3.9.2-eg-1 - Example Find the linear approximation L(x) to...

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Unformatted text preview: Example Find the linear approximation L(x) to the function f (x) = tan π 2 x 3π at the point x = 1/2 and use it to approximate tan 16 . Solution: The linear approximation to π x 2 f (x) = tan at the point x = 1/2 is given by L(x) = f (1/2) + f (1/2)(x − 1/2), where, using the Chain Rule, f (x) = sec2 so f (1/2) = sec2 π π x· , 2 2 π1 π · · 22 2 and f (1/2) = tan = √ 2 2 π 2 = π, π1 · = 1. 22 It then follows that L(x) = 1 + π (x − 1/2). To approximate tan 3π 16 using L(x), we need to first determine which value of x gives f (x) = tan 3π . 16 Clearly x must be such that π 3π x= , 2 16 or x= 2 3π 3 · =. π 16 8 Then tan 3π 16 3 3 ≈L , 8 8 =f where 3 31 π − =1+π =1− . 8 82 8 π 3π is 1 − . So the desired approximation to tan 16 8 L . (Note that the approximation satisfies L(3/8) = .61 while the exact value is given by . tan(3π/16) = .67.) 2 ...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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3.9.2-eg-1 - Example Find the linear approximation L(x) to...

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