Calculus: Analytic Geometry and Calculus, with Vectors

318 Functions, graphs, and numbers

for which If(x) 1 5 M when -1 5 x 5 1. Let x = 1/ lam. Then x F-- 0

and-1-<-<x<_1,butf(x)=1/x2=IMI+ 2, so lf(x)l>IMI>M.

6 Read Theorem 5.45. Then construct a figure which illustrates the mean-

ing of the following remark. If a < b and M > 0, then condition

(1) I.f(x)I < M (a<x<b)

is satisfied if and only if f is defined over the interval a 5 x S b and the graph of

y = f(x) over the interval a < x < b lies between the graphs of the lines having

the equations y = -111 and y = M. Note that this gives a "geometrical mean-

ing" to Theorem 5.45. Note that the inequality (1) holds if and only if M > 0

and-M<f(x)<MwhenaSx<_<b.

7 Sketch a graph of the function f for which f(0) = 0 and f(x) = 1/x when

x 0 0 and -1 < x 5 1. Show that there is no M such that the graph of f

over the interval -1 5 x <= 1 lies entirely above the line having the equation

y= -M.

8 Give an example of a function which has an upper bound over the interval

-1 <= x < 1 but has no lower bound.

9 Show that the function f for which f(x) = x has upper and lower bounds

over the open interval 0 < x < 1 but possesses neither a maximum nor a mini-

mum over this interval.

10 Show that the function f defined over the closed interval 0 5 x < 2 by

the formula

AX) = x - [x],

in which [x] denotes the greatest integer less than or equal to x, has an upper

bound but does not have a maximum.

11 Prove that there is a number x* for which a < x < b and f'(x*)

[f(b) - f(a)]/(b - a) when

(a) f(x) = x2, a = 0, b = 1

(b) f(x)=xe-7x2+3x+40,a=-1,b=1

12 Without undertaking extensive calculations that are easily made when

appropriate computing equipment is available, we call attention to the Newton

(1642-1727) method by which zeros of reasonable

functions are approximated in decimal form.

The method is based upon the elementary obser-

vation that, in many cases more or less like the

one shown in Figure 5.491, if x, (where n may

initially be 1) is a reasonably good approximation

x x to a number z for which f (z) = 0, then the tangent

Yn+1 to the graph of y = f (x) at the point (x,,, f

Figure 5.491 will intersect the x axis at a point (xn,+i,0) for

which x,+a is a much better approximation to z.

The Newton method is normally applied in cases where f has many continuous

derivatives and f'(x) 0 0 when x is near z but x 9& z. In such cases the equation

of the tangent at (x,,, f(x.)) is

y - AX-) - J'(x,,)(x - x)