(10/8/08)Math 10C. Lecture Examples.Section 12.4. Linear functions†Theorem (a) (The slope-intercept equat ion of a plane) Suppose that the z-intercept of aplane is b, the slope of its vertical cross sections in the positive x-direction is m1, an d theslope of its vertical cross sections in the positive y-direction is m2(Figure 1). Then the planehas the equation,z = m1x + m2y + b. (1)(b) (The point-slope equation of a plane) Suppose that a plane contains the point (x0, y0, z0),the slope of its vertical cross sections in the positive x-direction is m1, and the slope of itsvertical cross sections in the positive y-direction is m2(Figure 2). Then the plan e has theequation,z = z0+ m1(x − x0) + m2(y − y0). (2)The slope-intercept equation The point-slope equationFIGURE 1 FIGURE 2Example 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.Answer: z = −6x + 7y + 10Example 2 Give an equation of the plane throug h the point (1,2,3) with slope 4 in thepositive x-direction and slope −5 in the positive y-direction.Answer: z = 3 + 4(x − 1) − 5(y − 2)†Lecture notes to accompany Section 12.4 of C alculus by Hughes-Hallett et al.1Math 10C. Lecture Examples. (10/8/08) Section 12.4, p. 2Example 3 Find a formula for the linear function z = g(x, y) whose values are given inthe following table.Values of z = g(x, y)x = −3 x = 0 x = 3y = 2 8 14 20y = 0 14 20 26y = −2 20 26 32Answer: g(x, y) = 2x − 3y + 20.Example 4 Find a formula for the linear function z = h(x, y) who se level curves aregive n in Figure 3.814 202632x3−3 6−6y33FIGURE 3Answer: h(x, y) = 2x − 3y + 20. (Notice that h is the same as the function g from Example 3.)Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 14.6: Examples 1 and 2‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sectionsof the textbook for the
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