Advice for beginners in calculus

[plumberbabu]

Hello Guys,
I am a rookie to this forum and need some guidance in studying calculus. I am a medical graduate and master in epidemiology but will be enrolling for doctoral studies in epidemiology. I wish to take more of biostatistical courses and out of my interest I am contemplating to take math courses to build a foundation particularly in Calculus.

From what i gather, I have to brush up Trigonometry and Algebra-----> Then study calculus 1, 2 and 3 in that order and take Linear algebra at the end.
Q 1) Is this a correct way to study? Is there any overlap in Calculus 1 2 or 3.
Q 2) At what stage do I take Introduction to statistics classes.

FYI I have 1 year to study these topics and my employer will pay for 18 credits in 1 year to study ...Kindly guide me as to what is the ideal way so that i do not waste my effort in wrong way

My apologies for such a long post...Any advice will be appreciated

 

[micromass]

Hi pumberbabu!

As for Q1, your plan seems sound. You can even take linear algebra sooner (after calc II for example).

As for Q2, I'm not sure. Can you list the contents of your statistics course? Normally, a knowledge of Calc I and Calc II would be enough for basic statistics classes. However, you might do some multivariate things in it, so a knowledge of linear algebra could be handy. But it depends on the course in question...

 

[plumberbabu]

Hey Micromass,
Thanks for the quick reply. As of now I am not sure what are the contents of introduction to statistics but will find out from the school where i intend to study. It is a 3 credit course and all i know currently is that it has no prerequisites. Do you think I should wait for all other courses (Calculus+linear algebra) to finish before I study statistics?

Pertaining to my question 1)...How different is Precalculus from plain Algebra and trigonometry??

i hope I am asking questions correctly?

 

[micromass]

Hey Micromass,
Thanks for the quick reply. As of now I am not sure what are the contents of introduction to statistics but will find out from the school where i intend to study. It is a 3 credit course and all i know currently is that it has no prerequisites. Do you think I should wait for all other courses (Calculus+linear algebra) to finish before I study statistics?

Ah, as it has no prerequisites, then it's probably a very basic introduction to probability and stuff. So, you can start this immediately. But if you've done Calc I and Calc II first, you might be better prepared for the course (i.e. be a little bit "mathematically mature").

Also, I would advise to take Linear algebra before (or together with) Calc III, since Calc III will probably use some ideas from linear algebra.

Pertaining to my question 1)...How different is Precalculus from plain Algebra and trigonometry??

It's not much more difficult. If you can do Algebra and trig, then you can do precalc. It's calculus where things get serious (not that it's difficult, but it's "different", it takes some adjusting).

i hope I am asking questions correctly?

Yes, don't worry. However, you may have posted this in the academic guidance forum. You'll get more answers there. (click on "report" and ask to move this post in academic guidance.)

 

[plumberbabu]

Thank you for your guidance. This will definitely give me an idea how to go about things when studying Math and calculus.
Like you said, I will also move this post to academic guidance forum

 

[chiro]

Hey there plumberbabu and welcome to the forums.

Usually before you take a year long course in introductory statistics, you need a year of calculus. The reason is that for any kind of continuous probability result (that is figuring out probabilities for continuous distributions like exponential, normal etc), you are calculating integrals.

Hypothesis testing also involves you doing pretty much the same thing (although many integrals can not be calculated analytically so instead you get tables with certain resolution that you can use to get probability values)

Multivariate calculus is used when you have distributions with more than one variable.

Linear algebra is important because the math in linear algebra is used extensively in the area of linear models. In experiments you typically set up an experiment with particular blocking factors, or a factorial design, and you have to use the design and its output data to test various inferences.

In a computational sense, you are basically solving some kind of a linear model and using the results to test various inferences. The math that is used in this uses a lot of linear algebra including idempotent matrices (projections onto some subspace where the sum of all of these is the identity) and least squares to name two.

The assumptions for this framework is that all effects are normally distributed. In Generalized Linear Models, this restriction is lifted with the exception of error terms (which have to be normally distributed).

If you want to do experiments and clinical trials, you're going to at some point, have to take a course on linear models. If you have a good background in linear algebra, it will help you immensely when you look at experimental design.

My advice is you do Calc I and Calc II first, then do Linear Algebra and Calc III (together if you can), then start your introductory statistics (any time after Calc II), then move on to higher level statistics relevant to bio-statistics.

On top of this, make sure you understand optimization in multiple variables, how to interpret probabilities in terms of inequalities for multiple variable (ie P(X < Y < Z)) in terms of the required integral representation, and everything in linear algebra.

If you understand all the stuff you learn in your introductory year statistics course, you should find the transitions from that to higher level statistics (like the GLM framework) a lot easier. Everything builds up on one another so if you feel unsure, clarify it as soon as possible.

I wish you all the best and I hope you enjoy statistics as much as I am enjoying it myself.

 

[plumberbabu]

Hello Chiro,
Thank you for your wholesome reply. The things that you have mentioned is what I had in my mind when I was going through various forums. As of now starting this fall, I am planning to take following courses for the next one year-
Calculus 1 - 4 credits
Calculus 2- 4 credits
Calculus 3- 4 credits
Linear algebra (introductory) - 3 credits.
Like you correctly mentioned I will need a strong basics for Biostatistics considering I will be participating in clinical trials, designing studies and analyzing them.
Now with all these facts I have a very basic doubt.
Do you think I should study precalculus at home before calculus 1 or do I need to study algebra an trigonometry separately and then precalculus. Since I am changing my majors from medicine to math...please pardon my ignorance about the subject stratification.

 

[lisab]

I would advise you to *really* hit the algebra and trig hard before you start calculus. Keep in mind, every calculus problem you will face will require algebra and trig. You don't want to be fumbling with your tools as you try to use them to solve the problem.

 

[plumberbabu]

Thanks lisab,
That certainly sounds perfect. It's very important to start any subject with correct concepts. I am glad you guys have made everything crystal clear for me.

 

[QuarkCharmer]

What stats class are you taking exactly? I took a statistics course that required only college algebra, and touched on some set theory, but mainly focused on graphs and deviation, with introduction to sigma et al. It was intended for like nurses and that sort of thing. I took it because I had no idea what any of that stuff was. I never understood what a "spread" was when I heard people talk about football or whatever. Anyway, knowing which class it is would help here.

 

[plumberbabu]

I had an introductory class of Biostatistics during my masters...it was all about chi square, t-test, analysis of variance, Kaplan Meier survival analysis. But if I have to study at doctoral level I am expected to study in detail about these tests where they would use various algebraic expressions something like what Chiro has mentioned in the previous post...i do not know much wat I will be learning in these (bio) statistical classes so I m not sure wat I m looking at...I have seen researchers discussing about parametric and non parametric methods, general linear models if that makes any sense...

 

[TylerH]

1 year is a really short time to accomplish as much as you want to, mainly because of the interdependent nature of the calculus sequence. How many terms do you have ie, does our school operate on semesters, trimesters, quarters? Even if you skip precal, you'll need at least 3 to get calc I, 2, and 3, because each relies heavily on those before it. As a person studying physical trends, it may behove you to also have knowledge of (ordinary) differential equations. And, a proof class helps for linear algebra, they require it at my school.

Precalculus, for me, was a review of HS algebra I and II, with some trigonometry, and various other things (conic sections, polar, parametric functions, sequences and series, stuff you really won't need to know ahead of time, because they are introduced in calc II).

Intro stats, if it's the class I think it is, should be easy for someone who can pass precalculus. What course number is it ie, what year do math undergrads take it? The one I'm thinking of is a 100 or 200 level class.

My suggested schedule in 4 terms:
1: Precalculus, intro stats
2: calc I, ?(It would be a shame to leave this open, if you decide to skip precal, intro stats goes here)
3: calc II, proof class(they usually have calc I as prereq)
4: calc III, linear algebra, optional: diff eq

That 4th semester will probably give you a few mental breakdowns.

EDIT: Addition:
Differential equations (de's), although largely considered harder than calc III, may be more beneficial to you than calc III.

To give you a simple explanation, de's allow one to describe a variable in terms of it's change or the change of another variable. Populations can be modeled with de's eg, change in prey = -change in predator, which basically says that, as the number of predators decreases, the number of prey increases. Now, given you know the actual number to plug in for the change in one of them, and the initial number of each, then you can get an equation that models their populations.

http://en.wikipedia.org/wiki/Differential_equation#Biology

 

[plumberbabu]

Thanks TylerH,
You have added a new perspective with your point on DEs. I will definitely consider it. Your schedule for taking the courses sounds very sensible to me. But like I said with limited funding for 18 credits in a year, I might study precalculus at home and reimburse for tuition for higher classes.

Now for your question, my masters program operated on semester basis with courses in Spring and Fall (4 months each) and vacation in summer; while the undergrad courses on math are offered every semester (Fall, Summer and Spring - 4 months in each semester, not sure if that is a trimester??).

the Intro to Stats is course number 180 so i guess its a first year course for undergrad in maths. Calculus 1 & 2 have a course code 125 and 126 while calculus 3 has course code 227. Intro to linear algebra is 260.

I Intend to finish all these courses by Summer 2012. I have 3 semesters to finish them -Fall 2011, Spring 2012 and Summer 2012.

With that in mind I am also looking for recommendation on books for precalculus to get started on my own. Currently I have "precalculus" by "larson and Hosteller" for self study from my library and will then hop on to Calculus when the courses start.

Correct me if anyone thinks otherwise.

 

[TylerH]

the Intro to Stats is course number 180 so i guess its a first year course for undergrad in maths. Calculus 1 & 2 have a course code 125 and 126 while calculus 3 has course code 227. Intro to linear algebra is 260.

Then I assumed correctly. That course is generally considered less challenging than calc I.

I Intend to finish all these courses by Summer 2012. I have 3 semesters to finish them -Fall 2011, Spring 2012 and Summer 2012.

Good, that will fit perfectly with my proposed schedule, assuming the classes still have openings. Just forget the precal, and shift stats to term 2.

With that in mind I am also looking for recommendation on books for precalculus to get started on my own. Currently I have "precalculus" by "larson and Hosteller" for self study from my library and will then hop on to Calculus when the courses start.

Correct me if anyone thinks otherwise.

I'm not one for self learning from a book. I do mine from online sources. I found themathpage.com to be excellent and more to the point than my textbook. All the stuff I listed in parentheses when mentioning precal before were in the book, but not at all necessary for entrance into calculus. You do need to know basic trig. the Math Page is good for that, too. Don't get too bogged down on identities (your professor will be impressed if you know the double angle identities, which you can derive from the sum identities). Just remember the circle, and be able to graph sinx, cosx, tanx, and their reciprocals. AND, VERY IMPORTANT: know that sin²x+cos²x=1 and be able to manipulate that.

NOTE: I should say: I'm only in calc II, myself. I'm taking it and a proof class this summer. But, if you can look past the fact my education doesn't fully represent the knowledge I've gained from my extensive self guided learning, I think I can be of help.

 

[plumberbabu]

You have been great help TylerH. Thanks for the link. If you prefer online resources you might want to consider "MIT open courseware" too. They have some good classes for every subject uploaded online and for free...my 2 cents
I hope you are enjoying calculus II...I am myself eagerly waiting to get back to my favorite subject (maths)

 

[MathWarrior]

One of the biggest things I think I had to overcome when taking my first Calculus class at least. Was you must understand properties of exponents, I cannot stress how important this is in Calculus. Then in Calculus 2 its very important to understand the problem, which makes it much harder to set up the problem then it is to do the math.

 

[TylerH]

You have been great help TylerH. Thanks for the link. If you prefer online resources you might want to consider "MIT open courseware" too. They have some good classes for every subject uploaded online and for free...my 2 cents
I hope you are enjoying calculus II...I am myself eagerly waiting to get back to my favorite subject (maths)

MIT's OCW is a great resource. I advocate its use in many threads (I should've here), and even used it to self study calculus while taking precal. It may be a good way to evaluate your ability to understand calculus lectures. Have you tried watching some of the early lectures of 18.01? Keep in mind that it is both calc I and II, and you will not be expected to learn all of that in 1 term.

 

[plumberbabu]

Thanks Mathwarrior,
I will definitely remember wat you have mentioned. having a clear thought and concepts is very important in maths.

 

[plumberbabu]

TylerH,
I have not had a chance to listen to the lecture by MIT, but when I was trying to figure out what I need to do to prepare myself for this switch in my major, I came across it as a great resource for self studies. It is the same way I reached this forum and boy am I happy, you guys have been terrific help.

 

[Sankaku]

1: Precalculus, intro stats
2: calc I, ?(It would be a shame to leave this open, if you decide to skip precal, intro stats goes here)
3: calc II, proof class(they usually have calc I as prereq)
4: calc III, linear algebra, optional: diff eq

Being a math junkie myself, I like to recommend intro proof courses. However, for your purposes, it may not be the best choice.

I might suggest:

1. Calc I, Intro Stats
2. Calc II, Linear Algebra
3. Calc III, ODEs

Linear algebra is very useful for both Calc III and ODEs. It can be taught either as an applied course or as a theoretical course (proof heavy). As long as it is not a proof-heavy course and you can keep on top of the work, this schedule will get you through the core material in 3 terms. Check with your school to see if this is a usual progression for the courses you will be taking.

These online videos are good:

Calc I and II:
http://press.princeton.edu/video/banner/

Calc III:
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/
http://webcast.berkeley.edu/course_details_new.php?seriesid=2009-D-54296&semesterid=2009-D

ODEs:
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/

I didn't like MIT's Linear Algebra (Gilbert Strang) but YMMV. There are a number of videos here as well:
http://www.uccs.edu/~math/vidarchive.html
Some of the instructors are better than others...

Pick up any Precalc book second hand and work through the exercises. I think you can get an older copy of Blitzer for a couple bucks from abebooks.

 

[TylerH]

Being a math junkie myself, I like to recommend intro proof courses. However, for your purposes, it may not be the best choice.

I agree, it is far from optimal for someone learning linear algebra for its applications to take a proof class. I just assumed all universities have their proof class as a prereq to their linear algebra. That's how it is at my school. But, by all means, if it isn't required, you don't need it for your intended use.

 

[plumberbabu]

Thanks Sankaku for your insight.
The reference links you have suggested look helpful. I am not sure though if my school offers classes on proof, but I have a book on proofs by "Vellemen" which has had nice reviews. So I guess if I have to do good in linear algebra, I will have to be thorough with proofs.