Calculus: Analytic Geometry and Calculus, with Vectors

1.1 Real numbers 9

more elements of a sequence xi, x2, xs, of successively better approxima-

tions. If we suppose that xn (where n = 1 when we start) is one of these num-

bers which is different from N/"-a, we can absorb and prove the idea that

should lie between xn and a/xn. Do it. We then examine the tentative but

sensible suggestion that the average of x and a/xn may be a better approxima-

tion to N/-a. With this motivation, let

(1) xn+1 =2Cxn +xn

and prove that

(2)

and hence that

(3)

xn+1 - V r* xn - 2 \/a + - .'

xn+1 - / Ca = 1 (xn - 1/ a)2.
2xn
If we have not already picked up the idea that squares of small numbers are much
smaller, we can start by observing that (0.2)2 = 0.04, (0.04)2 = 0.0016, (0.0016)2
= 0.00000 256, and (0.00000 3)2 = 0.00000 00000 09. This leads us to the idea
that if xn is a good approximation to Va-, then is much better. In fact,
if one approximation xn is correct to k decimal places, we expect the next approxi-
mation xn+1 to be correct to about 2k decimal places. Jumps from 3 to 6 to 12
to 24 are quite amazing. Calculations based upon (1) can be made very rapidly.
When xn and a/xn agree to 10 decimal places and - lies between them, we have
very solid information. The method has another feature that even professional
computers like. Mistakes made before the final calculation do not produce an
incorrect answer, because using an erroneously calculated approximation is
equivalent to starting off with a different first approximation. There is even a
possibility that mistakes may be helpful.
48 Supposing that 0 < a < b, prove that
0<a+b_ <(b-a)2
2 a 8a

Hint: Obtain and use the equality

alb-1'aba2b-I-

(b a) 2

1 a b 2(a + b) + 4 Vah

2

Use the fact that if a quotient has a positive numerator and a positive denomi-
nator, we obtain a greater quotient when we replace the denominator by a smaller
positive number.

Remark: Persons who study science and philosophy can learn that noble but
basically ineffective efforts have been made to prove that points, lines, planes,
and numbers really exist in our physical universe, and to tell precisely what these
things are. It is the opinion of the author that discussions of such matters have
no place in a calculus textbook. As the preface says, we assume that these
mathematical things exist (at least as "mathematical models") and we make the