MATH 13100/58 Class 6

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1 MATH 13100/58 Class 6 Minh-Tam Trinh Today and Friday, we cover the equivalent of Chapters in Purcell Varberg Rigdon. 1. Calculus is about two kinds of operations on functions: differentiation and integration. Differentiating a function produces its derivative, while integrating a function produces its integral. We will study derivatives this quarter and integrals the next. Very roughly, you can think of a derivative as some kind of rate of change, while an integral is some kind of area below the graph of a function. The common idea needed to define both derivatives and integrals is the idea of a it. Notoriously, its are easy to understand at an informal level, but hard to pin down in a formal definition. The modern definition of a it uses something I like to call the -ı game. ( For every > 0 I toss to you, you have to toss back a ı > 0 that makes a certain condition true... ) Today, we just want to get a feeling at the informal level, through very vague protodefinitions and lots of examples. 2. Let f be a function, and let be your favorite real number. The philosophy behind its, and indeed all of calculus, is this: Instead of studying the value f. / directly, we can also study how f.x/ behaves... When x is very close to, i.e., approximates,. In a given interval around, albeit not at itself. When x slowly approaches, from either the left or the right on the number line. There is an Emily Dickinson poem that goes: Tell all the truth But tell it slant Success in circuit lies Roughly, the it of f at is the number that f.x/ approaches, as x approaches. The fact that we can approach from two directions on the number line will lead to two different notions of a one-sided it. Throughout what follows, we will keep asking ourselves the following three questions, in the following order: (1) Does the left-sided it exist? (2) Does the right-sided it exist? (3) If both exist, do they have the same value? Having said all this, don t worry if the following definitions don t make sense to you yet. Skip to the examples and with each one, compare back to the definitions.

2 MATH 13100/58 Class 6 2 Proto-Definition 1. We say L is the left-sided it, or it from below, of f at, iff f.x/ is close to L whenever x is close enough to on the left (i.e., from below). In this case, we write L D x! f.x/. Proto-Definition 2. We say L is the right-sided it, or it from above, of f at, iff f.x/ is close to L whenever x is close enough to from the right (i.e., from above). In this case, we write L D x! C f.x/. Proto-Definition 3. We say L is the it of f at iff it is both the left-sided it and the right-sided it of f at. In this case, we write L D x! f.x/. Example 4. The graph of the floor function f.x/ D bxc looks like: In the interval 1 < x < 0, i.e., directly to the left of 0, we see that f.x/ always equals 1. So the left-sided it of f.x/ at 0 equals 1: (1) x!0 bxc D 1: But in the interval 0 < x < 1, i.e., directly to the right of 0, we see that f.x/ equals 0. So the right-sided it of f.x/ at 0 equals 0: (2) bxc D 0: x!0 C Since the left- and right-sided its at 0 do not agree, the it of f at 0 does not exist. This statement captures the intuition that the floor function is broken or discontinuous at 0. Again, note that in discussing these concepts, we don t care about the value of the function at 0 itself (b0c D 0). Example 5. The graph of the sawtooth function s.x/ D x bxc is

3 MATH 13100/58 Class 6 3 As x approaches 0 from below, s.x/ approaches 1, so (3) x!0 s.x/ D 1: As x approaches 0 from above, s.x/ approaches 0, so (4) s.x/ D 0: x!0 C Once again, the left- and right-sided its do not match. Example 6. Instead of looking at the behavior of the sawtooth function at 0, let s look at its behavior at 1=2. As x approaches 1=2 from the left, s.x/ approaches 1=2. But this is also true when x approaches 1=2 from the right: (5) x!1=2 s.x/ D 1 2 D x!1=2 C s.x/: Since the left- and right-sided its exist and are equal, the it of s.x/ at 1=2 exists and equals their common value: (6) s.x/ D 1=2: x!1=2 So the behavior of its depends on which point on the number line you are considering, not just on the function you are considering. Example 7. Most functions you know and love are not as pathological as the floor function or the sawtooth function. For example, polynomials f.x/ D a 0 C a 1 x C C a n x n are well-behaved: At any point on the number line, the it of a polynomial exists and equals the value of the polynomial at that point. In other words, their behavior always resembles Example 6, not Example 5. More concretely, here s the graph of f.x/ D x 3 C x 2 : You can check for yourself that (7) f.x/ D 0 D f. 1/; x! 1 f.x/ D 0 D f.0/; x!0 x!2 f.x/ D 12 D f.2/; etc.,

4 MATH 13100/58 Class 6 4 and in general, x! f.x/ always exists and equals f. /. Later in the course, we will call this property continuity; our discussion above is saying that polynomials are examples of continuous functions. Example 8. Not every continuous function is a polynomial. Consider A.x/ D 1=.1 C x 2 /: This is also continuous. For example, x!0 A.x/ D 1 D A.0/. Intuitively, the graph of A never breaks or leaps the way the graphs of the floor or sawtooth functions do: You can draw the graph without lifting your pen(cil). Example 9. A piecewise function is one whose formula depends on the location on the number line. Here is an example of a piecewise function: 0 x D 0 (8) B.x/ D 1 x 0 1Cx 2 In other words, B.x/ is exactly the same as the function A.x/ in Example 8, except I ve changed its value at 0 to be B.0/ D 0. It turns out B has the same its everywhere as A. In particular, consider what happens at 0: (9) B.x/ D 1 D A.x/; x!0 x!0 even though B.0/ D 0. This example emphasizes how the definitions of x!0 B.x/ and x!0 C B.x/ don t depend on the value of B.0/, but only on the behavior of B close to 0, where B is the same as A. Example 10. We ve seen that its can fail to exist when the left- and right-sided its do not match. However, left- and/or right-sided its can themselves fail to exist. Consider the hyperbola f.x/ D 1=x:

5 MATH 13100/58 Class 6 5 As you approach 0 from the left, the function plummets to negative infinity, which is not a number. Similarly, as you approach 0 from the right, the function soars to positive infinity. So neither x!0 1=x nor x!0 C 1=x exists. A fortiori, x!0 1=x does not exist. Example 11. Here is an even more horrifying way that left- and/or right-sided its can fail to exist. Consider the function h.x/ D 1=x b1=xc, an obscure perversion of the sawtooth function: (In lecture, I gave a slightly different example constructed from what I called mountains. The basic idea is similar, though: The function oscillates infinitely in the vicinity of 0.) Neither the left- nor right-sided its exist at 0. The function h simply does not settle on a number as it gets close to 0. To use the language of the proto-defnitions, there does not exist a single number L such that h.x/ remains close to L whenever x is close enough to 0 either on the left or the right. Certainly h.x/ becomes close to various values between 0 and 1 infinitely often, but it doesn t stay near any of them. Purcell Varberg Rigdon calls this something like too much oscillation. Example 12. I want to avoid doing too much trigonometry for now, but if you remember the sine function, then you can cook up the following function with similar behavior to the previous example: f.x/ D sin.1=x/: Again, neither of the one-sided its at 0 exists, and thus the it at 0 does not exist. Example 13. Consider this variant of the previous example: g.x/ D x sin.1=x/:

6 MATH 13100/58 Class 6 6 I claim that x!0 g.x/ exists and equals 0. Can you recognize this from the graph?