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Section14_2 - Section 14.2 Horizontal cross sections of...

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(3/23/08) Section 14.2 Horizontal cross sections of graphs and level curves Overview: In the last section we analyzed graphs of functions of two variables by studying their vertical cross sections. Here we study horizontal cross sections of graphs and the associated level curves of functions. Topics: Horizontal cross sections of graphs Level curves Estimating function values from level curves Topographical maps and other contour curves Horizontal cross sections of graphs In Section 14.1 we determined the shape of the surface z = x 2 + y 2 in Figure 1 by studying its vertical cross sections in planes y = x and x = c perpendicular to the y - and x -axes. In the next example we look at the cross sections of this surface in horizontal planes z = c perpendicular to the z -axis. FIGURE 1 Example 1 Determine the shape of the graph of z = x 2 + y 2 in Figure 1 by studying its horizontal cross sections. Solution Horizontal planes have the equations z = c with constant c . Consequently, the horizontal cross sections of the surface z = x 2 + y 2 are given by the equations, braceleftBigg z = x 2 + y 2 z = c. Setting z = c in the first of these equation yields the equivalent equations for the cross section, braceleftBigg c = x 2 + y 2 z = c. (1) 288
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Section 14.2, Horizontal cross sections of graphs and level curves p. 289 (3/23/08) If c is positive, then x 2 + y 2 = c is the circle of radius c in an xy -plane and the cross section (1) of the surface is the circle of radius c in the plane z = c with its center at c on the z -axis. If c = 0 then the curve (1) is the origin of xyz -space. The cross section is empty (has no points in it) if c is negative. Since the radius of the circular cross section at z = c increases as c > 0 increases and the plane z = c rises, the surface has the bowl shape in Figure 2. square 3 6 9 x 4 y 4 Level curves of f ( x, y ) = x 2 + y 2 FIGURE 2 FIGURE 3 Level curves The horizontal cross sections of the surface in Figure 2 are at z = 0 , 1 , 2 , 3 , . . . , 10. The lowest cross section is the point at the origin. The other cross sections are circles; if we drop them down to the xy -plane, we obtain the ten concentric circles in Figure 3. The point and circles are curves on which f ( x, y ) = x 2 + y 2 is constant. They are called level curves or contour curves of the function. The function has the value 0 at the origin and the values 1 , 2 , 3 , . . . , 10 on the circles in Figure 3. The numbers 3, 6, and 9 on three of the circles indicate the values of the function on those circles. To visualize the surface in Figure 2 from the level curves, imagine that the xy -plane in Figure 3 is horizontal in xyz -space, that the innermost circle in Figure 3 is lifted one unit to z = 1, the next circle is lifted two units to z = 2, the next circle, labeled “3” is lifted to z = 3, and so forth. This gives the horizontal cross sections of the surface which determine its shape in Figure 3.
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