Brackets or Parentheses?Date: 01/07/97 at 13:50:52 From: lowell mcdonald Subject: Calculus question Dr. Math, We have been discussing a problem in my Advanced Placement Calculus class. It concerns increasing/decreasing functions as well as concave up/concave down. Here is an example where the problem occurs: For the given function f(x), find the interval where the function is increasing/decreasing and concave up/down: f(x) = 3x^5 - 5x^3 f'(x) = 15x^4 - 15x^2 f"(x) = 60x^3 - 30x Therefore: derivative increases from negative infinity through -1 derivative decreases from -1 through 0 derivative decreases from 0 through 1 derivative increases from 1 through positive infinity When expressing these answers as an interval, should I use a bracket, symbolizing that the endpoint is included, or a parenthesis, symbolizing that the endpoint is not included? e.g. bracket: increasing (-oo,-1] U [1,+oo) decreasing [-1,1] parentheses: increasing (-oo,-1) U (1,+oo) decreasing (-1,0) U (0,1) I would appreciate an answer as well as a brief explanation as to why. Also, what about concavity? Should we use brackets or parentheses? Date: 01/07/97 at 18:10:50 From: Doctor Jerry Subject: Re: Calculus question Hi Lowell, I graded AP exams last summer, but that was the first time. Thus, I have some experience with what you are asking, but I'm not an old pro at AP. However, I think I can answer your question. I'll answer it in a specific context, but the idea is applicable to increasing/decreasing and concave up/concave down. Different books, teachers, and mathematicians use slightly different definitions of increasing functions, but this is not a matter of much consequence as long as one is consistent. Suppose your definition of an increasing function is: f is increasing on an interval I if for each pair of points p and q in I, if p < q, then f(p) < f(q). Note that I may be open (a,b), half-open [a,b) or (a,b], or closed [a,b]. Consider f(x) = x^2, defined on R. The usual tool for deciding if f is increasing on an interval I is to calculate f'(x) = 2x. We use the theorem: if f is differentiable on an open interval J and if f'(x) > 0 for all x in J, then f is increasing on J. Okay, let's apply this to f(x) = x^2. Certainly f is increasing on (0,oo) and decreasing on (-oo,0). What about [0,oo)? The theorem, as stated, is silent. However, one can go back to the definition of increasing. To show that f is increasing on I = [0,oo), let u and v be in I and u < v. If 0 < u, then the theorem applies. Otherwise, 0 = u < v and we see that f(u) = 0 < f(v) = v^2. Many instructors, books, and even AP exams often skip consideration of endpoints. If you want to be ultra-safe, then you can do the above kind of analysis. It just takes a few extra steps, usually easy, past the standard test. Concave down/concave up is fussier since the definition is more awkward. Often, this is defined by saying that f is concave up on an interval J if, on this interval, the graph of f lies above each of the secant lines of f on this interval. Same general idea though. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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