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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


    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
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Associated Topics:
High School Calculus
High School Functions

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