Calculus 2 and Real Analysis in one semester?

[member2357]

I am in my first semester of university and currently taking Linear Algebra. I was planning on majoring in EECS but I lost interest in EE and engineering in general (except software) and gained a lot of interest in maths (especially statistics and financial mathematics) so I will double major in maths and CS. Real Analysis is a core subject for both CS and statistics. It is a prerequisite for most second year maths subjects and almost all third year maths subjects.

I have the following two options:

1. Take Calculus 2 in first year semester 2 and Real Analysis in second year semester 1.
2. Take an accelerated subject in first year semester 2 which combines both Calculus 2 and Real Analysis into one subject. I (barely) gained admission into this subject as I got 90+ in HS calculus.

I have a 6 weeks break between semesters 1 and 2, so is it a good idea to pre-study for the accelerated subject or will I run the risk of burning out? Or should I take it slowly and do Calculus 2 and Real Analysis separately?

The only difference it makes is that if I do Calculus 2 and Real Analysis separately, I won't be able to take Graph Theory and I won't be able to take Complex Analysis in second year. I can still take Complex Analysis in third year but I won't be able to take Graph Theory at all. Takeing Calculus 2 and Real Analysis as one subject basically allows me to take one extra subject.

If I want to pre-study for the accelerated class effectively, do you have any advice? If you were in this situation and had 6 weeks to prepare, what would you do?

To be honest, I am kind of intimidated by the accelerated class, because it is considered to be one of the most difficult undergraduate classes.

 

[lurflurf]

You do not give many details. What are Calculus 2 and Real Analysis? What country are you in? Canada? Did you understand the high school calculus? I infer from your post that calculus 2 overlaps substantially with high school calculus and real analysis is a small step up from that perhaps with more proofs and harder examples. The combined course probably covers the calculus 2 very quickly and speeds most of the time on real analysis. What you should do depends on where you weaknesses lie.
case 1)Your basic mathematics and calculus are both weak as is your mathematics in general
slow down revise past subjects and put off real analysis until next year
case 2)Your basic mathematics and calculus are solid but proofs and theory give you some trouble
work on proofs and theory real analysis is probably very reasonable with a head start
in particular try to generalize and prove things from calculus
case 3)Your calculus is a little shaky but you are very able, perhaps your linear algebra class was very difficult and you did well
go over calculus pay particular attention to general results and proofs as they are easy and fun for you
case 4)Your basic mathematics and calculus are solid but proofs and theory give you no trouble as you are very able
you have nothing to worry about, but you might like to preview course contents

As for burning out you probably do not want to study for 500 hours in the six weeks, but 50-100 hours well spent might worth it so that you are well prepared especially since that time may not be as easy to find during the term if you do have some gaps that need filling.

 

[member2357]

Thank you very much for the reply.

Calculus 2 is basically the second half of Thomas' Calculus. Thomas' Calculus has 16 chapters, so it is chapter 8 on-wards. Since I did the highest level of maths available in high school, I can skip calculus 1.

Real analysis includes these topics:

• The Real Numbers
• Sequences and Series
• Convergence and Divergence
• Basic Topology of R
• Functional Limits and Continuity
• The Derivative
• Sequences and Series of Functions
• The Riemann Integral
• Fundamental Theorem of Calculus
• Mean Value Theorem
• Taylor's Theorem
• Other Topics

I did well in linear algebra but I thought the course was fairly easy so may not be a fair measure of how well I can do in real analysis.

You are right, the accelerated courses covers the calculus part very quickly in the first few lectures and the rest of the course is spent on real analysis.

I am good at calculus so I am not worried about it, I am just worried that I am not yet ready for the rigor of real analysis. I find proving to be quite difficult because I have done very little proving in high school.

I will pre-study, do the accelerated course and then after a few weeks (maximum 3 weeks), if I find the course too difficult and unmanageable, I can withdraw and go back to calculus 2. This may cause some problems with assignments and course structure though.

 

[verty]

I have an idea for a way to check whether you can handle this other course. I don't claim that this is scientific or wise in any way, but you could try this:

#1 Mathematical Logic notes
#2 Zakon basic analysis

Spend one week getting all you can out of #1, study it to death. Know it, love it. When you get stuck, start over. If you reach the end of the week and you haven't become despondent or distracted, this is a good sign.

Now you'll have 5 weeks for #2, it has 5 chapters so that is convenient. Pace yourself and do all the exercises.

-- edit --
Ooh, I see now that calculus 2 is multivariable, that is going to be very! difficult to learn at the same time as analysis. Now my strong recommendation is not to combine them unless you did multivariable in school.

Or I suppose you could cram even more...

 

[member2357]

I have an idea for a way to check whether you can handle this other course. I don't claim that this is scientific or wise in any way, but you could try this:

#1 Mathematical Logic notes
#2 Zakon basic analysis

Spend one week getting all you can out of #1, study it to death. Know it, love it. When you get stuck, start over. If you reach the end of the week and you haven't become despondent or distracted, this is a good sign.

Now you'll have 5 weeks for #2, it has 5 chapters so that is convenient. Pace yourself and do all the exercises.

Sounds good. This was the answer I was looking for, thank you!

 

[verty]

Sounds good. This was the answer I was looking for, thank you!

That was when I thought calculus 2 was integration. See the edit I made above.

 

[member2357]

Ooh, I see now that calculus 2 is multivariable, that is going to be very! difficult to learn at the same time as analysis. Now my strong recommendation is not to combine them unless you did multivariable in school.

Or I suppose you could cram even more...

Damn... I will cram anyway, either way, it will help me.

This means no graph theory :( I really wanted to do that subject but I guess I can self study it, may be after uni if I ever need it in CS.

 

[IGU]

This means no graph theory :( I really wanted to do that subject but I guess I can self study it, may be after uni if I ever need it in CS.

You know, graph theory is actually relevant to CS. Calculus and analysis aren't, or at best are barely relevant. If I were in your situation I would work on learning how to do proofs, then take the combined Calc 2 and analysis course. I don't see that it should be particularly hard for you, given how you describe yourself. It's pretty much stuff you feel comfortable with already, plus doing proofs.

 

[member2357]

You know, graph theory is actually relevant to CS. Calculus and analysis aren't, or at best are barely relevant. If I were in your situation I would work on learning how to do proofs, then take the combined Calc 2 and analysis course. I don't see that it should be particularly hard for you, given how you describe yourself. It's pretty much stuff you feel comfortable with already, plus doing proofs.

Yes, probability, discrete mathematics, linear algebra and graph theory are probably the most relevant maths subjects to CS. I will try my very best to do the combined course.

 

[IGU]

Yes, probability, discrete mathematics, linear algebra and graph theory are probably the most relevant maths subjects to CS. I will try my very best to do the combined course.

Actually, since I'm handing out reminders, I'll remind you that it's a little early to have any confidence that your decision to do a math major will stick. If you you haven't done proof-based math yet, then (at least by mathematicians' standards) you haven't even done any math. None at all. You've just been dabbling in learning how to use elementary mathematics as a tool to do other things. Once you learn some real math (where using it is nice, but proving it is everything), then is the time you can intelligently decide if you want to be a math major.

Taking the analysis course (if it's rigorous) should tell you that. I also encourage you to have a long talk with your advisor about what you want and how to find out if it's really what you want. Also how to find out what you're good at. Best of luck.

 

[member2357]

Actually, since I'm handing out reminders, I'll remind you that it's a little early to have any confidence that your decision to do a math major will stick. If you you haven't done proof-based math yet, then (at least by mathematicians' standards) you haven't even done any math. None at all. You've just been dabbling in learning how to use elementary mathematics as a tool to do other things. Once you learn some real math (where using it is nice, but proving it is everything), then is the time you can intelligently decide if you want to be a math major.

Taking the analysis course (if it's rigorous) should tell you that. I also encourage you to have a long talk with your advisor about what you want and how to find out if it's really what you want. Also how to find out what you're good at. Best of luck.

I actually would never do a pure maths major, I am simply not good enough or have the interest for a pure maths major.

The mathematics major I am talking about is statistics. The only 'pure' and proof-based subject I will do is real analysis (discrete maths may also be categorized as proof-based, I don't know) and after that the only subjects I will take are probability, statistics, financial mathematics, discrete mathematics, etc. I am not going to take abstract algebra, algebraic geometry, topology, or any analysis subject higher than the real analysis subject I am talking about because I am not good enough and not interested.

I am interested in machine learning, which requires quite a bit of probability and statistics. I am also very interested in quantitative finance, which also requires a lot of probability and statistics. My goal is to work as a quantitative analyst at a bank, but if that didn't happen, I am happy to work in machine learning research (at a company, not university) or programming/software engineering.

I talked to many quants, and they all said a double major in CS and statistics is highly appreciated by banks. I also know that a major in statistics is useful in machine learning. Do you think that isn't a good enough reason for me to major in statistics? Should I only major in anything maths related if I was a mathematical Olympiad champion?

Thank you a lot for your help.

 

[verty]

You know, graph theory is actually relevant to CS. Calculus and analysis aren't, or at best are barely relevant. If I were in your situation I would work on learning how to do proofs, then take the combined Calc 2 and analysis course. I don't see that it should be particularly hard for you, given how you describe yourself. It's pretty much stuff you feel comfortable with already, plus doing proofs.

I'm guessing this course is similar to MIT's 18.022. This is n-variables at once, pure math that is not entirely relevant to CS. And it could be the type of course where the student does all the work. This is a steep ramp.

 

[member2357]

I'm guessing this course is similar to MIT's 18.022. This is n-variables at once, pure math that is not entirely relevant to CS. And it could be the type of course where the student does all the work. This is a steep ramp.

Yes, the syllabus for 'calculus 2' is pretty much the same as MIT's 18.022. However, the combined accelerated course has more topics than MIT's 18.022.