# Thread: Calculus & Analysis Marathon & Questions

1. ## Re: First Year Uni Calculus Marathon

Originally Posted by seanieg89
$1. Suppose f and g are differentiable functions on the interval [a,b] such that g(a)\neq g(b). Prove that there exists c\in (a,b) such that\\ \\ \frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.\\ \\ Hint: The main ingredient is applying the extreme value theorem to a suitably chosen function. (Applying the extreme value theorem in this way is actually how one proves a different well-known theorem which can also be used in this question in place of the extreme value theorem).\\ \\2. (Openended) Using (1) and simpler results (standard theorems from differential calculus but nothing to do with integration), state and prove whatever versions of L'Hopital's rule that you can.$
Do you have any hints as to what the 'suitably chosen function' is? I'm struggling to find it

2. ## Re: First Year Uni Calculus Marathon

Originally Posted by porcupinetree
Do you have any hints as to what the 'suitably chosen function' is? I'm struggling to find it
It is slightly tricky to spot, but the core idea is the same as in the proof of the mean value theorem. So perhaps I should make this a leadup exercise.

The mean value theorem asserts we can find a c in (a,b) with f'(c)=(f(b)-f(a))/(b-a) if f is a differentiable function on [a,b].

B1. (Rolle's Theorem) Prove that if g is a differentiable function on [a,b] with g(a)=g(b), then g'(c)=0 for some c in (a,b). (Use the extreme value theorem to do this)

B2. (MVT) By choosing g appropriately show that f'(c)=(f(b)-f(a))/(b-a) for some c in (a,b). (The appropriate choice is easier to see here than in my original question, I think you are capable of finding it. We can also interpret this statement geometrically, which might help).

Then attempt the questions in my previous post. Aim to use Rolle's in a similar way to how it is used in Q2 of this post. With this target in mind, the choice of function should be easier for you to find.

3. ## Re: First Year Uni Calculus Marathon

Note: To be clear on nomenclature, when I say a function is differentiable on [a,b], I am being lazy. I actually mean that f is a continuous function on [a,b] that is differentiable on (a,b). No assumptions are made about the existence of one-sided derivatives or anything at the boundary of the interval. I doubt this will affect the way anyone approaches the problem though.

4. ## Re: First Year Uni Calculus Marathon

I kinda want to make this thread more accessible to some people (where possible) so here's a somewhat easy question. If it's ignored too bad back to reasonable difficulty.

$\\By considering the function g:[a,b]\rightarrow \mathbb R, g(x) = f(x) - \left(\frac{f(b)-f(a)}{b-a}(x-a)+f(a) \right) where f is continuous on [a,b] and differentiable on (a,b), prove the statement of the mean value theorem.$

Notes:
 Spoiler (rollover to view): Some other theorem should be assumed.

5. ## Re: First Year Uni Calculus Marathon

Originally Posted by leehuan
I kinda want to make this thread more accessible to some people (where possible) so here's a somewhat easy question. If it's ignored too bad back to reasonable difficulty.

$\\By considering the function g:[a,b]\rightarrow \mathbb R, g(x) = f(x) - \left(\frac{f(b)-f(a)}{b-a}(x-a)+f(a) \right) where f is continuous on [a,b] and differentiable on (a,b), prove the statement of the mean value theorem.$

Notes:
 Spoiler (rollover to view): Some other theorem should be assumed.
This is basically a hint to one of seanieg89's hint exercises.

6. ## Re: First Year Uni Calculus Marathon

Originally Posted by InteGrand
This is basically a hint to one of seanieg89's hint exercises.
I, did not see that coming.

7. ## Re: First Year Uni Calculus Marathon

It's just providing the function for my question B2 (2nd question in second post) which is not too much of a spoiler, but now someone should definitely be able to prove that.

B1 is a bit trickier, but for any student who wants to assume Rolle's and doesn't care where it comes from, you can move straight on to the questions in my first post which was the main point (to prove L'Hopital's).

8. ## Re: First Year Uni Calculus Marathon

And some much easier ones for students who don't want to do the above questions:

E1. Prove that if a function f: (a,b) -> R is differentiable and f'(x) is non-negative in this interval, then f is non-decreasing in this interval.

E2. Prove that if a function f: (a,b) -> R is differentiable and f'(x) = 0 in this interval, then f is constant.

9. ## Re: First Year Uni Calculus Marathon

I thought E2 was kinda intuitive but I'm finding it hard to word my final bit.

Rolle's theorem dictates that if f is continuous on [a,b] and differentiable on (a,b), and we have f(a)=f(b), then there exists at least one value of c such that f'(c)=0 for c in (a,b)

But if f'(c)=0 for all (a,b) (or alternatively f has a horizontal tangent for all points on the interval), then as we have an infinite number of values satisfying f'(c)=0, f must be constant for every c in [a,b]

10. ## Re: First Year Uni Calculus Marathon

Originally Posted by leehuan
But if f'(c)=0 for all (a,b) (or alternatively f has a horizontal tangent for all points on the interval), then as we have an infinite number of values satisfying f'(c)=0, f must be constant for every c in [a,b]
What do you mean by saying f must be constant for every c? What does it mean to be constant at a point?

11. ## Re: First Year Uni Calculus Marathon

I wasn't too sure how to argue it cause both statements seemed trivial corollaries of specific theorems or definitions. I just wasn't sure how to argue about how if the tangent is always horizontal, the curve never increases or decreases

(And yeah that language was what I meant by I don't know how to word it)

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12. ## Re: First Year Uni Calculus Marathon

Originally Posted by leehuan
I wasn't too sure how to argue it cause both statements seemed trivial corollaries of specific theorems or definitions. I just wasn't sure how to argue about how if the tangent is always horizontal, the curve never increases or decreases

(And yeah that language was what I meant by I don't know how to word it)

Sent from my iPhone using Tapatalk
I wouldn't say they are trivial. (But they are easy consequences of specific theorems. A correct proof should be very short.)

You can't rely on intuition too much here as funny things can happen in analysis. I can tell you that just changing your wording in that final paragraph won't really lead to a proof. Also, you mention Rolle's theorem in your second para but then make no use of it later...did you mean to?

13. ## Re: First Year Uni Calculus Marathon

Originally Posted by seanieg89
I wouldn't say they are trivial. (But they are easy consequences of specific theorems. A correct proof should be very short.)

You can't rely on intuition too much here as funny things can happen in analysis. I can tell you that just changing your wording in that final paragraph won't really lead to a proof. Also, you mention Rolle's theorem in your second para but then make no use of it later...did you mean to?
Oh. Whoops at not properly using the theorem I quoted.

But nah I clicked submit with the mindset that the proof was invalid so I was open to comment. I was too stuck on wording

14. ## Re: First Year Uni Calculus Marathon

Originally Posted by seanieg89
$A1. Suppose f and g are differentiable functions on the interval [a,b] such that g(a)\neq g(b). Prove that there exists c\in (a,b) such that\\ \\ \frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.\\ \\ Hint: The main ingredient is applying the extreme value theorem to a suitably chosen function. (Applying the extreme value theorem in this way is actually how one proves a different well-known theorem which can also be used in this question in place of the extreme value theorem).\\ \\A2. (Openended) Using (1) and simpler results (standard theorems from differential calculus but nothing to do with integration), state and prove whatever versions of L'Hopital's rule that you can.$
A1:

$Define a new function h(x) = f(x) - rg(x) such that h(a) = h(b):$
$f(a) - rg(a) = f(b) - rg(b)$
$\Rightarrow r = \frac{f(a) - f(b)}{g(a) - g(b)}$
$\therefore h(x) = f(x) - \frac{f(a) - f(b)}{g(a) - g(b)} g(x)$
$h is continuous on [a, b] and differentiable on (a, b) due to limit/differentiation laws (given that both f and g are).$
$Hence we can apply Rolle's Theorem to h: There is c \in (a, b) such that h'(c) = 0, i.e.:$
$f'(c) - \frac{f(a) - f(b)}{g(a) - g(b)} g'(c) = 0$
$\Rightarrow \frac{f'(c)}{g'(c)} = \frac{f(a) - f(b)}{g(a) - g(b)}$

15. ## Re: First Year Uni Calculus Marathon

That's more like it .

16. ## Re: First Year Uni Calculus Marathon

This question got asked in my calculus tutorial. Took me 30 seconds of staring at it before I realised what was going on but Paradoxica could probably do it in 1 second.
(Especially since I have a feeling this got asked on BoS before...)

$\lim_{x\rightarrow \infty}e^{-x^2}\int_0^x{e^{t^2}dt}$

17. ## Re: First Year Uni Calculus Marathon

Originally Posted by leehuan
This question got asked in my calculus tutorial. Took me 30 seconds of staring at it before I realised what was going on but Paradoxica could probably do it in 1 second.
(Especially since I have a feeling this got asked on BoS before...)

$\lim_{x\rightarrow \infty}e^{-x^2}\int_0^x{e^{t^2}dt}$
Where on BOS do you think it got asked before? As in in some marathon from the past?

18. ## Re: First Year Uni Calculus Marathon

Originally Posted by InteGrand
Where on BOS do you think it got asked before? As in in some marathon from the past?
Feels like some marathon or extracurricular thread. Might not have been the exact same one but it looked too familiar.

@Para yes the answer was 0

19. ## Re: First Year Uni Calculus Marathon

Originally Posted by leehuan
This question got asked in my calculus tutorial. Took me 30 seconds of staring at it before I realised what was going on but Paradoxica could probably do it in 1 second.
(Especially since I have a feeling this got asked on BoS before...)

$\lim_{x\rightarrow \infty}e^{-x^2}\int_0^x{e^{t^2}dt}$
I don't see the answer jumping out at me.

Imaginary Error Function lol

I'm thinking of using a bounding argument on the integral and using that to find the limit.

20. ## Re: First Year Uni Calculus Marathon

I don't see the answer jumping out at me.

Imaginary Error Function lol

I'm thinking of using a bounding argument on the integral and using that to find the limit.
 Spoiler (rollover to view): LH

21. ## Re: First Year Uni Calculus Marathon

Originally Posted by leehuan
Feels like some marathon or extracurricular thread. Might not have been the exact same one but it looked too familiar.

@Para yes the answer was 0
Well that was based on me accidentally seeing that as -t2

So no.

22. ## Re: First Year Uni Calculus Marathon

Originally Posted by InteGrand
 Spoiler (rollover to view): LH
*blindly bashes everything with L'hôpital's Rule*

23. ## Re: First Year Uni Calculus Marathon

$\noindent Find the n-th order Taylor polynomial for f(x) = \frac{1}{1+x} . Hence find the n-th order Taylor polynomial for g(x) = \tan^{-1}x .$

24. ## Re: First Year Uni Calculus Marathon

Originally Posted by porcupinetree
$\noindent Find the n-th order Taylor polynomial for f(x) = \frac{1}{1+x} . Hence find the n-th order Taylor polynomial for g(x) = \tan^{-1}x .$
Has USYD already started Taylor series ?

25. ## Re: First Year Uni Calculus Marathon

Originally Posted by Drsoccerball
Has USYD already started Taylor series ?
Yup. I'm guessing you guys haven't?

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