Lect19 - Implicit vs. Explicit Functions: The equation y =...

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92.131 Lecture 19 1 of 14 Ronald Brent © 2008 All rights reserved. Implicit vs. Explicit Functions: The equation x x y + = 2 explicitly defines y as a function of x . We write ) ( x f y = to denote explicit functions. Consider the equation 1 3 2 = + y x If 0 = x , 1 3 = y , or 1 = y . If 1 = x , 0 3 = y , or 0 = y . If 2 = x , 1 2 3 = + y , or 1 3 = y , and so 1 = y .
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92.131 Lecture 19 2 of 14 Ronald Brent © 2008 All rights reserved. Any equation in x and y implicitly defines y as a function of x . Example: Consider 1 2 2 = + y x , which is a circle of radius 1, centered about the origin. Knowing the graph of the circle one immediately knows it fails to define a function in the usually sense since it does not pass the vertical line test. x -2 -1 1 2 -2 -1 0 1 2 y
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92.131 Lecture 19 3 of 14 Ronald Brent © 2008 All rights reserved. x -2 -1 1 2 -2 -1 0 1 2 y In solving 1 2 2 = + y x for y it becomes apparent why this is the case. 1 2 2 = + y x 2 2 1 x y = 2 1 x y ± = . Except for 1 ± = x , every value of x in the domain yields 2 values for y . Picking the + ” sign yields a value on the upper part of the circle and choosing the “ “ sign gives the lower part of the circle. 2 1 1 ) ( x x y = 2 2 1 ) ( x x y =
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92.131 Lecture 19 4 of 14 Ronald Brent © 2008 All rights reserved. While one doesn’t view the graph of 1 2 2 = + y x as coming from a function, one can still wonder about lines tangent to the graph of the circle at points ) , ( b a . The slope of these lines is viewed as the derivative ) , ( ) , ( b
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Lect19 - Implicit vs. Explicit Functions: The equation y =...

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