mthsc206-fall-2010-notes-15.3

Mthsc206-fall-2010-n - 15.3 Partial Derivatives Consider the function f x y = 4 x 2 y 2 If we fix the y-coordinate(i.e set y = b for some value b

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Unformatted text preview: 15.3 Partial Derivatives Consider the function f ( x, y ) = 4- x 2- y 2 . If we fix the y-coordinate (i.e., set y = b for some value b ), what do we get? − 3 − 2 − 3 − 5.0 − 1 − 2 − 2.5 − 1 x 0.0 1 y z 1 2 2.5 2 3 3 5.0 What would happen if we fix x = a , but let y be free? How does the function f change when we allow one variable to “vary” while we fix the other? − 3 − 2 − 3 − 5.0 − 1 − 2 − 2.5 − 1 x 0.0 1 y z 1 2 2.5 2 3 3 5.0 If we set y = b and then ask how the function f changes at the point ( a, b ) if we hold y constantly at b but allow the values for x to vary. The instantaneous rate of change at the point ( a, b ) in the “ x ”-direction (or with respect to the horizontal distance) is given by lim h → f ( a + h, b )- f ( a, b ) h . This should look somewhat familiar: it looks like a derivative. Definition 15.3.1 Let f ( x, y ) be a real-valued function of 2 variables. We say that the (first) partial derivative of f with respect to x at the point...
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This note was uploaded on 10/11/2010 for the course MTHSC 206 taught by Professor Chung during the Fall '07 term at Clemson.

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Mthsc206-fall-2010-n - 15.3 Partial Derivatives Consider the function f x y = 4 x 2 y 2 If we fix the y-coordinate(i.e set y = b for some value b

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