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Chapter 14: Problem 58

In Exercises \(53-60,\) sketch a typical level surface for the function. $$f(x, y, z)=y^{2}+z^{2}$$

Short Answer

Expert verified
The typical level surface is a circular cylinder with radius \( \sqrt{c} \) aligned along the \( x \)-axis.

Step by step solution

01

Understand the Function

The given function is \( f(x, y, z) = y^2 + z^2 \). This function does not depend on \( x \), indicating that the level surfaces will be the same in every \( x \)-plane, which points to a cylindrical symmetry.
02

Consider the Level Surface Equation

To sketch a level surface, set \( f(x, y, z) = c \), where \( c \) is a constant. Thus, we have \( y^2 + z^2 = c \). This equation represents a circle in the \( yz \)-plane for a fixed \( x \).
03

Recognize the Cylindrical Shape

Since the equation \( y^2 + z^2 = c \) depicts circles in the \( yz \)-plane for any constant \( c \), when combined with the fact that \( x \) does not affect \( f \), the level surface is a cylinder along the \( x \)-axis with radius \( \sqrt{c} \).
04

Sketch the Typical Level Surface

For a typical value of \( c > 0 \), sketch a right circular cylinder aligned along the \( x \)-axis. The circles forming this cylinder have radius \( \sqrt{c} \) on planes parallel to the \( yz \)-plane.

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Key Concepts

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

Cylindrical Symmetry
Cylindrical symmetry occurs when a structure looks the same around a central axis. In mathematical terms, this means that the form or graph of a function does not change as you move around or along this specific axis. For the function you are given, \(f(x, y, z) = y^2 + z^2\), notice how there is no \(x\) in the equation. This means that whatever shape the function forms in one plane (for a fixed \(x\) value), remains the same for all other \(x\) values. Essentially, the level surfaces are `cylindrical` parallel to the \(x\)-axis, maintaining identical cross-sections at every point along the axis. This characteristic is the hallmark of cylindrical symmetry.
  • Each cross-section parallel to the \(yz\)-plane is a circle.
  • The radius is constant for any given level surface value.
  • The symmetry makes it easier to sketch as the shape is repetitive across the \(x\)-axis.
Level Surface Equation
A level surface equation derives from taking a multivariable function and setting it to a constant. In this context, it simplifies a 3D scenario down to a more manageable graph or shape at certain values. For \(f(x, y, z) = y^2 + z^2\), if you set \(f\) to a constant \(c\), you get \(y^2 + z^2 = c\). This is a known equation of a circle. With the absence of \(x\) in the equation, you get a collection of these circles stacked infinitely along the \(x\)-axis. This leads to the creation of a cylinder.
  • Equation: \(y^2 + z^2 = c\), where \(c > 0\).
  • This shows how a 2D concept (the circle) becomes a 3D structure (the cylinder).
  • Each \(c\) value represents a different cylinder radius of \(\sqrt{c}\).
Circular Cross-Sections
In mathematics, cross-sections help to visualize and understand the shape of a solid figure better by taking a 'slice' perpendicular to an axis. For the function \(f(x, y, z) = y^2 + z^2\), any plane parallel to the \(yz\)-plane will have a circular cross-section. When you imagine cutting through the cylinder perpendicular to its axis, each slice, or cross-section, is a circle.
  • Each circle has a center at the origin of the \(yz\)-plane (0,0).
  • The radius of these circles is \(\sqrt{c}\), derived from \(y^2 + z^2 = c\).
  • These circles repeat consistently as you move along the \(x\)-axis.
Makes it intuitive to sketch and understand cylinder geometry by looking at the repeating pattern of circles.
Mathematical Visualization
Visualizing multivariable functions helps in understanding complicated structures and how different variables affect them. With \(f(x, y, z) = y^2 + z^2\), constructing a mental or physical model involves understanding how the components form a shape through symmetry, equations, and cross-sections. For this scenario:
  • The absence of \(x\) compresses variety into two dimensions, making a cylinder in 3D.
  • Visualizing cylinders revolves around repetitive circular patterns (as seen in cross-sections).
  • Visual models can be helpful. Draw multiple circular cross-sections along \(x\)-axis to emphasize symmetry.
Don't forget that mathematical visualization enhances problem-solving and deeper comprehension.

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Most popular questions from this chapter

Find \((\partial u / \partial y)_{x}\) at the point \((u, v)=(\sqrt{2}, 1)\) if \(x=u^{2}+v^{2}\) and \(y=u v .\)

Minimum distance to the origin Find the point closest to the origin on the line of intersection of the planes \(y+2 z=12\) and \(x+y=6 .\)

In Exercises \(49-54\) , use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function \(h=f-\lambda_{1} g_{1}-\lambda_{2} g_{2},\) where \(f\) is the function to optimize subject to the constraints \(g_{1}=0\) and \(g_{2}=0\) b. Determine all the first partial derivatives of \(h\) , including the partials with respect to \(\lambda_{1}\) and \(\lambda_{2},\) and set them equal to \(0 .\) c. Solve the system of equations found in part (b) for all the unknowns, including \(\lambda_{1}\) and \(\lambda_{2}\) . d. Evaluate \(f\) at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Maximize \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) subject to the constraints \(2 y+4 z-5=0\) and \(4 x^{2}+4 y^{2}-z^{2}=0\)

In Exercises \(49-54\) , use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function \(h=f-\lambda_{1} g_{1}-\lambda_{2} g_{2},\) where \(f\) is the function to optimize subject to the constraints \(g_{1}=0\) and \(g_{2}=0\) b. Determine all the first partial derivatives of \(h\) , including the partials with respect to \(\lambda_{1}\) and \(\lambda_{2},\) and set them equal to \(0 .\) c. Solve the system of equations found in part (b) for all the unknowns, including \(\lambda_{1}\) and \(\lambda_{2}\) . d. Evaluate \(f\) at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Minimize \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) subject to the constraints \(x^{2}-x y+y^{2}-z^{2}-1=0\) and \(x^{2}+y^{2}-1=0\)

Suppose that $$ w=x^{2}-y^{2}+4 z+t \quad \text { and } \quad x+2 z+t=25 $$ Show that the equations $$ \frac{\partial w}{\partial x}=2 x-1 \quad \text { and } \quad \frac{\partial w}{\partial x}=2 x-2 $$ each give \(\partial w / \partial x,\) depending on which variables are chosen to be dependent and which variables are chosen to be independent. Identify the independent variables in each case.
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