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331 Math Homework: Day 9 Practice and Reading Read Section 2.1 (including the problems) on functions, which should largely be review. Read Section 2.2 on limits. Pay particular to De nition 2.2.1 (memorize) and Examples 2.2.1 6. Be prepared to discuss the questions below on Monday. I will call on you to answer. Class Preparation for Next Time 1. From Math 135 (and 331), a function f is de ned as relation be between two sets A (called the domain) and B (called the codomain) where 1. for each a A, there is a b B so that (a, b) f , and 2. if (a, b) and (a, c) are both in f , then b = c. If A = R and B = R, then which of the following collections are functions. d) e) f) g) h) a) f = {(a, b) | b = a2 } b) g = {(a, b) | a = |b|} c) h = {(a, b) | b = tan a} True or false: f = {(a, b) | b = sin a} is not a function because 2 B = R, but there is no a A = R so that (a, 2) f . When using the usual y = f (x) function notation for a function (which we almost always will) what do we assume about the domain of f ? Describe the function f (x) = x2 + 1 as a collection of ordered pairs, as above. What is the domain of f g if f (x) = x 4 and g(x) = x2 ? What is the domain of g f if f (x) = x 4 and g(x) = x2 ? 2 i) Why do my calculus students think f (x) = x + 1 and g(x) = x 1 are the same function? Are they? x 1 2 are not the same function? Are they? j) My calculus students think f (x) = |x| and g(x) = x 2. Draw a graph of the Dirichlet function D(x) on page 48. 3. Which of the following are polynomials? Which are rational functions? a) p(x) = 6x8 x3 + 11 x2 1 d) x+1 b) p(x) = 6x8 x3/2 + 11 e) 6 c) p(x) = 6x 8 + x3 11 2 x2 3 f) + x x4 + 9 4. In De nition 2.2.1 of limit, what is the importance of the condition 0 < |x a|? 5. a) On the axes on the left below, draw (shade) the set of all points (x, y) that satis es both of these 1 conditions: |x 1| < 2 and |y 1.5| < 1. 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 a 1 b) How would modify your answer if the rst condition changed: 0 < |x 1| < 2 and |y 1.5| < 1. c) On the axes on the right above, draw (shade) the set of all points (x, y) that satis es both of these conditions: 0 < |x a| < and |y L| < . (Your choice of and .) L 1 Hand In By Thursday at 5 PM: Problems from Section 1.4 The rst ve are completely straightforward. 1. Problem 1.4.1, but use the interval [1, 5] instead. 2. Problem 1.4.2 (a and c). No proof, just make sure the cover works. 3. Problem 1.4.4 (a). 4. Problem 1.4.5 5. Problem 1.4.11. No need to show work. 6. Problem 1.4.9. This picture may help you think about the problem. ( x s4 s2 s3 s5 x s6 s7 s1 . . . sn ) x+ 7. Problem 1.4.10. Hint: Use a previous problem on this set. This can be done in one sentence. 8. a) Suppose that is the least upper bound of a set S and that is not in S. Show that is an accumulation point of S. Hint: You need to show that for any > 0, there is a point s S with s = so that such that < s < + . Explain why you can actually nd s so that < s < . b) Give an example of a set S with a least upper bound that is not an accumulation point of S. 9. Problem 1.4.14. The argument is virtually identical to the one suggested in Problem 1.4.13. 10. Page 64 #2.2.9(a). 2

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