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Linear Approximations
In this section we will look at an application not of derivatives but of the tangent line to a function. Certainly, to get the tangent line we do have to take derivatives, thus in some way this is an application of derivatives as well.
Given a function, f ( x ) , we can determine its tangent at x = a . The equation of the tangent line, that we'll call L ( x ) for this discussion, is,
L ( x ) = f ( a ) + f ′ ( a ) ( x - a )
Take a look at the given graph of a function & its tangent line.
From the graph we can illustrates that near x = a the tangent line & the function have closely the similar graph. On instance we will utilizes the tangent line, L ( x ) , as an approximation to the function, f ( x ) , near x = a . In these cases we call the tangent line the linear approximation to the function at x = a .
If f(x) is an infinitely differentiable function so the Taylor Series of f(x) about x=x 0 is, Recall that, f (0) (x) = f(x) f (n) (x) = nth derivative of f(x)
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