Endpoints of Intervals Where a Function is Increasing or Decreasing
Date: 05/01/2009 at 23:36:06 From: Bizhan Subject: No equation or formula applies. Why do some calculus books include the ends when determining the intervals in which the graph of a function increases or decreases while others do not? I feel that the intervals should be open and the ends should not be included as they may be, for example, stationary points where a horizontal tangent can be drawn. I noticed that the AP Central always include the ends in the formal solutions of such problems (one can see many examples there), but an author like Howard Anton never does. Could you clarify this for me please? Can both versions be correct?
Date: 05/02/2009 at 01:07:07 From: Doctor Minter Subject: Re: No equation or formula applies. Hello Bizhan, You pose an excellent question, and I agree that the discrepancy among the various textbooks is quite misleading. I completely agree with your claim that the intervals should be open (that is, should not include the endpoints). Let me attempt to give comprehensive reasoning as to why this should be. We use derivatives to decide whether a function is increasing and/or decreasing on a given interval. Intervals where the derivative is positive suggest that the function is increasing on that interval, and intervals where the derivative is negative suggest that the function is decreasing on that interval. Recall that the definition of the derivative f'(x) of a function f(x) is given by f(x+h) - f(x) f'(x) = lim -------------. h->0 h For the function to be differentiable, this limit MUST EXIST. The concept of the existence of this limit has some very major implications. To find a derivative using the limit definition (which ALL derivatives must satisfy--the familiar, simpler rules of differentiation such as the power rule can all be derived from this definition), we must consider what happens as "h" approaches zero from BOTH SIDES. Recall that for a limit to exist, the expression must be the same when the value is approached from either side. Thus, in this case, f(x+h) - f(x) f(x+h) - f(x) f'(x) = lim ------------- = lim -------------. h->0+ h h->0- h where the first limit involves values of "h" that are positive, and the second limit involves values of "h" that are negative. These one-sided limits must be equal for the derivative to exist. Thus, for intervals where the function is increasing, it must be true that the function is increasing around some small neighborhood that encloses each point in the interval. However, let's consider the right endpoint of a given interval on which the function f(x) is increasing. At values greater than this endpoint, one of two things occurs. Either the function ceases to increase, and (like you mentioned) there is a horizontal tangent at this point (which is then called a "critical point"), or there is a discontinuity in the derivative function f'(x), arising from a discontinuity in the function itself, or a "cusp," in which there is an abrupt change in slope. For a discontinuity, the derivative does not exist at that point. Something that does not exist cannot be labeled as positive, so the function cannot be said to be increasing at this point. Either way, critical point or discontinuity, we CANNOT define a neighborhood around the endpoint in which the derivative is strictly positive! Thus, it is not safe to include endpoints on intervals of increasing or decreasing functions. In summary, for a function to be increasing (all of these concepts are similar for decreasing intervals as well), we have to be able to show that the function is greater for larger values of "x," and less for smaller values of "x" in a small neighborhood around each point in the interval. An endpoint cannot have both of these properties. I hope this helps. Please feel free to write again if you need further assistance, or if you have any other questions. Thanks for using Dr. Math! - Doctor Minter, The Math Forum http://mathforum.org/dr.math/
Date: 05/02/2009 at 02:30:03 From: Bizhan Subject: Thank you (No equation or formula applies.) Dear Dr. Minter, Your explanations were very clear and helpful. I enjoyed reading them. Thank you for helping me in such a thorough manner.
Date: 10/22/2014 at 13:51:50 From: Kevin Subject: Endpoints of intervals for increasing/decreasing (redux) Dear Dr. Math, Above, you gave an argument for why endpoints should *not* be included when determining intervals where a function is increasing or decreasing. Implicit in your answer is that "increasing at a point" means "has a positive derivative in a neighborhood of that point." I wonder if it makes sense to define increasing at a point. I also wonder about another definition that I came up with (which, I acknowledge, doesn't work for a point). We could define increasing for an interval [a, b] as: whenever x and y are in [a, b] then f(x) < f(y). This makes no reference to derivatives, so you could still talk about a function being increasing even if it fails to be differentiable some places (e.g., we could say x^(1/3) is increasing everywhere). I'm also thinking about piecewise functions with jumps; again, it seems like we should be able to say they're increasing even if the derivative doesn't exist everywhere. With this second definition, it seems to me that if a function is increasing on (a, b) and continuous at a and b, then it would be guaranteed to be increasing on [a, b]. What do you think? Do we need to have a definition of "increasing at a point" for some reason? Is there any way to reconcile these two definitions? There doesn't seem to be consensus here. It's a basic calculus concept and there seems to be two (very convincing) ways of looking at it that are in conflict.
Date: 10/22/2014 at 17:04:02 From: Doctor Peterson Subject: Re: Endpoints of intervals for increasing/decreasing (redux) Hi, Kevin. Dr. Minter is talking within the realm of calculus, which looks at individual points. As you mention below, this is not really sufficient for the general case. > We could define increasing for an interval [a, b] as: whenever x and y > are in [a, b] then f(x) < f(y). > > This makes no reference to derivatives, so you could still talk about a > function being increasing even if it fails to be differentiable some > places (e.g., we could say x^(1/3) is increasing everywhere). I'm also > thinking about piecewise functions with jumps; again, it seems like we > should be able to say they're increasing even if the derivative doesn't > exist everywhere. This is the proper definition of increasing on an interval, which applies to any function, and is found, for example, here: http://mathworld.wolfram.com/IncreasingFunction.html A function f(x) increases on an interval I if f(b) ≥ f(a) for all b > a, where a, b in I. If f(b) > f(a) for all b > a, the function is said to be strictly increasing. ... If the derivative f'(x) of a continuous function f(x) satisfies f'(x) > 0 on an open interval (a, b), then f(x) is increasing on (a, b). However, a function may increase on an interval without having a derivative defined at all points. For example, the function x^(1/3) is increasing everywhere, including the origin x = 0, despite the fact that the derivative is not defined at that point. > With this second definition, it seems to me that if a function is > increasing on (a, b) and continuous at a and b, then it would be > guaranteed to be increasing on [a, b]. > > What do you think? Do we need to have a definition of "increasing at a > point" for some reason? Is there any way to reconcile these two > definitions? I agree that these are two different things. The last paragraph From MathWorld above reflects the relationship between them. > There doesn't seem to be consensus here. It's a basic calculus concept > and there seems to be two (very convincing) ways of looking at it that > are in conflict. There are actually two different concepts: a precalculus concept, applicable to any function; and a calculus concept, applicable to differentiable functions. I would prefer not to confuse them. Much as we distinguish uniform vs. pointwise continuity, these notions of increasing could be better distinguished like this: The function f is increasing on the interval [0, 1], meaning that comparing any two points, the one on the right is higher. The function f is increasing at every point on the interval (0, 1), meaning that the derivative is positive everywhere. Your question reminds me of an exchange I had last year about a similar issue. A student wrote in: I have y = f(x). On the x-axis there are points a, b, and c. When x = a, y = 0; when x = b, y = 4; when x = c, y = 1. I realise the function increases on [a, b]. I also realise the function decreases on [b, c]. But why is the b in brackets? I know that they indicate closed intervals; that's no problem. If the graph increases from point a to point b, that is [a, b], but then the graph *MUST* decrease on (b, c]. If it increases TO "b," it decreases FROM "b" ... EXCEPT "b"? I responded: I'll suppose you know that a < b < c, so the picture is something like this: + o | + | + | + o | +-----o------+------+----- a b c We can't be sure of this just from only the facts you've given me; I'll suppose you were told this explicitly, or that you were given a graph that makes it clear, like this: + o | + o o | + o o | + o o | +-----o------+------+----- a b c I think what you are asking is why they include the endpoints in the intervals. That's strange, because we've had other questions about why the endpoints are NEVER included in the interval of increase or decrease! Different texts have different policies on this. How does your text DEFINE "increasing on an interval"? Can you show me the first example they give? See these pages, which emphasize this variability among texts: Brackets or Parentheses? http://mathforum.org/library/drmath/view/53566.html Endpoints of Intervals Where a Function is Increasing or Decreasing http://mathforum.org/library/drmath/view/73202.html I'm inclined to agree more with the first of these than the second; but I think in pre-calculus, it's a good idea to ignore this detail and either always use open intervals or always use closed intervals. It's not really an important issue; but your concern that a function can't be increasing AND decreasing at the same point would tilt me in the direction of using open intervals just to avoid confusing students like you! Really, however, you need to notice that your definition probably is only about increasing or decreasing IN AN INTERVAL, not AT A POINT. That is, they are not saying the function is increasing at b -- only that it is increasing in the interval [a, b]. So no claim is being made that the function is both increasing and decreasing at b! Once I see your book's definition, I can be more clear on that. This student never replied with his text's definition, so I didn't get to explore the details with him. One such detail would have been the distinction between saying that a function is increasing on an interval and saying that that is a maximal interval, in the sense that there is no containing interval (open or closed) on which it is increasing. In my experience, texts leave a lot unstated. While these omitted details might keep things simple for the less mature student, they would be worth exploring with a curious, capable one like you! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 10/22/2014 at 17:13:02 From: Kevin Subject: Thank you (Endpoints of intervals for increasing/decreasing (redux)) Thanks. The link you posted seemed to confirm what I thought was right. And you pinpointed the issue: it usually arises from vague definitions in texts. Thanks for your time!
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