Calculus: Analytic Geometry and Calculus, with Vectors

322 Functions, graphs, and numbers

Hence (5) shows that d2 is a digit and

(11) d2<102(B-ip)<d2+1.

Dividing by 102 and transposing give

(12) 0<8-10-

d2 1

so (3) holds when n = 2. This procedure enables us to prove (3) by induction.

If d1, d2, d are digits and (3) holds, then

n

(13) 0<10n+'(8-10 U22 ... lp")<10

and (7) shows that d"+1 is a digit and

(14) d"}1<10n+1(0- -- d2 10 102 ...

Dividing by 10n+t and transposing give the result of replacing it by it + 1 in (3).

This proves (3) by induction, that is, by use of the principle of mathematical

induction of Problem 15.

19 Let F(8) be the temperature or pressure at the place P where a circle

having its center at the origin of an x,y coordinate system is intersected by the

ray from the origin which makes the angle 0 (as in trigonometry) with the positive

x axis. It is supposed that F is continuous and F(8 + 2a) = F(8) for each 0.

Prove that there are two diametrically opposite points of the circle at which F

has equal values. Hint: Apply the intermediate-value theorem to the function

f for which f (8) = F(8) - F(8 + a). Observe that if f (Bo) > 0, then f (0o + a) =

-f(8o) < 0. Remark: While we do not yet have equipment required for proof,

we can learn an interesting property of continuous functions defined over sur-

faces like spheres. There are two antipodal (or diametrically opposite) places

on the surface of the earth having both equal temperatures and equal atmos-

pheric pressures.

20 Suppose that a world has existed so long and so favorably to fish that an

infinite number of fish have existed but that only a finite number of fish have

existed at any one time because the world contains only a finite number of atoms.

Prove that there is a least number mo such that the mass m (measured in some

standard system) of each past and present and future fish is less than or equal to

mo.

21 It is easy to presume that if f is differentiable over the interval -1 S x < 1

and if f'(0) = 1, then there must be a positive number Is such that f is increasing

over the interval -h 5 x S h. Use the function f for which f(0) = 0 and

f(x) = x + x2 sin x= (x 3-1 0)

to show that the presumption is false. Hint: Show that f'(0) = 1 and that,

when x 0 0,

f'(x) = I + 2x sin^1

Z-zcos xi