We have that L 1 01 f 1 f 11 01 1 01 Whereas the true value of ln1 01 0 00995

We have that l 1 01 f 1 f 11 01 1 01 whereas the true

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We have that: L (1 . 01) = f (1) + f 0 (1)(1 . 01 - 1) = . 01 Whereas the true value of ln(1 . 01) = 0 . 00995 - our approximation is pretty close. The key is that 1.01 is ”close” to 1 - that is (1.01 - 1) is small. This linear approximation paradigm breaks down if our value of x is far from our value of a . Consider the following example. 3
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Example 4 Use linear approximation to approximate the values cos(2) , cos(18) using f ( x ) = cos(2 x ) around the point a = 1 2 . Sol. Notice that cos(2) = f (1) and cos(18) = f (9). Now, all we need to do is find f 0 ( a ) and f ( a ), which aren’t hard: f 0 ( a ) = - 2 sin(2 a ) = - 2 sin(1) , f ( a ) = cos(1) So our linear approximation function is: L ( x ) = cos(1) - 2 sin(1) x - 1 2 If we plug in x = 1, we get L (1) = cos(2) = - . 3011. The actual value of cos(2) is roughly -.4161, giving us an error around 25% (not too bad, but not great). If we plug in x = 9, we get L (9) = cos(18) = - 13 . 764, which is preposterous because cosine should give us a value between -1 and 1. The error in this case is in the thousands of percent. Why does this occur? Because x = 9 is not close to x = 1 2 , whereas at least 1 and 1/2 were somewhat close. 4
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