Calculus: Analytic Geometry and Calculus, with Vectors

1.1 Real numbers 7

25 Iflx-5l <2,then 3 <x<7?

26 If lx-2l <0.01,then 1x2-4l <0.0401?

27 If lx- 31 <1,then lx2-9l < 5?

28 If1 <x<2and1 <y<2,then lx-yI <1?

29 1(3.05)(3.06) - 91 < $?

30 There is no real number x such that x2 = -I?

31 If 0 < x < 1, then -1 < -x < 0?

32 Ifa<x<b,then-b<-x<-a?

33 If0<x<Iand0<y<1,then 0<x+y<1?

34 If 0 < x < 1, then 0 < 2x < 1?

35 If Ix - al 5 S, then Ix - al < S?

36 If Ix - al < 8,then Ix -al 5 S?t

37 If S = I and 0 < x < 1, then Ix2 - -11 < S?

38 If3= and 05x<1,then Ix2-1I<S?

39 Ifs = and0<x<1,then lx2--1I <S?

40 If S = - and 0 < x < 1, then ix2-l 5 S?
Answers, 0 for false and 1 for true:
5 10 15 20 25 30 35 40
11011 01100 11011 11101 00011 10111 11000 10111

41 We learned while studying arithmetic and algebra that the product of
either two positive numbers or two negative numbers is positive, while the product
of a positive number and a negative number is negative. Supposing that x and y
are nonnegative, use the identity

(1) (Y - x)(Y + x) = Y2 - x2

to show that x 5 y if x2 S y2. Hence show that the inequality

(2) j a + bI < Iai + IbI

will be true if la + b12 < (lal + lb1)2 or

(3) (a + b)2 < a2 + 21allbl + b2.

Finally, show that (3) is true and hence that (2) is true.

42 Sketch several figures which lead to the conclusion that if x and e are num-

bers for which

(1) Ix-2I+Ix-3I<e,

then e > 1. Remark: Without using figures, we can prove the result by observing
that if (1) holds, then

1=I3-2I=I(x-2)-(x-3)I<Ix -2I+Ix -3l<e

and hence e > 1.

43 Using the ideas of the preceding problem, prove that if a, b, x, and a are

numbers for which

lx-al+Ix-bl<6,

then e > lb - al.