1
$\begingroup$

My calculus teacher assigns us online homework to do. He never went over any question that looks like this (he in fact said we shouldn't be concerned with this):What is this?

Yet, I need to answer this right to progress with my homework. It stinks because if I get it wrong, I lose points on my homework average.

So, could someone help explain to me what's going on here, and perhaps guide me to a point where I can try to figure out the solution myself? I'm not asking for a straight answer (although if thats what you want to provide, go for it [since I was told I don't have to know this stuff]), but this stuff really confuses me. Thank you for your help.

Oh, and in case you need it, here's the original prompt (the question I posted above is just a part of a series of questions that go along with this prompt):

enter image description here

2 Answers 2

3

I think the question is quite confusingly worded. It took me several minutes to figure out what it meant -- and it's not as if I don't know the subject matter well.

What must be going on is that you're supposed to imagine reading something like this in a proof:

bla bla bla, and therefore we know that $f$ is continuous, and that $f(c)\ne 0$. We can then apply the definition of continuity with $\varepsilon = $ _______ to find a $\delta$ such that $f(x)$ has the same sign as $f(c)$ for every $x\in(c-\delta,c+\delta)$. Thus, bla bla bla

One of $|f(c)|$ and $|c|$ will make this into a valid argument if you fill it into the blank, and one will produce nonsense. Your task is to select the valid one.

In order to answer the question you need (1) to remember the definition of continuity, and (2) to be able to distinguish a nonsense argument from a valid one. The second of these abilities is often considered too advanced a skill to demand of pre-university students these days (they're supposed to be satisfied with accepting the teacher's judgement in each case), which is probably why your teacher is not allowed to say you must be able to do it...

  • 0
    Well, another problem I have is that I have no knowledge of what $\delta$ or $\epsilon$ are/means. We were never taught about that matter, and we were even told "not to worry about it." Would you mind elaborating on these terms a little bit?2011-09-30
  • 0
    Well, then your options seem to be either to read up on it on your own, or to forego the points here. Is the epsilon-delta definition of continuity present in your textbook even if it has not been discussed in class? If so, go read that section. If not, then luckily the symbol usage is fairly well standardized here; [Wikipedia's rendering](http://en.wikipedia.org/wiki/Continuous_function#Weierstrass_definition_.28epsilon-delta.29_of_continuous_functions) of the definition (under the "Weierstrass definition") should give you what you need.2011-09-30
  • 0
    Upon reading this, it seems to me that the solution to my question would then be $\epsilon$ = $|f(c)|$.2011-09-30
  • 0
    @Mike, correct.2011-09-30
  • 0
    Indeed, you are right. But the online stuff is obviously intended for a different type of course than the one you are actually taking.2011-09-30
  • 0
    I agree with @André that it sounds very strange that this question would be part of homework for a class where the $\epsilon$-$\delta$ definition hasn't even been discussed. Which educational level is this at? Is the teacher reusing an online test designed for something else entirely?2011-09-30
  • 0
    There's one more question this follows this, a simple yes or no: If the distance traveled away from f(c) is less than the absolute value of f(c), is it possible for f to change sign?2011-09-30
  • 0
    That would make most sense as a hint for the question you've already answered, I think.2011-09-30
  • 0
    Actually, the professor didn't design the homework. We covered a section on continuity in the textbook, and he likes to skip parts of each section (obviously the part we've been talking about). The textbook company (I believe Pearson?) has an online homework system established for each section that my professor requires us to do. The educational level - well, this is Calculus 1.2011-09-30
  • 0
    @Henning The question mentions 'show that there is interval... blah blah blah... where f has the same sign as f(c)', so I have a feeling the answer to that question is 'No'.2011-09-30
  • 0
    That explains it. Hopefully the teacher has a way to ignore the answers to non-taught sections when grading. I don't even know what "Calculus 1" signifies -- such terms vary from country to country, and it's not even clear from your description whether this is secondary school or university freshman level.2011-09-30
  • 0
    I am a freshman at a university in the United States - this is my first semester, 5th week. Regarding my most previous comment @Henning, am I correct?2011-09-30
  • 0
    Yes that should be no. (And I would have guessed you were somewhere in the UK, based on your gravatar...)2011-09-30
  • 0
    @Henning: I'm mostly of English decent, but not from the UK - I apologize for the confusion though. Thank you so much for your time and help.2011-09-30
1

Without loss of generality we may assume that $f(c)$ is positive.

Let $\epsilon =f(c)$. By the definition of continuity of $f$ at $c$, there is a $\delta>0$ such that if $|x-c|<\delta$ (and $a\lt c-\delta$, and $c+\delta \lt b$, to make sure we stay in our interval) then $|f(x)-f(c)|<\epsilon$.

Now if you have some experience with inequalities, you should be able to reach the conclusion.

If the "without loss of generality" is not persuasive, if $f(c)<0$, let $g(x)=-f(x)$, apply the above argument to $g(x)$, and see what this says about $f(x)$.

I did not use precisely the language of the question. But you should now be able to see enough of what is going on to be able to answer that question.

Comment: It will be helpful to draw a picture while figuring out what the second paragraph is saying.

  • 0
    Would you mind elaborating on what the terms $\delta$ and $\epsilon$ even mean? My professor told us "not to worry about those", and the textbook doesn't define them. Some university...2011-09-30
  • 0
    @Mike: What university, if you don't mind my asking?2011-09-30
  • 1
    If you have not had any prior exposure to these things, it would take a couple of lectures' worth, at least, to get going. The question really cannot make any sense without a fair bit of background.2011-09-30
  • 0
    @mixedmath: I'd rather not mention it, as not to really trash it's reputation. It's really got a great Computer Science program (and is #1 in its region) [which is my major], but the math department seems lacking.2011-09-30