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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 (or contour curves) for this surface are given by the equation are found by substituting \(z = k\). In the case of our example this is, \[k = \sqrt {{x^2} + {y^2}} \hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}{x^2} + {y^2} = {k^2}\]
Level Curves. If hikers walk along rugged trails, they might use a topographical map that shows how steeply the trails change. A topographical map contains curved lines called contour lines. Each contour line corresponds to the points on the map that have equal elevation .
When working with functions , the level sets are known as level curves. When we are looking at level curves, we can think about choosing a -value, say , in the range of the function and ask “at which points can we evaluate the function to get ?”
For example, if $c=-1$, the level curve is the graph of $x^2 + 2y^2=1$. In the level curve plot of $f(x,y)$ shown below, the smallest ellipse in the center is when $c=-1$. Working outward, the level curves are for $c=-2, -3, \ldots, -10$.
Dec 29, 2020 · A level curve at \(z=c\) is a curve in the \(x\)-\(y\) plane such that for all points \((x,y)\) on the curve, \(f(x,y) = c\). When drawing level curves, it is important that the \(c\) values are spaced equally apart as that gives the best insight to how quickly the "elevation'' is changing.
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3.3 Level Curves and Level Surfaces. Topographic (also called contour) maps are an effective way to show the elevation in 2-D maps. These maps are marked with contour lines or curves connecting points of equal height. Figure 1: Topographic map of Stowe, Vermont, in the US.
