A little homework help (mean value theorem)

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pietertje
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A little homework help (mean value theorem)

Can somebody tell me if this is correct?

Q: Show that the function [imath]f(x) = x +\frac{1}{x}[/imath] satisfies the hypotheses of the mean value theorem on the interval [2, 3] and find all numbers c in that interval that satisfy the conclusion of that theorem.

A: [imath]f(x) = x +\frac{1}{x}[/imath]
[imath]f'(x) = 1 - \frac{1}{x^2}[/imath]
[imath]f'(c) = \frac{f(b) - f(a)}{b - a}[/imath]
[imath]1 - \frac{1}{c^2} = \frac{3 + \frac{1}{3} - 2 - \frac{1}{2}}{1}[/imath]
[imath]1 - \frac{1}{c^2} = \frac{5}{6}[/imath]
[imath]\frac{1}{c^2} = \frac{1}{6}[/imath]
[imath]c^2 = 6[/imath]
[imath]c = \sqrt{6}[/imath] or [imath]c = -\sqrt{6}[/imath] <- not valid.

So [imath]c = \sqrt{6}[/imath]

Thanks!

skullturf
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Re: A little homework help (mean value theorem)

Those look like the right calculations for finding c. Note that if we're being precise, there's a little bit more to the question: your first task is to show that f satisfies the hypotheses of the mean value theorem.

If you refer back to the statement of the mean value theorem that you learned, it'll say that the theorem applies to functions that satisfy certain conditions. Are those conditions true in this case?

pietertje
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Re: A little homework help (mean value theorem)

skullturf wrote:Those look like the right calculations for finding c. Note that if we're being precise, there's a little bit more to the question: your first task is to show that f satisfies the hypotheses of the mean value theorem.

If you refer back to the statement of the mean value theorem that you learned, it'll say that the theorem applies to functions that satisfy certain conditions. Are those conditions true in this case?

Well, in the case of the mean value theorem the fact that there is a defined value for c automatically means that the function satisfies the theorem. But you're right, I should indeed add something along the lines of "c is defined, thus function f(x) satisfies the theorem."

Thanks, I always forget to add these kind of things, which has cost me quite a few points in the past.

Talith
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Re: A little homework help (mean value theorem)

What do you mean c is defined? The MVT doesn't work on certain kinds of functions, one example is the step function f(x)=floor(x). What part of f(x) means that you can apply the MVT to it? The function in your question happens to be one of these functions - show why. c shouldn't appear in this part of your answer anywhere, we haven't started looking for it yet.

pietertje
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Re: A little homework help (mean value theorem)

Talith wrote:What do you mean c is defined? The MVT doesn't work on certain kinds of functions, one example is the step function f(x)=floor(x). What part of f(x) means that you can apply the MVT to it? The function in your question happens to be one of these functions - show why. c shouldn't appear in this part of your answer anywhere, we haven't started looking for it yet.

Doesn't the MVT apply to all functions which are "curved" and have a continuous line going between point A and B. Or am I missing something?

Talith
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Re: A little homework help (mean value theorem)

From wikipedia the statement of the MVT is:

Let f : [a, b] → R be a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that [imath]f'(c)=\frac{f(b)-f(a)}{b-a}[/imath].

You need to show that your function and interval agree with the first sentence in this statement. Do you know how to show that f is continuous on an interval, How about differentiable? I'm not really sure what level you're being taught at so this might be asking too much if you're not taking an introduction to real analysis course, in which case forget I asked.

skullturf
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Re: A little homework help (mean value theorem)

pietertje wrote:Doesn't the MVT apply to all functions which are "curved" and have a continuous line going between point A and B. Or am I missing something?

Almost.

For an informal description of the types of functions to which MVT can be applied, one might say something like "the graph of the function has to be smooth, and it has to consist of one unbroken piece." Or words to that effect.

That's an informal paraphrase. In the precise statement of the MVT, look at the hypotheses, and think about how they relate to the informal description in the previous paragraph.

How do you show that a function is continuous? That may depend a lot on the course. In some introductory calculus courses, it may be considered enough to write something like "we previously learned that such-and-such types of functions are continuous wherever they are defined." You might want to ask your instructor.

pietertje
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Re: A little homework help (mean value theorem)

Talith wrote:From wikipedia the statement of the MVT is:

Let f : [a, b] → R be a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that [imath]f'(c)=\frac{f(b)-f(a)}{b-a}[/imath].

You need to show that your function and interval agree with the first sentence in this statement. Do you know how to show that f is continuous on an interval, How about differentiable? I'm not really sure what level you're being taught at so this might be asking too much if you're not taking an introduction to real analysis course, in which case forget I asked.

Well, I'm taking calculus A on university level, so I should indeed know this. Unfortunately I kind of forgot how to show that a function is continuous on a geven interval, luckily there's always my textbook.

Thanks for pointing me in the right direction.

Eastwinn
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Re: A little homework help (mean value theorem)

^ Well, if f and g are both continuous on [a,b], f + g is continuous as well. So in the case of x+1/x, it's a matter of whether x is continuous on [2,3] (yup) and whether 1/x is continuous on [2,3] (yup). I can't imagine them wanting more detail than that.
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Talith
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Re: A little homework help (mean value theorem)

Well you need to show differentiability too, but that follows from the additional rule as well. Your lecturer could expect you to answer this question with a varying degree of assumptions.

You might be allowed to simply assume that f is continuous and differentiable on the given interval because it's defined everywhere except at 0 and it's a standard continuous and differentiable function on every point that it's defined - I don't think this is the case though because it specifically says in the question that you need to show that f satisfies the hypotheses of the MVT.

You might be allowed to only assume that the addition law holds and/or x and 1/x are continuous and differentiable on the given interval - this is the most likely case as once you've got to the MVT in an analysis course, the lecturer tends to assume you can work fine (or at least have done enough of) with epsilon-delta proofs.

If the lecturer is a real hard arse, or you've not long been doing them, he may expect you to assume nothing but the definitions of continuity and differentiability and want you to show that f is both from epsilon-delta proofs - I really don't think this is the case though because it's long winded and distracts you from the real point of the question; making sure you understand the statement of the MVT and any applications it has.

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Re: A little homework help (mean value theorem)

pietertje wrote:
skullturf wrote:Those look like the right calculations for finding c. Note that if we're being precise, there's a little bit more to the question: your first task is to show that f satisfies the hypotheses of the mean value theorem.

If you refer back to the statement of the mean value theorem that you learned, it'll say that the theorem applies to functions that satisfy certain conditions. Are those conditions true in this case?

Well, in the case of the mean value theorem the fact that there is a defined value for c automatically means that the function satisfies the theorem. But you're right, I should indeed add something along the lines of "c is defined, thus function f(x) satisfies the theorem."

That shows it satisfies the conclusion of the theorem. You were asked to show that it satisfies the hypotheses. Many theorems in mathematics are of the form "if [object] satisfies [conditions A, B, and C] then [object] satisfies [conclusion]." The hypothesis of the theorem is that you have an object which satisfies conditions A, B, and C. So for the MVT, you have to show that you have a function which satisfies the conditions at the beginning of the statement of the theorem.
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Yakk
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Re: A little homework help (mean value theorem)

All [class] are [conclusion]
[instance] is a [conclusion]
Thus [instance] is a member of [class].

Ie:
All men are gits.
Alice is a git.
Thus Alice is a man.

Showing that your instance satisfied the conclusion (or has the properties that the theorem implies) does not show that it satisfied the theorem.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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