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HH_14_3 - Math 10C Lecture Examples Section 14.3 Local...

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(10/7/08) Math 10C. Lecture Examples. Section 14.3. Local linearity and the differential Zooming in on level curves of a nonlinear z = f ( x , y ) If a function y = f ( x ) of one variable has a derivative at x 0 and the graph y = f ( x ) is generated by a calculator or computer in a small enough window containing the point ( x 0 , f ( x 0 )), the displayed portion of the graph will look like a line. This occurs because the graph is closely approximated by the tangent line near that point. Graphs of functions of two variables with continuous first derivatives are closely approximated by planes in small windows. Consequently, their level curves at equal z -increments look like equally spaced parallel lines in small windows. This is illustrated by the level curves of K ( x, y ) = 3 x 2 y 3 + x in Figures 1 through 3. The level curves look more like equally spaced parallel lines in Figure 2 than Figure 1, and even more like equally spaced parallel lines in Figure 3. These approximate closely the level lines of the
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