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Home | 18.013A | Chapter 10 |
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The linear approximation to is exactly true if is constant for that means that is linear. The inaccuracy of the linear approximation to at at argument arises from the changes to between arguments and .
If is differentiable in the interval between and we can get a better approximation to at by making a linear approximation to and using it to estimate the change to in the interval.
In short if is differentiable in that interval we can compute its derivative, called the second derivative of with respect to and written as or as or sometimes as and use it to improve the estimate of .
All of our standard functions have differentiable derivatives and even differentiable second derivatives, etc on forever wherever they are defined, except perhaps at specific singular points.
They are said to be "infinitely differentiable" because we can keep differentiating them as long as we like. We may therefore compute second derivatives, and also third and higher derivatives and generate a sequence of better and better approximations to any such function.
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