3.9.6-eg-1

# 3.9.6-eg-1 - f x for x near 2 For example L(1 99 = 11 14(1...

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Example Find the linear approximation L ( x ) to the function f ( x ) = x 3 + 2 x - 1 at the point x = 2 and use it to approximate f (1 . 99). Solution: Recall that for any diﬀerentiable function f , the linear approximation to f at the point x = x 0 is given by L ( x ) = f ( x 0 ) + f 0 ( x 0 )( x - x 0 ) , where in this case, f ( x ) = x 3 + 2 x - 1 and x 0 = 2 . Then f 0 ( x ) = 3 x 2 + 2 , and since f (2) = 2 3 + 2 · 2 - 1 = 11 , f 0 (2) = 3 · 2 2 + 2 = 14 , it follows that the linear approximation of f ( x ) at x 0 = 2 is L ( x ) = 11 + 14( x - 2) . We can then use L ( x ) to ﬁnd approximations of
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Unformatted text preview: f ( x ) for x near 2. For example, L (1 . 99) = 11 + 14(1 . 99-2) = 10 . 86 , which could be used as an approximation to f(1.99). (Note that in fact we can easily calculate the exact value of f (1 . 99), f (1 . 99) = 1 . 99 3 + 2 · 1 . 99-1 = 10 . 8606 , so in this case the linear approximation correctly determines the ﬁrst 4 digits of f (1 . 99). )...
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