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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. Returning to the function g(x, y) = √9−x2 −y2 g (x, y) = 9 − x 2 − y 2, we can determine the level curves of this function.
courses.lumenlearning.com/calculus3/chapter/level-curves/
Learn how to find level curves of a function in Calculus 3.
- 13 min
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- The Math Sorcerer
Feb 28, 2021 · Calculus 3 video that explains level curves of functions of two variables and how to construct a contour map with level curves. We begin by introducing a typical temperature map as an example of...
- 21 min
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- Houston Math Prep
. THEOREM 15.12. The Gradient and Level Curves. Given a function. f. differentiable at. (a,b) , the line tangent to the level curve of. f. at. (a,b) is orthogonal to the gradient. ∇f(a,b) , provided. ∇f(a,b)≠0. . Proof: Consider the function. z=f(x,y)
Sep 29, 2023 · A level curve of a function \(f\) of two independent variables \(x\) and \(y\) is a curve of the form \(k = f(x,y)\text{,}\) where \(k\) is a constant. A level curve describes the set of inputs that lead to a specific output of the function.
For example, if $c=1$, the equation is $x^2-y^2=1$. If $c=-1$, the equation is $y^2-x^2=1$. A number of level curves are plotted below. We can “stack” these level curves on top of one another to form the graph of the function. Below, the level curves are shown floating in a three-dimensional plot. Drag the green point to the right.
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Recall that the level curves of a function \(f(x, y)\) are the curves given by \(f(x, y) =\) constant. Recall also that the gradient \(\nabla f\) is orthogonal to the level curves of \(f\)
