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Two lines
- The level curve equation x2 −y2 = 0 x 2 − y 2 = 0 factors to (x − y)(x + y) = 0 (x − y) (x + y) = 0. This equation is satisfied if either y = x y = x or y = −x y = − x. Both these are equations for lines, so the level curve for c = 0 c = 0 is two lines.
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Level curves of the function g(x,y)=√9−x2−y2 g (x y) = 9 − x 2 − y 2, using c=0,1,2 c = 0 1, 2, and 3 3 (c=3 c = 3 corresponds to the origin). A graph of the various level curves of a function is called a contour map.
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 \(f\left( {x,y} \right) = k\).
The level curve $y=f(x,z)=c$ is given by \[x^2+(z-c)^2=c^2 \] for $c\geq 0$. The above equation describes a circle of radius $c$ centered at $x=0$ and $z=c$. Note that here $x$ and $z$ are the independent variables and $y$ is the dependent variable. The level curves are circles in the $xz$-plane.
The level curve equation $x^2-y^2=0$ factors to $(x-y)(x+y)=0$. This equation is satisfied if either $y=x$ or $y=-x$. Both these are equations for lines, so the level curve for $c=0$ is two lines.
Given a function f (x, y) f (x, y) and a number c c in the range of f, a f, a level curve of a function of two variables for the value c c is defined to be the set of points satisfying the equation f (x, y) = c. f (x, y) = c.
Dec 29, 2020 · The level curve for \(c=0\) is the set of all points \((x,y)\) such that \(0=\sqrt{1-\frac{x^2}9-\frac{y^2}4}\). Squaring both sides gives us \[\frac{x^2}9+\frac{y^2}4=1,\]
Jan 28, 2022 · By definition, a level curve of \(f(x,y)\) is a curve whose equation is \(f(x,y)=C\text{,}\) for some constant \(C\text{.}\) It is the set of points in the \(xy\)-plane where \(f\) takes the value \(C\text{.}\)
