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Unformatted text preview: (3/23/08) Section 14.6 Tangent planes and differentials Overview: In this section we study linear functions of two variables and equations of tangent planes to the graphs of functions of two variables. Then we discuss differentials with two variables and their use in estimating errors. Topics: Linear functions of two variables Level curves of linear functions Zooming in on level curves of a nonlinear z = f ( x , y ) Equations of tangent planes Normal vectors Differentials and error estimates Linear functions of two variables Recall that a function y = f ( x ) of one variable is linear if its graph in an xyplane is a line, and that in this case its derivative is constant and equals the slope of the line. In studying such functions, we frequently use either the slopeintercept equation y = mx + b for the line, where m is the slope and b the yintercept of the line (Figure 1), or the pointslope equation y = y + m ( x x ) where m is the slope and ( x ,y ) a point on the line (Figure 2). x y b Slope = m x y ( x ,y ) Slope = m The pointslope equation The slopeintercept equation y = mx + b y = y + m ( x x ) FIGURE 1 FIGURE 2 A function z = f ( x,y ) of two variables is linear if its graph in xyzspace is a plane. We found equations of planes in Section 13.5 by using their normal vectors. Here we will need the next theorem, which gives equations for planes in terms of the slopes of their cross sections in the x and ydirections. 342 Section 14.6, Tangent planes and differentials p. 343 (3/23/08) Theorem 1(a) (The slopeintercept equation of a plane) Suppose that the zintercept of a plane is b , that the slope of its vertical cross sections in the positive xdirection is m 1 , and that the slope of its vertical cross sections in the positive ydirection is m 2 (Figure 3). Then the plane has the equation, z = m 1 x + m 2 y + b. (1) (b) (The pointslope equation of a plane) Suppose that a plane contains the point ( x ,y ,z ) , that the slope of its vertical cross sections in the positive xdirection is m 1 , and that the slope of its vertical cross sections in the positive ydirection is m 2 (Figure 4). Then the plane has the equation, z = z + m 1 ( x x ) + m 2 ( y y ) . (2) The slopeintercept equation The pointslope equation FIGURE 3 FIGURE 4 Proof of part (a): We consider the case where the zintercept b , the slopes m 1 and m 2 , and x and y are all positive, as in Figure 3. To obtain an equation for the plane we need to find the zcoordinate of the point R with xycoordinates ( x,y ). The zcoordinate of the point P is b . The horizontal run from P to Q is x and the line segment PQ , which is on a vertical cross section of the plane in the xdirection, has slope m 1 . Consequently the zcoordinate of Q is z = b + m 1 x . The horizontal run from Q to R is y and the line segment QR , which is on a vertical cross section in the ydirection, has slope m 2 , so that the zcoordinate of R is z = b + m 1 x + m 2 y , as stated in...
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 Summer '08
 Eggers
 Math, Equations

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