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You can infer all sorts of data from level curves, depending on your function. The spacing between level curves is a good way to estimate gradients: level curves that are close together represent areas of steeper descent/ascent.
Figure 2. 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.
Mar 2, 2022 · But you can infer certain shape variants based on the concentrics. For example, if in your ContourPlot there is only one concentric behavior, i.e. a sole set of concentric circles (ellipses, ovals, etc.), then this is an indication of a global minimum or maximum.
Level curves and contour plots are another way of visualizing functions of two variables. If you have seen a topographic map then you have seen a contour plot. Example: To illustrate this we first draw the graph of z = x2 + y2. On this graph we draw contours, which are curves at a fixed height z = constant.
Figure 5(a): The graph of the function $ z=f(x,y)=x+2y+1$ and the level curves Figure 5(b): The level curves of $ f$$ with different values of $c$. Recall that $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is the equation of a hyperbola with two vertices at $(\pm a,0)$.
A function has many level curves, as one obtains a different level curve for each value of $c$ in the range of $f(x,y)$. We can plot the level curves for a bunch of different constants $c$ together in a level curve plot, which is sometimes called a contour plot.
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What data can you infer from a level curve?
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What is the relationship between a level curve and a gradient?
There is a close relationship between level curves (also called contour curves or isolines) and the gradient vectors of a curve. Indeed, the two are everywhere perpendicular. This handout is going to explore the relationship between isolines and gradients to help us understand the shape of functions in three dimensions.
