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- Given a function f (x, y) f (x, y) and a number c c in the range of f 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.
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Functions of two variables have level curves, which are shown as curves in the x y-plane. x y-plane. However, when the function has three variables, the curves become surfaces, so we can define level surfaces for functions of three variables.
The level curves of a function \(z=(x,y)\) are curves in the \(xy\)-plane on which the function has the same value, i.e. on which \(z=k\text{,}\) where \(k\) is some constant. Note: Each point in the domain of the function lies on exactly one level curve.
Nov 17, 2020 · Definition: level curves. Given a function \(f(x,y)\) and a number \(c\) in the range of \(f\), a level curve of a function of two variables for the value \(c\) is defined to be the set of points satisfying the equation \(f(x,y)=c.\)
3.5: Level Curves. Page ID. Jeremy Orloff. Massachusetts Institute of Technology via MIT OpenCourseWare. Recall that the level curves of a function f(x, y) f (x, y) are the curves given by f(x, y) = f (x, y) = constant.
The level curves are given by $x^2-y^2=c$. For $c=0$, we have $x^2=y^2$; that is, $y=\pm x$, two straight lines through the origin. For $c=1$, the level curve is $x^2-y^2=1$, which is a hyperbola passing vertically through the $x$-axis at the points $(\pm 1,0)$.
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Dec 29, 2020 · Given a function \(z=f(x,y)\), we can draw a "topographical map'' of \(f\) by drawing level curves (or, contour lines). A level curve at \(z=c\) is a curve in the \(x\)-\(y\) plane such that for all points \((x,y)\) on the curve, \(f(x,y) = c\).
