Ques 1 of preliminary excercise with reasons 2.9

SOLUTION Let 0 be such that 0 # . For any b, consider the line L(0) drawn at angle 0 to the x axis starting at (0, b). has formula y = (tan 0)x + b. Let A(b) be the amount of ham above the line minus that below the line. Let A > 0 be the area of the ham. We have to accept the following (reasonable) assumptions: For low enough b = bo, the line L(0) lies entirely below the ham, so that A(bo) = A - 0 = A. For high enough b1, the line L(0) lies entirely above the ham, so that A(b1) = 0 - A = -A. . A(b) is continuous as a function of b. Under these assumptions, we see A(b) is a continuous function satisfying A(bo) > 0 and A(b1) 0. By the IVT, there is some c E [0, c1] such that A(c) = 0. 30. Figure 4(B) shows a slice of ham on a piece of bread. Prove that it is possible to slice this open-faced sandwic each part has equal amounts of ham and bread. Hint: By Exercise 29, for all 0 0, we apply the IVT on [0, x] with B(m) 0. If B(0) = 0, we are also done; L(0) is the appropriate line. 158 CHAPTER 2 | LIMITS 2.9 The Formal Definition of a Limit Preliminary Questions 1. Given that lim cos x = 1, which of the following statements is true? (a) If [cos x - 1| is very small, then x is close to 0. (b) There is an E > 0 such that |x | 0 such that [cos x - 1| 0 such that (cos x| 0 such that (cos x - 1| (d) If (x) - 21 0 such that the following statement holds: If (x) - L| 0 such that the following statement holds: If(x) - L| 0, If(x) - 35| 0). (b) Let E > 0. Let 8 = /8 and suppose 0 0, we found an appropriate o so that 0