(10/7/08)Math 10C. Lecture Examples.Section 14.3. Local linearity and the differential†Zooming in on level curves of a nonlinearz=f(x,y)If a functiony=f(x) of one variable has a derivative atx0and the graphy=f(x) is generated by acalculator or computer in a small enough window containing the point (x0, f(x0)), the displayed portionof the graph will look like a line. This occurs because the graph is closely approximated by the tangentline near that point.Graphs of functions of two variables with continuous first derivatives are closely approximated byplanes in small windows. Consequently, their level curves at equalz-increments look like equally spacedparallel lines in small windows. This is illustrated by the level curves ofK(x, y) = 3x2y3+xin Figures 1through 3. The level curves look more like equally spaced parallel lines in Figure 2 than Figure 1, andeven more like equally spaced parallel lines in Figure 3. These approximate closely the level lines of the
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