Unformatted text preview: yx we conclude that such a function f does not exist. 53. Let 1 f = f ( a + h , b + k ) − f ( a , b ) be the change in f at P = ( a , b ) . Set 1 v = h h , k i . Show that the linear approximation can be written 1 f ≈ ∇ f P · 1 v 6 SOLUTION The linear approximation is 1 f ≈ f x ( a , b ) h + f y ( a , b ) k =f x ( a , b ), f y ( a , b ) ® · h h , k i = ∇ f P · 1 v 54. Use Eq. (6) to estimate 1 f = f ( 3 . 53 , 8 . 98 ) − f ( 3 . 5 , 9 ) assuming that ∇ f ( 3 . 5 , 9 ) = h 2 , − 1 i . SOLUTION By Eq. (6), 1 f ≈ ∇ f P · 1 v The vector 1 v is the following vector: 1 v = h 3 . 53 − 3 . 5 , 8 . 98 − 9 i = h . 03 , − . 02 i Hence, 1 f ≈ ∇ f ( 3 , 5 , 9 ) · 1 v = h 2 , − 1 i · h . 03 , − . 02 i = . 08...
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 Fall '09
 Park
 Calculus, Derivative, Multivariable Calculus, Continuous function, S E V E R A L VA R

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