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Chapter 10: Problem 4

Describe in words the set of all points in \(\mathbf{R}^{3}\) that satisfy the following expressions: (a) \(x=0\) (b) \(y=0\) (c) \(z=0\) (d) \(z \geq 0\) (e) \(y \leq 0\)

Short Answer

Expert verified
Points on yz-plane, xz-plane, xy-plane, half-space above xy-plane, and half-space below xz-plane.

Step by step solution

01

Understanding Expression (a)

The expression \(x = 0\) describes all points in three-dimensional space where the x-coordinate is zero. This means that the points lie on the yz-plane. The yz-plane is the plane parallel to the x-axis that contains the y-axis and z-axis.
02

Understanding Expression (b)

The expression \(y = 0\) describes all points where the y-coordinate is zero. These points lie on the xz-plane. The xz-plane contains all points that are parallel to the y-axis and lies in the direction of the x and z axes.
03

Understanding Expression (c)

The expression \(z = 0\) describes all points where the z-coordinate is zero. These points lie on the xy-plane. The xy-plane contains all x and y points and is parallel to the z-axis.
04

Understanding Expression (d)

The expression \(z \geq 0\) defines all points in space where the z-coordinate is greater than or equal to zero. This includes the entire half-space above and including the xy-plane.
05

Understanding Expression (e)

The expression \(y \leq 0\) describes the set of points where the y-coordinate is less than or equal to zero. This includes the entire half-space on or below the xz-plane.

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Key Concepts

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

Understanding the yz-plane
In three-dimensional space, when we talk about the yz-plane, we are referring to a very specific set of points. These are all the points where the x-coordinate is equal to zero. Imagine it as a flat sheet that stretches across the y and z axes, leaving out the x-axis. You can think of it like a page in a book that lies between the two book covers (in this case, the y-axis and z-axis), standing upright without touching the x-axis.
  • The yz-plane is parallel to the x-axis.
  • It includes points such as (0, 1, 2), (0, -3, 5), and (0, 4, -2).
  • For any point on this plane, the equation is defined by setting the x-coordinate to zero: \(x = 0\).
  • This plane is crucial for graphing and visualizing concepts in 3D space.
Understanding the xz-plane
The xz-plane is another major reference plane in three-dimensional space. This plane is all about the points where the y-coordinate is zero. Picture it as the flat surface spanning between the x-axis and the z-axis like a perfectly horizontal sheet on a table.
  • The xz-plane runs parallel to the y-axis.
  • Examples of points on the xz-plane are (1, 0, 2), (-3, 0, 5), and (4, 0, -2).
  • To find the xz-plane, set the y-coordinate to zero: \(y = 0\).
  • Understanding this plane helps in visualizing changes happening in x and z directions only.
Understanding the xy-plane
The xy-plane is a fundamental plane in 3D space that includes points where the z-coordinate is zero. Imagine it as a big flat stage that the axes x and y create, extending infinitely in those directions.
  • The xy-plane is parallel to the z-axis.
  • Points like (1, 2, 0), (-3, 5, 0), and (4, -2, 0) are all on the xy-plane.
  • The equation for this plane is expressed by \(z = 0\).
  • This plane is widely used in mathematical modeling and drawing projections.
Understanding half-space in 3D
In three-dimensional geometry, a half-space is one part of space divided by a plane. You can think of it as one half of a sandwich where a plane acts like the bread dividing it.

Positive and negative half-space:

  • The expression \(z \geq 0\) defines a positive half-space above the xy-plane, encompassing all points where the z-coordinate is zero or positive. It includes places like (1, 2, 0) or (-3, 5, 4).
  • Similarly, \(y \leq 0\) forms a negative half-space along or below the xz-plane, covering points such as (4, 0, -1) or (-2, -3, -4).

Significance of half-spaces:

  • Half-spaces are useful in inequalities and optimization problems in mathematics.
  • Dividing spaces into half-spaces helps in analyzing and describing geometric locations.

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Most popular questions from this chapter

The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by \(N\), the predator density by \(P\), the time available for searching for prey by \(T\), and the handling time of each prey item per predator by \(T_{h}\), then the number of prey encounters per predator as a function of \(N, T\), and \(T_{h}\) can be expressed as $$ f\left(N, T, T_{h}\right)=\frac{b^{2} N^{2} T}{1+c N+b T_{h} N^{2}} $$ where \(b\) and \(c\) are positive constants. (a) Investigate how an increase in the prey density \(N\) affects the function \(f\left(N, T, T_{h}\right)\) (b) Investigate how an increase in the time \(T\) available for search affects the function \(f\left(N, T, T_{h}\right)\). (c) Investigate how an increase in the handling time \(T_{h}\) affects the function \(f\left(N, T, T_{h}\right)\) (d) Graph \(f\left(N, T, T_{h}\right)\) as a function of \(N\) when \(T=2.4\) hours, \(T_{h}=0.2\) hours, \(b=0.8\), and \(c=0.5\)

The negative binomial model can be reduced to the Nicholson-Bailey model by letting the parameter \(k\) in the negative binomial model go to infinity. Show that $$ \lim _{k \rightarrow \infty}\left(1+\frac{a P}{k}\right)^{-k}=e^{-a P} $$ (Hint: Use l'Hospital's rule.)

Refer to the negative binomial host-parasitoid model. Problems \(7,8,11\), and 12 are best done with the help of a spreadsheet, but can also be done with a calculator. The negative binomial model is a discrete-generation host- parasitoid model of the form $$ \begin{array}{l} N_{t+1}=b N_{t}\left(1+\frac{a P_{t}}{k}\right)^{-k} \\ P_{t+1}=c N_{t}\left[1-\left(1+\frac{a P_{t}}{k}\right)^{ k}\right] \end{array} $$ for \(t=0,1,2, \ldots\) When the initial parasitoid density is \(P_{0}=0\), the negative binomial model reduces to $$ N_{t+1}=b N_{t} $$ as shown in the previous problem. For which values of \(b\) is the host density increasing if \(N_{0}>0 ?\) For which values of \(b\) is it decreasing? (Assume that \(b>0 .\) )

Find the Jacobi matrix for each given function. $$ \mathbf{f}(x, y)=\left[\begin{array}{c} x+y \\ x^{2}-y^{2} \end{array}\right] $$

Find a linear approximation to each function \(f(x, y)\) at the indicated point. $$ \mathbf{f}(x, y)=\left[\begin{array}{c} (x+y)^{2} \\ x y \end{array}\right] \text { at }(-1,1) $$
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