Approximations and
Inequalities
7.1 Interpolation and Extrapolation
A scientist wishes to know the index of refraction of pure water relative
to air for sodium light at a temperature of 54’C. A table provides the
following information:
Temperature (“C.) Index (r(T))
20 1.33299
30 1.33192
40 1.33051
50 1.32894
60 1.32718
70 1.32511
80 1.32287
90 1.32050
100 1.31783
Likely, the number sought lies between 1.32718 and 1.32894. Can we be
more precise? Can we find a function which expresses the index in terms
of the temperature? If not, can we sensibily approximate a functional re-
lation between the variables by means of a polynomial? We will look at
some possible approaches, and assess the effectiveness for this particular
situation.
Exercises
- One way to find the required index of refraction from the given table
is to argue that, as 54 is 4/10 of the way from 50 to 60, the index
lies 4/10 of the way from 1.32894 to 1.32718. Verify that this leads
to the result
1.32894 - (0.4)(0.00176) = 1.32824.
Does this seem reasonable?- The approach in Exercise 1 amounts to assuming that, between 50’
and 60°, the index r(T) is a linear function of T:
r(T)=aT+b (5O<T<60).