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$\noindent Don't need integration. Just write $g(x) = e^x -x -4$ and show by direct substitution (and use of a calculator) that $g(0)$ and $g(2)$ have opposite signs. The claim then follows, i.e. we have $g\left(\xi\right)=0$ for some $\xi \in \left(0,2\right)$ (by the Intermediate Value...

You do understand what the Q's mean though, right?

The meanings of Q.1's parts are clear too (and you haven't told us what f(x) is for that one).

Which parts don't you understand? Like for question 2), a) is standard (differentiate f and show the derivative is always positive), b)'s meaning is clear, and so are the meanings of c) and d).

Re: First Year Uni Calculus Marathon Ah, OK.

If your steps are reversible, then it is definitely logically fine.

Re: First Year Uni Calculus Marathon Oh, I thought this was the first claim you were referring to (i.e. not the "other" one). So the one without the "other" was the one about the well-defined limit (which was of course true)?

If your steps in the proof are all reversible (so "<=>" type steps), then to prove an inequality, it is logically valid to start with the given to-prove inequality and simplify it down using such reversible steps to something that you know is clearly true (e.g. (x-y)^2 >= 0), and then...

To get a mark of 60 in 4U, I think a really low raw mark suffices. Haven't checked the Raw Marks Database recently but from memory something like 65-70 raw scales to E4 (so 90), so imagine what'd be needed raw-wise to scale to 60 (quite low probably!). Furthermore, a lot of the Q's in HSC 4U...

Scaled mark of 60 in 4U? Isn't that achievable with essentially a very low raw mark? (What do you mean by scaled mark of 60, do you mean the HSC examination mark?)

According to http://www.atarcalculator.com.au, a 96 in Legal Studies is a 99.75 ATAR-equivalent, whereas 89 in HSC 4U Maths is 99.35 ATAR-equivalent. According to that ATAR Calculator, the minimum 4U mark needed to get an ATAR-equivalent above that of a 96 Legal Studies mark is 93 4U HSC mark...

Re: First Year Uni Calculus Marathon OK. So essentially the limit of that trig. expression, defined with domain D being what you said (basically can just take it to be some punctured neighbourhood of 0 with all points where sin(1/t) = 0 also removed, i.e. remove all the numbers of the form...

How much do you plan on studying in the holidays before start of Term 1 Year 12? If you do too much you may burn yourself out before the year's even begun. Wouldn't you want to relax a bit in the holidays, seeing as it's a long break (and the last break) before Year 12 officially begins?

Re: First Year Uni Calculus Marathon Ah, thanks for those explanations! :) Regarding the counterexample thing, the claim I think there should be counterexamples to is essentially the following (let's just stick to functions from R to R for now). $\noindent Let $g\colon \mathbb{R}\to...

Re: First Year Uni Calculus Marathon Ah, I see. I was (evidently mistakenly) thinking the limit wouldn't exist (or even be meaningful to talk about), because the expression in question (the thing with the t*sin(1/t)'s) isn't even defined in any neighbourhood of 0, is it ('cos it blows up...

Re: First Year Uni Calculus Marathon Invertibility is a sufficient but not actually quite necessary condition. Might post about it later or someone else might.

Re: First Year Uni Calculus Marathon The "informal approach" (change of variables in a limit) actually does work if certain conditions are satisfied (which usually is the case, but isn't here). And what was this bait thing?

Re: First Year Uni Calculus Marathon The purpose of this Q. was actually to show that this sort of change of variables in limits does not always work. The last limit in fact doesn't exist (first two parts are correct :)). Can you see why?

Re: First Year Uni Calculus Marathon One last Q. from me for now. $\noindent a) Show $\lim _{t \to 0}t \sin \frac{1}{t} = 0$. $ $\noindent b) Find $\lim _{x \to 0} \frac{\left(1+x\right)^2 -1}{x}$.$ $\noindent c) Find $\lim _{t\to 0} \frac{\left(1+t\sin \frac{1}{t}\right)^2 -1}{t\sin...

Re: First Year Uni Calculus Marathon Oh yeah, whoops, sorry.