My questions are from lecture 9, MIT OCW SV Calculus, Jerison, 2009;(adsbygoogle = window.adsbygoogle || []).push({});

At 27:50 he is deriving the linear approximation for the function

e^(-3x)(1+x)^(-1/2)≈(1-3x)(1-1/2x)≈1-3x-1/2x+3/2x^2≈1-7/2x, for x near 0.

In the last step he drops the x squared term since it is negligible(no questions so far). He then says that dropping the quadratic term is ok since in an earlier analysis, the quadratic and higher terms were ignored and that methodology is just being consistently applied. However, that was not done, at least explicitly. He points to an earlier approximation done, e^x≈1+x.

This was done using the expression f(x)≈f(0) + f'(0)x where f(x)=e^x. However, no non linear terms were generated and and therefore none were dropped. Although I believe him, I don't see the how the dropped terms are generated in the earlier analysis. Does anyone know how one would generate those terms? I'm not trying to needlessly complicate things, just fully understand what he was saying.

My other question concerns the example at 29:50 in the same lecture. He shows how the time dilation expression for a GPS satellite is given by the lorentz transformation approximated by a fairly simple linear expression. The approximation is what was used by the engineers who designed the satellite. My question here is really about the motivation of the engineers. Is it simply easier to write the code for a linear calculation, instead of the original expression? Does it use less memory? Any Ideas?

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# Linear approximation higher order terms

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