Taylor Series

# This is illustrated in the diagram below where we

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Unformatted text preview: erms of the size of | f (x) |. Generally speaking, the further x is away from a and the more curved that the graph of f (x) is, the larger the potential for error in using linear approximation. This is illustrated in the diagram below where we have two diﬀerent functions f (x) and g (x) with the same tangent line at x = a. The error in linear approximation is the length of the vertical line joining the graph of the function and the linear approximation. You will notice that in the diagram above, the graph of g (x) is much more curved near x = a than is the graph of f (x). You can also see that at the chosen point x the error Error(1) =| f (x) − La (x) | in using La (x) to estimate the value of f (x) is extremely small, where as the error Error(2) =| g (x) − La (x) | in using La (x) to estimate the value of g (x) is noticably smaller. The diagram also shows that for both f (x) and g (x), the further away x is from a, the larger the error is in the linear approximation process. In the case of the function g (x) in our diagram, the graph looks more like a par...
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