linear_functions

linear_functions - Linear Functions In single-variable...

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L inear F unctions In single-variable calculus, we studied linear equations because they allowed us to approximate complicated curves with a fairly simple linear function. Our goal will be to use linear functions in two dimensions to approximate complicated surfaces. A linear function in one variable could be written in point-slope form. That is, we said that y = y 0 + m ( x x 0 ). Once we knew a point on the linear and the slope, we had all the information necessary to specify the equation. When dealing with two variables, we have a similar point-slope form. This time, however, we have two slopes: one in the x -direction and another in the y -direction. Equation of a Linear Function If a linear function (a plane) passes through the point ( x 0 , y 0 , z 0 ) and has slope m in the x -direction and slope n in the y -direction, the equation of the linear function (plane) is given by: z = f ( x , y ) = z 0 + m ( x x 0 ) + n ( y y 0 ) Letting c = z 0 mx 0 ny 0 , we can also write f ( x , y ) in the form z = f ( x , y ) = c + mx + ny Just as two points uniquely determine the equation of a line in 2-space, three points uniquely determine the equation of a plane in 3-space. Example 1: Find the equation of the plane passing through the points (1, 2, 3), (1, 4, 1), (2, 2, 5). Solution: Notice that the first two points have the same x -value. So, we can use them to find the slope of the plane in the y -direction. As y changes from 2 to 4, the z -value changes from 3 to 1. Thus, the slope in the y -direction is given by n = D z / D y = (1 – 3)/(4 – 2) = -1.
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