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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 .
Extended product rule : As a last topic let's note that the product rule can be extended to more than two functions, for instance. ( f g h )′ = f ′ gh + f g ′ h+ f g h′ ( f
7 3/4-3 5/6=
3x+y=9 5x-y=7
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