/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Draw the level curves (in the \(... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 13

Draw the level curves (in the \(x y\) plane) for the given finction \(f\) and specified values of \(c .\) Sketch the graph of \(z=f(x, y)\) $$f(x, y)=\left(100-x^{2}-y^{2}\right)^{1 / 2}, c=0,2,4,6,8,10$$

Short Answer

Expert verified
Level curves are concentric circles decreasing in radius; surface is a hemisphere with radius 10.

Step by step solution

01

Understanding Level Curves

To draw level curves for the function \( f(x, y) = \sqrt{100 - x^2 - y^2} \), set \( f(x, y) = c \), where \( c \) are the given values 0, 2, 4, 6, 8, 10. This gives \( \sqrt{100 - x^2 - y^2} = c \), which implies \( 100 - x^2 - y^2 = c^2 \). Thus, \( x^2 + y^2 = 100 - c^2 \).
02

Identify Possible Radius

The equation \( x^2 + y^2 = 100 - c^2 \) represents a circle centered at the origin (0, 0) with radius \( \sqrt{100 - c^2} \). Calculate this radius for each \( c \) value: \( c = 0, 2, 4, 6, 8, 10 \).
03

Calculate Radii for Level Curves

For each given \( c \) value: - When \( c = 0 \), radius is \( \sqrt{100} = 10 \).- When \( c = 2 \), radius is \( \sqrt{100 - 4} = \sqrt{96} \).- When \( c = 4 \), radius is \( \sqrt{100 - 16} = \sqrt{84} \).- When \( c = 6 \), radius is \( \sqrt{100 - 36} = 8 \).- When \( c = 8 \), radius is \( \sqrt{100 - 64} = 6 \).- When \( c = 10 \), radius is \( \sqrt{100 - 100} = 0 \).
04

Sketch the Level Curves

Using these radii, draw circles in the \( xy \)-plane for each value of \( c \): - For \( c = 0 \), a circle with radius 10.- For \( c = 2 \), a circle with radius \( \sqrt{96} \), approximately 9.8.- For \( c = 4 \), a circle with radius \( \sqrt{84} \), approximately 9.2.- For \( c = 6 \), a circle with radius 8.- For \( c = 8 \), a circle with radius 6.- For \( c = 10 \), a point at the origin.
05

Sketch the Surface

To visualize the surface \( z = f(x, y) = \sqrt{100 - x^2 - y^2} \), imagine a hemisphere with its base in the \( xy \)-plane and the origin as its center. The top of the hemisphere is at \( z = 10 \), gradually decreasing to the edge of the base where \( x^2 + y^2 = 100 \).

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Key Concepts

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

Circular Level Sets
Level curves represent the connection of points where a function takes a specific constant value. In this problem, we look at the function given by \( f(x, y) = \sqrt{100 - x^2 - y^2} \). By substituting values of \( c \) into the equation \( f(x, y) = c \), we derive level curves, forming circles in the plane.
These curves center at the origin with a radius derived from the equation \( x^2 + y^2 = 100 - c^2 \). Each circle represents a distinct level set, capturing the measure of the function鈥檚 value across the xy-plane. Here, the level set turns into a point at the origin when \( c = 10 \), signifying a collapse as the height of the function, \( z \), reaches zero.
Function Plotting
Plotting \( f(x, y) = \sqrt{100 - x^2 - y^2} \) involves a combination of visualizing its level curves and overall surface in three-dimensional space. Here's how this is broken down:
  • **Level Curves** form concentric circles on the xy-plane, shrinking in radius as \( c \) increases.
  • **Surface View** takes a step further by imagining these circles as horizontal slices of the surface.
    This surface is like a dome (or hemisphere) sitting on the base at the xy-plane.
The visualization allows perception not just of particular values, but the general structure and behavior of the function over its domain. Observing that at \( z = 10 \), the surface reaches its apex, we understand this as the maximum height of the hemisphere.
Radius Calculation
To identify the radius of circular level curves, solve the equation \( x^2 + y^2 = 100 - c^2 \). For each constant \( c \), you calculate \( \sqrt{100 - c^2} \):
  • When \( c = 0 \), the radius calculation is straightforward: \( \sqrt{100} = 10 \).
  • For \( c = 2 \), it becomes \( \sqrt{96} \), which is approximately 9.8, just slightly less than 10.
  • Increasing \( c \) to 4, the radius reduces to \( \sqrt{84} \), about 9.2.
  • At \( c = 6 \), the radius equals 8, reflective of more shrinkage.
  • With \( c = 8 \), the curve tightens further to \( \sqrt{36} = 6 \).
  • Finally, when \( c = 10 \), the radius is zero: a single point at the origin.
Calculating these radii defines how each level defines the size and reach of our circles in the plane. The radii provide tangible measures of drop or elevation of the curves as the value of \( c \) shifts.

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