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Chapter 11: Problem 37

Identify the surface and make a rough sketch that shows its position and orientation. $$ z=(x+2)^{2}+(y-3)^{2}-9 $$

Short Answer

Expert verified
The surface is an upward-opening paraboloid with a vertex at \((-2, 3, 0)\).

Step by step solution

01

Identify the Surface Type

Start by recognizing the given equation form. The equation is \( z = (x+2)^2 + (y-3)^2 - 9 \). This resembles the standard form of a paraboloid \( z = ax^2 + by^2 + c \). Since both \( x^2 \) and \( y^2 \) terms are present and positive, this is an upward-opening paraboloid.
02

Find the Vertex of the Paraboloid

The vertex of the paraboloid can be found by examining the squared terms. Transforming these terms, the vertex is located at \( (x+2) = 0 \) and \( (y-3) = 0 \), resulting in the vertex being at \( (-2, 3) \). The z-coordinate of the vertex can be found by setting \( x = -2 \) and \( y = 3 \) giving \( z = 0 \). Thus, the vertex is at \((-2, 3, 0)\).
03

Determine the Orientation

Since both squared terms \((x+2)^2\) and \((y-3)^2\) are positive, and the linear term is added to these positive terms, the paraboloid opens upwards along the z-axis.
04

Sketch the Surface

With the vertex known and the upward orientation established, execute a rough sketch. Plot a point at the vertex \((-2, 3, 0)\). Draw a three-dimensional box representing the x, y, and z axes. Sketch the parabolic shape opening upwards from the vertex. Indicate the orientation in positive z-direction.

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Key Concepts

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

Paraboloid
A paraboloid is a type of surface in three-dimensional space. It can be thought of as the 3D version of a parabola. Paraboloids come in two variants: elliptic and hyperbolic. The surface presented in the exercise is an elliptic paraboloid. In an elliptic paraboloid, the cross-sections obtained by slicing it parallel to the x-y plane are circles or ellipses. Moreover, the cross-sections parallel to the x-z plane and y-z plane are parabolas.

In the given equation \[ z = (x+2)^2 + (y-3)^2 - 9 \],both \( (x+2)^2 \) and \( (y-3)^2 \) are positive squares. This corresponds to the shape of an elliptic paraboloid that opens upwards along the z-axis, making it easy to visualize if you imagine a bowl's shape.

This type of paraboloid is commonly used in engineering and physics to describe various phenomena, like the shape of satellite dishes, or for approximations in calculus when studying local behavior of functions. Understanding the nature of the paraboloid helps in comprehending its applications in real-world scenarios.
Vertex of a Surface
The vertex is a key feature of a paraboloid, similar to the peak of a mountain or the lowest point of a valley, depending on its orientation. For the paraboloid described by the equation \[ z = (x+2)^2 + (y-3)^2 - 9 \],the vertex is obtained by examining the squared terms. Replace \( (x+2) \) and \( (y-3) \) with zero to find the coordinates.

  • The equation \( (x+2) = 0 \) gives \( x = -2 \).
  • The equation \( (y-3) = 0 \) gives \( y = 3 \).
Thus, the vertex's coordinates are \( (-2, 3) \). By substituting these coordinates into the equation, \( z = 0 \), which fully reveals the vertex at \((-2, 3, 0)\).
Knowing the vertex helps determine the surface's position in space. It acts as a reference point for graphing and analyzing its orientation. The vertex also holds significance in problems involving optimization, where it might represent a minimum or maximum value.
Graphing Surfaces
Graphing multivariable surfaces can initially seem daunting, but breaking it down into steps makes it manageable and insightful. Let's look at graphing the surface given by \[ z = (x+2)^2 + (y-3)^2 - 9 \]:

  • Understand the Basics: Recognize that you're dealing with a 3D shape; it is an elliptic paraboloid opening upwards.
  • Locate the Vertex: With the vertex at \((-2, 3, 0)\), plot this point first as it will ground your sketch.
  • Sketch the Axes: Draw three-dimensional axes to provide context 鈥� x, y, and z.
  • Indicate the Parabolic Curve: Starting from the vertex, draw a parabolic shape stretching upwards along the z-axis. Remember that the symmetrical nature implies similar expansion in all directions from the vertex.
Graphing involves visualizing and interpreting mathematical relationships in spatial terms, a fundamental skill in multivariable calculus used to solve problems involving constraints and intersections of surfaces. Such graphs can effectively illustrate real-world models like satellite dishes or mountain ranges and enhance comprehension of how mathematical equations model spatial phenomena.

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Most popular questions from this chapter

Express the vector \(\mathbf{v}\) as the sum of a vector parallel to \(\mathbf{b}\) and a vector orthogonal to \(\mathbf{b}\). (a) \(\mathbf{v}=\langle-3,5\rangle, \mathbf{b}=\langle 1,1\rangle\) (b) \(\mathbf{v}=\langle-2,1,6\rangle, \mathbf{b}=\langle 0,-2,1\rangle\) (c) \(\mathbf{v}=\langle 1,4,1\rangle, \mathbf{b}=\langle 3,-2,5\rangle\)

Let \(\mathbf{u}\) be a unit vector in the \(x y\) -plane of an \(x y z\) -coordinate system, and let \(\mathbf{v}\) be a unit vector in the \(y z\) -plane. Let \(\theta_{1}\) be the angle between \(\mathbf{u}\) and \(\mathbf{i}\), let \(\theta_{2}\) be the angle between \(\mathbf{v}\) and \(\mathbf{k}\), and let \(\theta\) be the angle between \(\mathbf{u}\) and \(\mathbf{v}\). (a) Show that \(\cos \theta=\pm \sin \theta_{1} \sin \theta_{2}\). (b) Find \(\theta\) if \(\theta\) is acute and \(\theta_{1}=\theta_{2}=45^{\circ}\). (c) Use a CAS to find, to the nearest degree, the maximum and minimum values of \(\theta\) if \(\theta\) is acute and \(\theta_{2}=2 \theta_{1}\).

Suppose that the temperature \(T\) at a point \((x, y, z)\) on the line \(x=t, y=1+t, z=3-2 t\) is \(T=25 x^{2} y z .\) Use a CAS or a calculating utility with a root-finding capability to approximate the maximum temperature on that portion of the line that extends from the \(x z\) -plane to the \(x y\) -plane.

Sketch the region enclosed between the surfaces and describe their curve of intersection. The ellipsoid \(2 x^{2}+2 y^{2}+z^{2}=3\) and the paraboloid \(z=x^{2}+y^{2}\)

Express the vector \(\mathbf{v}\) as the sum of a vector parallel to \(\mathbf{b}\) and a vector orthogonal to \(\mathbf{b}\). (a) \(\mathbf{v}=2 \mathbf{i}-4 \mathbf{j}, \mathbf{b}=\mathbf{i}+\mathbf{j}\) (b) \(\mathbf{v}=3 \mathbf{i}+\mathbf{j}-2 \mathbf{k}, \mathbf{b}=2 \mathbf{i}-\mathbf{k}\) (c) \(\mathbf{v}=4 \mathbf{i}-2 \mathbf{j}+6 \mathbf{k}, \mathbf{b}=-2 \mathbf{i}+\mathbf{j}-3 \mathbf{k}\)
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