Calculus: Analytic Geometry and Calculus, with Vectors

120 Functions, limits, derivatives

if -2 5 x S 2 and, for each such x, y(x) is one or the other of the two numbers

-x - -\/3(4 - x2) -x + 3(4 - x2)

2

,

2

which are equal only when x = -2 and when x = 2. Hint: Use the quadratic

formula.

15 An integer n greater than 1 is said to be composite if, like 39, it is repre-

sentable as the product of two integers each greater than 1 and is said to be a

prime if, like 29, it is not so representable. The primes are 2, 3, 5, 7, 11, 13, 17,

19, 23, 29,- -- , there being aninfinite set of them. One of the famous func-

tions of number theory is x(x), the number of primes less than or equal to X.

It is easy to see that x(8.27) = 4. It has been proved that x(103) = 168,

x(106) = 78,498, and x(109) = 50,847,478. To graph x(x) over the whole inter-

val 0 S x 5 109 would be quite a chore. However, draw the graph over a

shorter interval, say 0 < x S 40, and try to pick up some ideas.

16 Another famous number-theoretic function has, for each positive integer

n, the value d(n), where d(n) is the number of positive integer divisors (including

I and n) of n. For example, the divisors of 6 are 1, 2, 3, and 6. Verify the

entries in the little table

n= 1 2 3 4 5 6 7 8 9 10 11 12 13

d(n)=122 324243 4 2 6 2

and calculate d(233252).

17 We take a brief preliminary look at some functions that play fundamental

roles in physics, mechanics, and statistics. Let n be a positive integer. For each

k = 1, 2, 3, --, n, let a particlePk of massMk be concentrated at the point

Pk(xk,yk). In what follows we use Z (xi, the Greek x) to denote a number which

can easily be considered to be the x coordinate of a point, and we use M with a

superscript to make us think of a moment. For each number t, the number

Mp?f defined by

(1) M`'e=ml(xI- )+ms(xs- )+ ... +mn(xn

is called the first moment of the mass system about the line having the equation

x = !;. Supposing that the total mass

(2) M = ml + m2 + .. + mn

of the system is positive, we can put (1) in the form

• .
  (3) M (maxi

+ m2x2 + ... + M.X.

M

Similarly, for each number rt (eta) the number M,('-',, defined by

(4) M, = ml(y, - il) + m2(y2 - 17) +. .. + mn(Y,- rl)

is called the first moment of the mass system about the line having the equation

y=n,and

(5) Mas = Mlm1Y1 + -2Y2

M

... + miYn- nl.