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A graph of the various level curves of a function is called a contour map. Example: Making a Contour Map Given the function [latex]f\,(x,\ y)=\sqrt{8+8x-4y-4x^{2}-y^{2}}[/latex], find the level curve corresponding to [latex]c=0[/latex].
Nov 16, 2022 · The level curves of the function \(z = f\left( {x,y} \right)\) are two dimensional curves we get by setting \(z = k\), where \(k\) is any number. So the equations of the level curves are \(f\left( {x,y} \right) = k\). Note that sometimes the equation will be in the form \(f\left( {x,y,z} \right) = 0\) and in these cases the equations of the ...
Sep 29, 2023 · Topographical maps can be used to create a three-dimensional surface from the two-dimensional contours or level curves. For example, level curves of the distance function defined by \(f(x,y) = \frac{x^2 \sin(2y)}{32}\) plotted in the \(xy\)-plane are shown at left in Figure \(\PageIndex{8}\).
A level curve of a function of two variables f (x, y) f (x, y) is completely analogous to a contour line on a topographical map. Figure 4.7 (a) A topographical map of Devil’s Tower, Wyoming. Lines that are close together indicate very steep terrain.
Dec 29, 2020 · Example \(\PageIndex{3}\): Drawing Level Curves. Let \(f(x,y) = \sqrt{1-\frac{x^2}9-\frac{y^2}4}\). Find the level curves of \(f\) for \(c=0\), \(0.2\), \(0.4\), \(0.6\), \(0.8\) and \(1\). Solution. Consider first \(c=0\). The level curve for \(c=0\) is the set of all points \((x,y)\) such that \(0=\sqrt{1-\frac{x^2}9-\frac{y^2}4}\).
Nov 17, 2020 · Sketch a graph of a function of two variables. Sketch several traces or level curves of a function of two variables. Recognize a function of three or more variables and identify its level surfaces.
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One way to collapse the graph of a scalar-valued function of two variables into a two-dimensional plot is through level curves. A level curve of a function $f(x,y)$ is the curve of points $(x,y)$ where $f(x,y)$ is some constant value. A level curve is simply a cross section of the graph of $z=f(x,y)$ taken at a constant value, say $z=c$. A ...
