local_linearity

local_linearity - Local Linearity and Tangent Plane...

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L ocal L inearity and T angent P lane A pproximations Recall, when we studied one-variable functions, as we zoomed in near a particular point, the function looked more and more linear. Using that fact coupled with the fact that the derivative at a point is the slope of the tangent line at that point, we were able to develop a tangent line approximation for any (possibly complicated) function at a particular point. For example, see Figure 1. 1 2 3 x - 1 1 y Tangent Line Approximation Figure 1: Tangent Line Approximation Something similar happens when we deal with functions of two variables. Consider what happens as we zoom in on the following surface. Figure 2: Zooming in on a Two-variable Function As we zoom in, notice that the surface is looking more and more like a plane. Recall, a plane is the natural extension of a line to two variables. We can see this behavior if we examine contour diagrams. (As we zoom in, we have added more contours.) 1
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Figure 3: Zooming in on the Contour Diagram of a Two-variable Function Notice that the last contour diagram looks like that of a linear function that we studied previously. This corresponds to what we observed with the graph of the function in Figure 2. The plane that we say in Figure 2 is called the tangent plane to the surface that point. Figure 4 illustrates this. Figure 4: The tangent plane to the surface z = f ( x , y ) at the red dot There is still one important left to answer: what is the equation of this tangent plane? At the point (
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local_linearity - Local Linearity and Tangent Plane...

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