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

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

1. 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 explainations 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)

2. Homework Equations

• Conditions for continuity
• Definition of a derivative

3. The Attempt at a Solution

"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."

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Hurkyl
Staff Emeritus
Gold Member
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.

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
Staff Emeritus
Gold Member
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
Homework Helper
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?

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HallsofIvy