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Chapter 12: Problem 12

Describe and sketch the graphs of the equations given in Problems 1 through 30 . \(x=1+y^{2}+z^{2}\)

Short Answer

Expert verified
The graph is a circular paraboloid opening along the x-axis.

Step by step solution

01

Identify the Shape of the Equation

The given equation is in the form of a three-dimensional equation: \[ x = 1 + y^2 + z^2 \]This resembles the equation of a circular paraboloid centered at \((1, 0, 0)\) that opens in the positive x-direction.
02

Determine the Cross Sections

To understand the shape better, consider cross sections by setting \(x = c\), where \(c\) is a constant:\[ c = 1 + y^2 + z^2 \] These are circles centered at \((0, 0)\) in the \(yz\)-plane with radius \(\sqrt{c-1}\) for each fixed value of \(x\).
03

Draw the Graph Sketch

Start by drawing the \(yz\)-plane as the base. For increasing values of \(x\), draw circles with increasing radius starting from a point when \(x=1\). This forms a parabolic-shaped surface that spreads wider as \(x\) increases.

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Key Concepts

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

3D Graph Sketching
Creating a sketch for a 3D graph might seem daunting, but it's quite like painting a picture with your mathematical tools. For our equation, the first step is to visualize the overall shape. Here, the equation \(x = 1 + y^2 + z^2\) tells us we're dealing with a circular paraboloid. Imagine a bowl turned on its side, opening along the positive x-axis. The surface curves and widens as it moves away from the vertex, centering at point \((1,0,0)\).
To sketch such an object successfully:
  • Begin with the coordinate axes, focusing on the \(x\), \(y\), and \(z\) directions.
  • Visualize the central point where \(x=1\) and draw outward, as this is where your paraboloid begins.
  • Recognize that this isn't a perfect circle when viewed directly on the xy or yz planes, but it assumes more of an elongated, symmetrical shape known as the paraboloid.
With these tips, a complex 3D surface can be illustrated with clarity and understanding.
Cross-Sectional Analysis
Understanding the cross-section of a 3D object is like breaking down the object slice by slice. With our paraboloid, setting \(x = c\) allows us to examine these slices. Each slice is a circle in the \(yz\)-plane with equations like \c = 1 + y^2 + z^2\.
When performing cross-sectional analysis:
  • For a specific constant \(c\), calculate the radius of the circle: \(\sqrt{c-1}\).
  • Notice that these circles get larger as \(c\) increases, meaning as you move further from the point \(x=1\).
  • These circular cross-sections give an intuitive sense of how the paraboloid spreads outwards.
This analysis helps to comprehend the increasing radius and how cross-sections form a bigger picture of the 3D graph.
Three-Dimensional Equation
Three-dimensional equations may appear intricate but break down into components we can understand more clearly. The equation \(x = 1 + y^2 + z^2\) is a classic example of this calculative simplicity.
Breaking it down:
  • The equation is a quadratic form in terms of \(y\) and \(z\), translating to a parabolic shape in 3D.
  • Notice the axially aligned setup where changes in \(y\) or \(z\) influence the equation equally, maintaining symmetry.
  • This symmetry creates the circular shape found in cross-sections, firmly associating the equation with real-world physics in forms like satellite dishes or in designing optics.
Understanding a 3D equation as distinct parts acting together helps demystify the overall structure and makes tackling similar equations approachable and less imposing.

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Most popular questions from this chapter

Describe the graph of the given equation. (It is understood that equations including \(r\) are in cylindrical coordinates and those including \(\rho\) or \(\phi\) are in spherical coordinates.) \(\rho^{3}+4 \rho=0\)

Describe the graph of the given equation. (It is understood that equations including \(r\) are in cylindrical coordinates and those including \(\rho\) or \(\phi\) are in spherical coordinates.) \(\phi=5 \pi / 6\)

Describe and sketch the surface or solid described by the given equations and/or inequalities. \(0 \leqq \rho \leqq 2,0 \leqq \phi \leqq \pi / 2\)

Describe and sketch the graphs of the equations given in Problems 1 through 30 . \(x^{2}+9 y^{2}+4 z^{2}=36\)

Describe and sketch the graphs of the equations given in Problems 1 through 30 . \(x=\sin y\)
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