Calculus & Analysis Marathon & Questions (1 Viewer)

leehuan · Apr 30, 2016

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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seanieg89 · Apr 30, 2016

Re: First Year Uni Calculus Marathon

leehuan said:
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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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?

leehuan · Apr 30, 2016

Re: First Year Uni Calculus Marathon

seanieg89 said:
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

porcupinetree · May 1, 2016

Re: First Year Uni Calculus Marathon

seanieg89 said:
$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)}$$

seanieg89 · May 1, 2016

Re: First Year Uni Calculus Marathon

That's more like it

.

leehuan · May 2, 2016

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...)

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

InteGrand · May 2, 2016

Re: First Year Uni Calculus Marathon

leehuan said:
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?

leehuan · May 2, 2016

Re: First Year Uni Calculus Marathon

InteGrand said:
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

Paradoxica · May 2, 2016

Re: First Year Uni Calculus Marathon

leehuan said:
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.

InteGrand · May 2, 2016

Re: First Year Uni Calculus Marathon

Paradoxica said:
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.

LH

Paradoxica · May 2, 2016

Re: First Year Uni Calculus Marathon

leehuan said:
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.

Paradoxica · May 2, 2016

Re: First Year Uni Calculus Marathon

InteGrand said:
LH

*blindly bashes everything with L'hôpital's Rule*

porcupinetree · May 7, 2016

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 $.$

Drsoccerball · May 7, 2016

Re: First Year Uni Calculus Marathon

porcupinetree said:
$$\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 ?

porcupinetree · May 7, 2016

Re: First Year Uni Calculus Marathon

Drsoccerball said:
Has USYD already started Taylor series ?

Yup. I'm guessing you guys haven't?

Paradoxica · May 7, 2016

Re: First Year Uni Calculus Marathon

porcupinetree said:
Yup. I'm guessing you guys haven't?

shhh....

We need to give UNSW students the impression that they're ahead....

leehuan · May 7, 2016

Re: First Year Uni Calculus Marathon

porcupinetree said:
Yup. I'm guessing you guys haven't?

Taylor series are second semester for us.

KingOfActing · May 7, 2016

Re: First Year Uni Calculus Marathon

porcupinetree said:
$$\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 $.$

$\frac{1}{1+x} = \frac{1}{1 - (-x)} = \sum_{n=0}^{\infty} (-x)^n\\\frac{1}{1+x^2} = \sum_{n=0}^{\infty} (-x^2)^n\\ \int_{0}^{x} \frac{1}{1+t^2}\,dt = \int_{0}^{x} \sum_{n=0}^{\infty} (-1)^nt^{2n}\,dt\\\arctan{x} = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{2n+1}$

The nth order Taylor polynomials are the just the infinite series truncated at the nth degree summand.

Paradoxica · May 7, 2016

Re: First Year Uni Calculus Marathon

KingOfActing said:
$\frac{1}{1+x} = \frac{1}{1 - (-x)} = \sum_{n=0}^{\infty} (-x)^n\\\frac{1}{1+x^2} = \sum_{n=0}^{\infty} (-x^2)^n\\ \int_{0}^{x} \frac{1}{1+t^2}\,dt = \int_{0}^{x} \sum_{n=0}^{\infty} (-1)^nt^{2n}\,dt\\\arctan{x} = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{2n+1}$

The nth order Taylor polynomials are the just the infinite series truncated at the nth degree summand.

Don't forget to state the radius of convergence.

Although in this case, it is fairly obvious.

InteGrand · May 7, 2016

Re: First Year Uni Calculus Marathon

$\noindent i) Show that the $n$-th order Taylor polynomial for $\mathrm{e}^{x}$ about 0 is $P_{n} (x) = 1+x+\frac{x^2}{2!}+\cdots + \frac{x^{n}}{n!}$.$

$\noindent ii) Using the Lagrange form for the error term (or otherwise), show that the Taylor series $1+x+\frac{x^2}{2!}+\cdots$ converges to $\mathrm{e}^{x}$ for all real $x$.$

$\noindent iii) For $x<0$, show using induction or otherwise that$

$$\mathrm{e}^{x} < 1+x+\frac{x^2}{2!}+\cdots + \frac{x^{n}}{n!}$ if $n \geq 0$ is even$

$\noindent and$

$$\mathrm{e}^{x} > 1+x+\frac{x^2}{2!}+\cdots + \frac{x^{n}}{n!}$ if $n \geq 0$ is odd.$

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