Calculus: Analytic Geometry and Calculus, with Vectors

3.1 Functional notation 121

The particular point (x,71) for which Mt # = 0 and .11v1', = 0 is called the cen-
troidt (thing like a center) of the mass system. The coordinates of the centroid
are denoted by z and y. Thus, t1fz = 0 and M411 = 0, and it follows from (3)
and (5) that

(6) z =m1x1 + m2r2 + ... + mnx, - M15'1+m2y2 + ... + mny,,

M

I

_

M

In case mk = 1 for each k, (2) shows that M = n and the formulas (6) reduce to

(7) =x1 + x2 + -- + xn, y = Y1+ Y2 + + Y.

n n

In this case the centroid (z,y) is called the centroid of the set of points P,, P2,
P,,. To prepare us for Section 4.7 and other sections where less simple
mass systems are considered, we should take brief cognizance of a more general
definition. Let p be a nonnegative integer. The number is defined by

(8) Mgt = m1(x1 + m2(x2 - s)" + ... + m,(xn - S)p

and is called the pth moment of the mass system about the line having the equa-

tion x Similarly, the number MC,7 defined by

(9) Mien = mi(Y1 - 7])p + m2(Y2 - 77)p +.. + m,,(Y, - 77)p

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

y = 71. In physics and mechanics (but not so often in statistics) the second

moment is called moment of inertia. Since we are studying functions, we can

observe that, if our mass system contains 40 particles, there is a sense in which

the moments in (8) and (9) are functions of 82 variables of which two are p and.

While this textbook does not require calculations of these moments, we can recog-

nize that there are many situations in which calculations must be made, and this

is one of the reasons why the world contains so many calculating machines and

computers of assorted mechanical and electronic varieties.

18 If f(x) = 1 + x + x2 + xa + x4, show that f(1) = 5 and

f(x)=x5-1=1-x5

x-1 1-x

when x 0 1.

19 Remark: This remark invites more complete comprehension of ideas and

terminology involving functions. The left-hand part of Figure 3.194 represents

Domain 0-I ) Black box

T Y egg

Figure 3.194

a set D of numbers or of vectors or of entities of some other kind that is called a

domain. The central part of the figure represents a mechanism that is sometimes

t We are being rather unrealistic if we suppose that everybody always chooses the same

coordinate system when studying a given system of particles. The coordinates of the cen-

troid depend upon the coordinate system used, but (we omit the proof) the location of the

centroid relative to the system of particles is the same for all coordinate systems. For

example, if three particles of equal mass lie at the vertices of a triangle, then the centroid

lies at the intersection of the medians of the triangle.