Endpoints of Intervals Where a Function is Increasing or DecreasingDate: 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. |
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