribution by computing \(M_{\infty} X_{0}\). (f) Check your answer by directly computing \(M^{n}\) for large specific values of M. (Hint: MATLAB can compute the powers of a matrix \(\mathrm{M}\) by entering \(\mathrm{M}^{\wedge} 10\), for example.) (g) You can alter the fundamental presumption in this problem by assuming, alternatively, that all members of the \(n\)th generation must mate only with a parent whose genotype is purely dominant. Compute the eventual population distribution of that model. Chapters 12-14 in Rorres and Anton have other interesting models. | aa | 0 | 0 | 0 | 1//4 | 1//2 | 1 |

Use contour to do the following. (a) Plot the level curves of the function \(f(x, y)=3 y+y^{3}-x^{3}\) in the region where \(x\) and \(y\) are between \(-1\) and 1 (to get an idea of what the curves look like near the origin), and in some larger regions (to get the big picture). (b) Plot the curve \(3 y+y^{3}-x^{3}=5\). (c) Plot the level curve of the function \(f(x, y)=y \ln x+x \ln y\) that contains the point \((1,1)\).

Create a 17-frame movie, whose frames show filled red circles of radius \(1 / 2\) centered at the points \((4 \cos (j \pi / 8), 4 \sin (j \pi / 8)), j=0,1, \ldots, 16\). Make sure that all the circles are drawn on the same set of axes, and that they look like circles, not ellipses.
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