Calculus & Analysis Marathon & Questions (6 Viewers)

Paradoxica · Jul 15, 2016

Re: First Year Uni Calculus Marathon

Here's a fun exercise

Evaluate the following limit without using L'Hôpital's rule.

Also no series expansions of any sort.

$\lim_{x \to 0} \frac{x-\sin{x}}{x^3}$

You may assume the following preliminary result:

$\lim_{x \to 0} \frac{\sin{x}}{x} = 1$

InteGrand · Jul 15, 2016

Re: First Year Uni Calculus Marathon

1/3! = 1/6, via Taylor expansion.

Paradoxica · Jul 15, 2016

Re: First Year Uni Calculus Marathon

InteGrand said:
1/3! = 1/6, via Taylor expansion.

stop ruining everything for me.

InteGrand · Jul 15, 2016

Re: First Year Uni Calculus Marathon

Paradoxica said:
stop ruining everything for me.

It's probably the first time I've done so for a while.

leehuan · Jul 15, 2016

Re: First Year Uni Calculus Marathon

Haven't even done Taylor series or ODEs yet. Heck, haven't even done integration techniques at uni.

Paradoxica · Jul 15, 2016

Re: First Year Uni Calculus Marathon

leehuan said:
Haven't even done Taylor series or ODEs yet. Heck, haven't even done integration techniques at uni.

The limit's proof is completely elementary.

Try using substitution, in the exact same way you would for integration.

leehuan · Jul 26, 2016

Re: First Year Uni Calculus Marathon

I have no idea where this question is meant to go but it's not SOS. Just an interesting question that came out of MATH1251 discussion.

$Let $S$ be a finite set such that $S \neq \emptyset\\ $a) Write down a set $S$ that satisfies closure under addition.$\\ $b) Prove that this is the only finite set that satisfies closure under addition.$

seanieg89 · Jul 26, 2016

Re: First Year Uni Calculus Marathon

Are you talking about subsets of the reals/complex numbers with the standard operation of addition? Then the only finite additively closed set is {0} because if some nonzero z is in it, then nz:=z+...+z (n copies) must be in it for all positive integers n, and these numbers are distinct.

Or are you talking about an abstract set with an addition operation satisfying some axioms?

leehuan · Jul 26, 2016

Re: First Year Uni Calculus Marathon

Nah that's right lol the set is {0}

InteGrand · Aug 7, 2016

Re: First Year Uni Calculus Marathon

Solve the differential equation $\frac{\mathrm{d}y}{\mathrm{d}t} = 2\sqrt{y}$ .

leehuan · Aug 7, 2016

Re: First Year Uni Calculus Marathon

InteGrand said:
Solve the differential equation $\frac{\mathrm{d}y}{\mathrm{d}t} = 2\sqrt{y}$ .

This is about the same as the one I asked for help with right?

$y=0$ is a solution. Otherwise $\int \frac{dy}{2\sqrt{y}}dy=\int dt \implies y=(t+C)^2$

Paradoxica · Aug 7, 2016

Re: First Year Uni Calculus Marathon

leehuan said:
This is about the same as the one I asked for help with right?

$y=0$ is a solution. Otherwise $\int \frac{dy}{2\sqrt{y}}dy=\int dt \implies y=(t+C)^2$

but what happens when t<-C ?

The solution you have provided is invalid.

leehuan · Aug 7, 2016

Re: First Year Uni Calculus Marathon

Shit. Forgot about restrictions already even though I asked about it the other day.

Paradoxica · Aug 7, 2016

Re: First Year Uni Calculus Marathon

Evaluate the following integral by using the Leibniz Rule.

$$\noindent$ \int_0^{\pi} \frac{\log{(1+\cos{x}\sin{\theta}})}{\cos{x}} \, \text{d}x \\ $Where $0 \le \theta \le \frac{\pi}{2}$

seanieg89 · Aug 8, 2016

Re: First Year Uni Calculus Marathon

$J(t):=\int_0^\pi \frac{\log(1+t\cos(x))}{\cos(x)}\, dx\\ \\ = J(0)+\int_0^t J'(s)\, ds\\ \\ = \int_0^t \left(\int_0^\pi \frac{d\phi}{1+s\cos(\phi)}\right)\, ds\\ \\ = \int_0^t \frac{\pi}{\sqrt{1-s^2}}\, ds\\ \\ = \pi \sin^{-1}(t) $ for $t\in [0,1].\\ \\ $Hence $J(\sin(\theta))=\pi \theta $ for $\theta\in [0,\pi/2].\\ \\ $(Note that the $\phi$ integral is MX2 level and can be done by the Weierstrass substitution for example.)$$

Paradoxica · Aug 8, 2016

Re: First Year Uni Calculus Marathon

$$\noindent$ f $ is a differentiable function for all $x \in \mathbb{R} \\ $It is given that $f$ satisfies $ f'(\beta) = \frac{f(\beta)}{2-\beta} +\frac{f(\beta)}{4-\beta} \\ $Prove $\beta \in [ 2, 4 ]$

Hint: Use Rolle's Theorem.

seanieg89 · Aug 11, 2016

Re: First Year Uni Calculus Marathon

Here's a cool one to think about:

$$Given an arbitrary sequence $(a_k)_{k=0}^\infty$ of real numbers, does there exist a a smooth function $f\in\mathcal{C}^\infty(\mathbb{R})$ such that\\ \\ $\frac{d^kf}{dx^k}(0)=a_k$ for each $k$?$\\ \\$

Paradoxica · Aug 11, 2016

Re: First Year Uni Calculus Marathon

seanieg89 said:
Here's a cool one to think about:

$$Given an arbitrary sequence $(a_k)_{k=0}^\infty$ of real numbers, does there exist a a smooth function $f\in\mathcal{C}^\infty(\mathbb{R})$ such that\\ \\ $\frac{d^kf}{dx^k}(0)=a_k$ for each $k$?$\\ \\$

Construct a Taylor Series with pointwise convergence at 0?

seanieg89 · Aug 11, 2016

Re: First Year Uni Calculus Marathon

Paradoxica said:
Construct a Taylor Series with pointwise convergence at 0?

Taylor series are indeed the correct lines to think along here, but as an arbitrary Taylor series need not converge anywhere apart from the point it is centred at, this problem is not completely trivial.

Do you have a construction in mind?

Paradoxica · Aug 11, 2016

Re: First Year Uni Calculus Marathon

seanieg89 said:
Taylor series are indeed the correct lines to think along here, but as an arbitrary Taylor series need not converge anywhere apart from the point it is centred at, this problem is not completely trivial.

Do you have a construction in mind?

The coefficients of x^k will be a_k/k!...

If you wanted my naive first contruction...

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