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Rog_Sec_14_4

# Rog_Sec_14_4 - Math 20C Lecture Examples Section 14.4...

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(8/17/10) Math 20C. Lecture Examples. Section 14.4. Linear approximations and tangent planes A function z = f ( x,y ) of two variables is linear if its graph in xyz -space is a plane. Equations of planes were found in Section 12.5 by using their normal vectors. Here we will need instead equations given in the next theorem for planes in terms of the slopes of their cross sections in the x - and y -directions. Theorem 1(a) (The slope-intercept equation of a plane) Suppose that the z -intercept of a plane is b , the slope of its vertical cross sections in the positive x -direction is m 1 , and the slope of its vertical cross sections in the positive y -direction is m 2 (Figure 1). Then the plane has the equation z = m 1 x + m 2 y + b. (1) (b) (The point-slope equation of a plane) Suppose that a plane contains the point ( x 0 ,y 0 ,z 0 ) , the slope of its vertical cross sections in the positive x -direction is m 1 , and the slope of its vertical cross sections in the positive y -direction is m 2 (Figure 4). Then the plane has the equation z = z 0 + m 1 ( x - x 0 ) + m 2 ( y - y 0 ) . (2) The slope-intercept equation The point-slope equation FIGURE 1 FIGURE 2 Lecture notes to accompany Section 14.4 of Calculus, Early Transcendentals by Rogawski. 1

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Math 20C. Lecture Examples. (8/17/10) Section 14.4, p. 2 Example 1 Give an equation of the plane with slope - 6 in the positive x -direction, slope 7 in the positive y -direction, and z -intercept 10.
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