Yes it's true, but why ? A calculus riddle (of sorts) involving definitions

[jt103]

"Yes it's true, but why...?" A calculus riddle (of sorts) involving definitions

Homework Statement

Here's the story: I'm in an AP B/C Calculus class and our current activity is "engaging in an all out study frenzy before the AP exam." We've already gone over all of the material in the book, and now we're taking time to just go through and practice free-response questions and multiple choice, etc.

Anyway, my teacher got a new sample multiple choice packet from the College Board last week, put in an order for some copies, and worked one of the packets herself. No problems.

She went back to do proofs and explanations for each problem (so show the class should problems arise) and stubled upon a bit of a snafu; she invited any of the students who wanted to take a crack at the question a chance to; we all arrived at the same answer without too much trouble, and we had all come up with the same answer that was on they key. The problem was, and this was what troubled my teacher in the first place, was why/how is this true?

Below is the problem exactly as it appears in the sample booklet:

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(can't post urls yet, it would seem)

Homework Equations

• Conditions for continuity
  
• Definition of a derivative

The Attempt at a Solution

Now, we talked quite extensively about this, and we kicked around a few possibilities...

"Maybe the function is finite, and bounded at x=3 inclusively" among others...

But basically, here is the reasoning as it stands...

Answers a), and b) cannot be the correct response, because differentiablity implies continuity, and both would violate the conditions for continuity if they were ever false.

Answers c) and d) cannot be the correct response because c) represents a definition of the slope of a line drawn tangent to a curve (a derivative), and we have already been told that the derivative exists at x=3 and at that point is equal to 5 and d) cannot be true for essentially the same reason. That is, if either were false, f`(x) would not and could not equal 5, as we know it does.

So, we are left with is choice e), which agrees with the key provided. All that we are trying to figure out is, what would an example of this condition being false look like?

Now, this could be a misprint. It has happened before. However, I am loath to cry misprint right off the bat, because there have been times when we have done that, only to find out that we were just looking at it the wrong way.

Anyway, any help/insight would be greatly appreciated by my teacher and the other 4 or so students working on this "riddle."

 

[Hurkyl]

The simplest counterexample I can think of is that 3 might be the only place where f is differentiable, so f'(x) (and the requested limit) simply doesn't exist.

 

[jt103]

The simplest counterexample I can think of is that 3 might be the only place where f is differentiable, so f'(x) (and the requested limit) simply doesn't exist.

But wouldn't that make c) and d) false also?

Or am I missing something? Sorry...heh...

 

[Hurkyl]

But wouldn't that make c) and d) false also?

No; the existence of those limits require only that f be differentiable at 3 -- it doesn't matter whether or not f is differentiable anywhere else.

 

[Dick]

I'll give you an example to think about. Take f(x)=x^2*sin(1/x) and define f(0)=0. Can you show f'(0)=0? Next can you show lim(x->0) f'(x) does not exist? Can you relate this to your problem?

 

[HallsofIvy]

Dick's point is that while f(x) must be continuous at x= 3 in order to be differentiable, its derivative is not necessarily continuous there.

 

[DavidWhitbeck]

That is a good question by the way because at that level of calculus students can be tricked into thinking that they only deal with smooth functions, you know infinitely differentiable.

The thing is that I don't mean to be a curmudgeon but some teachers like to use the official AP practice test for their final, and many other teachers make that practice exam worth credit.

So would you mind not posting and discussing any more problems from it? A student can get questions about the test answered by their teacher, and teachers can get questions answered from the College Board's calculus edg. But a student in another class that knows they will have it for an exam, might google to find answers ahead of time and then find threads like this one.