How can I improve at real analysis and proving questions? (1 Viewer)

[Synchronised]

I noticed that almost all the more difficult maths extension 2 questions are either analysis or proving so how can I effectively improve at those? Maths extension 2 textbooks don't really cover those topics* very well so which textbooks are good? Should I consider buying university level textbooks? Is doing a lot of past papers enough? But how can I learn the theory behind the questions before doing them? I started year 12 with tutor in about June so all I have left is mechanics and harder 3U but when I do past papers I feel like the textbooks don't really cover everything, some questions are quite difficult, especially real analysis (series, sequences, summation, etc...) and proving.

 

[Carrotsticks]

The idea of the HSC is not to learn as many advanced topics as possible, to get ahead, but to apply what you already have.

Sure, a detailed course in Analysis may help you out a bit with finer concepts such as convergence, but no such thing is really introduced during HSC Examinations. For example, they wouldn't exactly 'lead you' into the Monotone Convergence Theorem, which is one of the most important theorems when it comes to limits and series etc.

If I were you, I would focus on the MX2 topics and perhaps dig through a couple of 'HSC technique-friendly yet more difficult than usual' problems, which you can often find in older textbooks on say Complex Numbers (which would eventually lead to Complex Analysis most of the time).

Learn to apply your techniques. You want to be good at Mathematics because you can apply them well, not because you have done 'similar questions in the past'.

 

[Synchronised]

Ok, I get what you mean, I should just focus on the MX1 and MX2 topics and then after I learn the topics very well I apply the techniques to harder questions.
The problem is that there is no topic in MX2/MX1 that directly covers real analysis so my only option is to just do as many past papers as possible and learn how to do these questions...

 

[Carrotsticks]

Ok, I get what you mean, I should just focus on the MX1 and MX2 topics and then after I learn the topics very well I apply the techniques to harder questions.
The problem is that there is no topic in MX2/MX1 that directly covers real analysis so my only option is to just do as many past papers as possible and learn how to do these questions...

The thing is, university-level Real Analysis is VERY different from analysis topics studied in the HSC.

For example, Calculus arises from Analysis, but the stuff you are used to doing (finding equation of tangent such that blah blah) is VERY different to the treatment of calculus in higher Mathematics. If you were to pick up a Real Analysis book right now, chances are you would be thinking 'wtf is this?' because so much of it is based on very STRICT FORMAL definitions, which is not something emphasised in the HSC.

Series and Sequences is a part of analysis, sure. The whole concept of "A limiting sum can only exist if -1 < r < 1", sure. However, to contrast the HSC and Uni... HSC just lets you asssume that a limiting sum exists if |r| < 1 but at a higher level, such a claim cannot be said lightly like that without a formal proof.

Your best bet ATM is to do lots of past papers to get a rough idea of the level of difficulty and the techniques used, and then those 'hard textbook questions' to REALLY get you to use these techniques properly, to their fullest potential.

And again, regarding the bolded part, you don't want to 'learn how to do questions'. You want to learn how to APPLY your knowledge to questions so then if the Board of Studies decides to throw in something never seen before, you're prepped for that kind of new and unfamiliar situation.

 

[Synchronised]

The thing is, university-level Real Analysis is VERY different from analysis topics studied in the HSC.

For example, Calculus arises from Analysis, but the stuff you are used to doing (finding equation of tangent such that blah blah) is VERY different to the treatment of calculus in higher Mathematics. If you were to pick up a Real Analysis book right now, chances are you would be thinking 'wtf is this?' because so much of it is based on very STRICT FORMAL definitions, which is not something emphasised in the HSC.

Series and Sequences is a part of analysis, sure. The whole concept of "A limiting sum can only exist if -1 < r < 1", sure. However, to contrast the HSC and Uni... HSC just lets you asssume that a limiting sum exists if |r| < 1 but at a higher level, such a claim cannot be said lightly like that without a formal proof.

Your best bet ATM is to do lots of past papers to get a rough idea of the level of difficulty and the techniques used, and then those 'hard textbook questions' to REALLY get you to use these techniques properly, to their fullest potential.

And again, regarding the bolded part, you don't want to 'learn how to do questions'. You want to learn how to APPLY your knowledge to questions so then if the Board of Studies decides to throw in something never seen before, you're prepped for that kind of new and unfamiliar situation.

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