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Chapter 29: Problem 43

In Exercises \(37-44,\) sketch the indicated curves and surfaces. Sketch the graph of \(x^{2}+y^{2}-2 y=0\) in three dimensions and in two dimensions.

Short Answer

Expert verified
The equation \(x^2 + y^2 - 2y = 0\) represents a circle center at \((0, 1)\), radius 1; the 3D surface is a cylinder along the z-axis.

Step by step solution

01

Understand the Equation

The given equation is \(x^{2} + y^{2} - 2y = 0\). This is a two-dimensional equation representing a circle. To simplify, recognize what form this equation represents and relate it to the standard circle equation.
02

Rearrange the Equation

Let's rearrange the equation \(x^2 + y^2 - 2y = 0\) to make it more recognizable. Group the terms involving \(y\): \(x^2 + (y^2 - 2y) = 0\).
03

Complete the Square

Complete the square for the \(y\)-terms. \(y^2 - 2y\) can be rewritten as \((y-1)^2 - 1\). Substitute this back into the equation: \[x^2 + (y-1)^2 - 1 = 0\].
04

Simplify the Equation

Add \(1\) to both sides to simplify the equation: \[x^2 + (y-1)^2 = 1\]. This is the standard form of a circle equation with center at \((0, 1)\) and radius \(1\).
05

Sketch in Two Dimensions

Plot the circle on the xy-plane. The center of the circle is at \((0, 1)\), and it has a radius of \(1\). Mark the center on a 2D graph and draw a circle around it extending 1 unit in all directions from the center.
06

Interpret in Three Dimensions

In three dimensions, the z-coordinate remains a constant variable forming a cylinder. The curve \(x^2 + (y-1)^2 = 1\) extends infinitely along the z-axis, creating a cylindrical surface with its axis parallel to the z-axis.
07

Sketch in Three Dimensions

To sketch in three dimensions, imagine or draw a circle in the xy-plane as done in two dimensions. Then, extend this circle vertically along the z-axis to represent a cylinder.

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Key Concepts

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

Completing the Square
Completing the square is a powerful mathematical technique used to transform a quadratic equation into a simpler, more manageable form. This involves reorganizing a quadratic expression so it becomes a perfect square trinomial, which simplifies the process of graphing or solving equations.Here's a breakdown of how it works:
  • Start by identifying the quadratic expression you want to simplify, like the terms involving a squared variable, such as in our equation: \(y^2 - 2y\).
  • To complete the square for \(y^2 - 2y\), notice that you can add and subtract the square of half the linear coefficient. In this case, take half of \(-2\) to get \(-1\), and then square it to get \(1\).
  • This turns the expression into a perfect square trinomial: \((y - 1)^2 - 1\).
By using this method, quadratic expressions are transformed into a form that more clearly reveals the properties of the equation, making it easier to draw graphs or identify key features like the vertex or center.
Three-Dimensional Graphs
Visualizing equations in three-dimensional space involves interpreting how relationships between variables change when an additional dimension is introduced. When we discuss three-dimensional graphs, we're looking at how a shape or a curve behaves and extends in the space beyond our usual 2D plane.To understand three-dimensional graphs:
  • Consider each variable as representing a different axis. In the equation \(x^2 + (y-1)^2 = 1\), the x and y variables interact to form a circular shape in the two-dimensional plane.
  • Adding another dimension, such as the z-axis, involves projecting or extending this 2D shape along the new axis. For example, if there's no z in the equation, every point on the circle maintains its z-coordinate unchanged, creating a cylindrical form.
The key to grasping 3D graphs is to imagine how flat, two-dimensional planes extend or transform when they gain a third dimension, adding depth and volume to their representation.
Cylindrical Surfaces
In geometry, cylindrical surfaces are fascinating structures that arise when a two-dimensional figure is projected along a line or axis in a three-dimensional space. The most common example of this is the cylinder created by extending a circle along a straight line, perpendicular to the plane of the circle.Here's how to understand cylindrical surfaces:
  • Start with a simple 2D shape, like the circle given by \(x^2 + (y-1)^2 = 1\). This represents a circular path on the xy-plane.
  • By extending this path along the z-axis, you form a cylinder where every cross-section perpendicular to the z-axis is identical to the original 2D shape.
  • The cylinder's axis is parallel to the z-axis, indicating that it stretches infinitely up and down along the z dimension.
Cylindrical surfaces help illustrate how simple 2D shapes can transition into more complex 3D structures. Recognizing these forms is crucial in fields ranging from architecture to mathematical modeling.

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