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From my understanding, linear approximations and differentials both use the tangent line to a function to estimate the value of the function at a point. I understand that their respective equations are different - but conceptually how are they different?

maddie
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  • The differential of a function at a point is the linear approximation of the increment of the function from this point. – Bernard Sep 22 '17 at 22:47
  • so the differential is not the linear approximation of the function at that point, but the linear approximation of the difference between two points (say y(0) and y(0.01) – maddie Sep 22 '17 at 22:58
  • If you know anything about Taylor series, it's worth noting that the linear approximation of $f(x)$ at $a$ is just the first order Taylor series approximation of $f(x)$ centered at $a$. – gf.c Sep 22 '17 at 23:03
  • Yes. This is quite obvious in the formula $; f(x_0+h)=f(x_0)+\mathrm d\mkern1mu f_{x_0} \cdot h+o(h)$. This applies also to functions from $\mathbf R^m$ to $\mathbf R^n$. – Bernard Sep 22 '17 at 23:04

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The differential of a function at a point is an idealization of that function — it is a gadget that remembers a little bit extra information about the behavior of that function than just its value at the point.

(to be clear, the differential itself doesn't remember the value, only the extra information)

A linear approximation is a linear function that approximates something. A typical formula for a good linear approximation uses the value of the function at a point along with the differential of the function at the same point to produce produce an estimate of the function at values near that point. (this approximation is often called the differential approximation)

For a brief but sloppy explanation of the difference:

  • The differential consists of just the slope
  • The linear approximation consists of both the value and the slope