7.1. Interpolation and Extrapolation 209
(f) Manipulating formally, we have Ek = (I+A)k. Expand the right
side by the binomial theorem and use it to obtain the resultf(t+k) = f(t)+kAf(Q+ ( ; ) A2f(t)+ ( ; ) A3f(t)+-..
(g) Make the substitution t = 1, k = n - 1 in (f) to determine the
polynomial in n whose values at n = 1,2,3,... are respectively
(i) 1,3,6,10,15 ,...
(ii) 2,11,35,85,175,322 ,....- Consider the function g(n) whose values for n = 1, 2, 3, 4, 5, 6 are
respectively 6, 50, 225, 735, 1960, 4536. Since we know six values of
g, we might suppose that g(n) is given by a polynomial of the fifth
degree in n. Make up a difference table, and determine what this
polynomial is. - Refer to the table for the index of refraction of water for sodium light
given at the beginning of this section. Let f(t) be the index when the
temperature is lot degrees Celsius.
(a) Determine: Af(2), A3f(3), A2f(5).
(b) Neglect fourth order differences, and use
f(5.4) = (I + A)1.4f (4)
= f(4) + (1.4)Af(4) + (1/2)(1.4)(0.4)A2f(4) + ...
to approximate the index of refraction for 54’C.
(c) Use f(5.4) = (I + A)-0,6f(6) to approximate the index of re-
fraction for 54’C.- Given the following table
number natural logarithm0.5 -0.69315
1.0 0.00000
1.5 0.40547
2.0 0.69315
2.5 0.91629
3.0 1.09861
3.5 1.25276
4.0 1.38629
interpolate to approximate the natural logarithms of 1.25, 0.75, 2.1,
2.71828. Extrapolate to find the natural logarithm of 0, 0.25, 5.0.
Use various methods. Check your answers with a pocket calculator
or from a table of logarithms.