What do I need to know to be ready to take Calculus?

[Relunctant Mathematics]

Hello,

Long time listener, first time poster.

As the title suggests, I'm looking for advice on what I need to know to be prepared to take Calculus. I'm thinking learning Geometry, Algebra I and II, and Trigonometry will set me up nicely. Anything I'm missing or are any of my choices not needed?

That paragraph serves as a tl;dr version, but I'm a nerd who tends to be inclined to write walls of text. If anyone is interested in the context, read on.

I am in my mid 30s and dropped out of college after a year my first time. I worked in production, music, theater and video, and built a decent multimedia skillset. I've worked at a number of companies that offered L&D programs, and leveraged them to build skills over the years.

I first become a presentation specialist based on strong knowledge of MS-Office. Later I picked up Photoshop, Illustrator and In Design. I learned Avid, Final Cut, Avid, After Effects, etc and put together a nice career around video production specializing in post-production. Not having a degree always felt like it held me back.

About 2 years ago, I got an offer to work at a fairly high-end engineering college. I accepted, and a benefit is free classes. I've engaged in a multimedia-based degree with a minor in CS, and have been cherry-picking classes as a non-matriculated student. I went with things that would help my job - animation, motion graphics, digital imaging, etc. Good times. Quality knowledge. Throw Maya on the 'known software' list.

But now I'm matriculating in the fall, and will have to take the general ed requirements, including Calculus. I dread math. Loathed it as a high school student. Passed Geometry and Algebra 1 and failed Algebra II. Think I got out of HS without having to take any more math. To be fair, my failure was born of boredom, laziness, and distraction rather than capability. I enjoyed understanding what mathematics can do, and was interested in the general concepts, but not interested in showing my work and repeatedly solving problems. I didn't do homework.

I've recently taken some history of sci/tech classes at this college as well, and I was inspired to learn that all great thinkers of antiquity tended to have a base in math. I particularly enjoyed learning of the sign, legend suggests hung outside of Plato's Academy, which read: 'Let none ignorant of Geometry enter".

The professors are my co-workers, so poor performance is out of the question, as it will negatively affect my reputation and career. I am committed to learning mathematics - to improve myself, to protect my career, and because it seems like a worthwhile pursuit.

Anyway, I have to play catch up. I'm planning on using Kahn Academy (http://www.khanacademy.org/) to facilitate my learning cheaply. A google search on 'Calculus prerequisites" suggests I should maybe not worry about geometry. Seems important, though, if Plato's sign is to be believed. I'm also going to have to take Physics at some point and the CS minor will require a bunch of algorithm business.

Any advice appreciated.

[Trinopus]

Trig, definitely. Memorize as many of the "Trig Identities" as possible -- such as 1/sin = cosec, and sin-squared plus cosine-squared = 1, etc. Better yet, try to get fairly good at moving around within these identities, deriving them, etc. The more familiarity you have with 'em, the happier you'll be.

Also...don't get discouraged by "epsilon delta proofs." Ugliest doggone things in the world, but, if you can get through 'em, then it's easier sledding for a while, and, better yet, you'll never have to use them again. They are the philosophical underpinning of calculus, and you have to understand it.

(Amusing to note that Charles Dodgson -- "Lewis Carroll" to literature fans) *didn't* get the infinitesimal calculus! He couldn't understand how a difference could be made very small but not eliminated. So, this is your chance to show you're smarter than a professor of maths nigh on to a couple centuries ago!)

Anyway, yes to Trig, Algebra I and II, and Geometry. Geometry was the path Newton used to invent Calculus, but, today, it isn't taught as a primarily geometric field.

Try to have some fun!

[Thudlow Boink]

If you don't have a pretty firm grounding of algebra, it will cause you a great deal of trouble in a Calculus course.

[leahcim]

Here are some questions for you, taken from memories my days as a Calc I TA:


Do you understand what a function is?
Do you understand what it means for a function to be invertable?
Do you understand the real line? The difference between the rational numbers and the reals?
Restricting the scope to functions from real numbers to real numbers, do you understand, in general, how to plot such a function on an x-y Cartesian plane?
Given a plot of an invertable function, can you sketch the plot of the inverse of that function?
Given the plot of a function y = f(x), can you sketch the plot of y = a*f(x), y=f(x) + b, y = f(cx), y=f(x+d), and y=1/f(x) for known quantities a, b, c, and d??
Can you sketch plots of the following functions, including labeling axis-intercepts and extrema: sin(x), cos(x), tan(x), exp(x), ln(x), x^n, 1/x^n?
Do you understand exponents? Polynomials?
Can you recognize and plot the equation of a line in it's various forms (e.g. y = mx+b, (y-y0)/(x-x0)=m, &c)? Can you identify the axis intercepts given such equations?
Given a quadratic polynomial, ax^2+bx+c, can you find its roots, minimum/maxiumum value and location of that value?


Those were the kinds of pre-calc activities we'd do before getting down to it. If you can answer "yes" to all of those, you're probably golden for Calc I.

[Lemur866]

Each layer of mathematics is pretty much just doing the layer below it. So, first you learn your numbers. Then you learn arithmetic, which means making the numbers dance. Then you learn algebra, in which you use arithmetic to solve algebra problems. And then you have calculus, where you use algebra to solve calculus problems.

So if you're terrible at algebra, calculus is going to be a horrible slog, because everything you do in a calculus class is applied algebra and trig.

[ultrafilter]

A google search on 'Calculus prerequisites" suggests I should maybe not worry about geometry. Seems important, though, if Plato's sign is to be believed.


Plato's been dead for a long time. You will need to know some basic geometry--in particular, you should know the formulas for the area and perimeter of some common shapes--but you don't need a detailed knowledge of geometry for a first semester calculus class. Algebra and trigonometry are much more important.

[Relunctant Mathematics]

Also...don't get discouraged by "epsilon delta proofs." That's all greek to me (zing!). Thanks for the confirmation and warnings.

Those were the kinds of pre-calc activities we'd do before getting down to it. If you can answer "yes" to all of those, you're probably golden for Calc I. Thanks, that's helpful. As I'm teaching myself, I can move on as soon as I'm comfortable with principles - not a slave to a book or the pace of a class. That list sounds like it will help to set milestones.

So if you're terrible at algebra, calculus is going to be a horrible slog, because everything you do in a calculus class is applied algebra and trig. I was terrible at Algebra because I never really wanted to try. I'm convinced that I am capable of learning it all, and coming to understand, enjoy and leverage mathematics.

We'll see how that works out. . .

[Relunctant Mathematics]

Plato's been dead for a long time. You will need to know some basic geometry--in particular, you should know the formulas for the area and perimeter of some common shapes--but you don't need a detailed knowledge of geometry for a first semester calculus class. Algebra and trigonometry are much more important.

This was what my research suggested, Geometry not that important. I'll look at those formulas at the very least.

Any suggestions on order to tackle these? I was thinking:

Geometry (even if narrowed focus)
Trig
Algebra 1
Algebra II

If I recall, that's the order it went in High School.

[Silophant]

You can probably tackle algebra and geometry in either order, but Trig requires a solid knowledge of both, and should definitely come after the other three.

[Senegoid]

You mentioned never wanting to study much math, but you think you are ready to tackle it now. How many hours a week do you think you can stand studying math? Since you've already passed Algebra I, are you planning to start now with II, or take I again to brush up? You might very well need to take I again if it's been a while.

I have an Associate of Science degree in math. I studied and practiced something like 20 hours a week outside class to get there (and that was just the math classes).

> Geometry (even if narrowed focus)
> Trig
> Algebra 1
> Algebra II

You absolutely need to have Alg. I and II first before Trig, since Trig is largely just a specialized application of algebra.

You definitely need to know as much as possible of each level in order to get anywhere at the next level. As you get into Calculus, you will see that all the Algebra and Trig textbook authors knew where it was heading, and their choices of topics to cover was not arbitrary. EVERYTHING (almost) in ALL those courses comes back to haunt you in calculus!

It's not like, say, History, where you can take U. S. History 1-A (beginnings to Civil War) and 1-B (Reconstruction to Present) in either order.

I have a theory: The grade you get in any level becomes the limit of what you can do thereafter. If you get a B in one class, you will have a hard time ever getting better than a B in any later class. If you get a C, you will be stuck a C math student. The things you don't know, that cause you to get anything less than A++++ will bite you ever after.

Is your ambition to become a really good math whiz, or just to get through the classes with your ass intact? If you really want to excel (and even if you don't), it will be helpful if you study all you can, well beyond the text and the class material, at every level.

There is much debate these days (that is, for the last 40 years at least) that classes are getting more and more dumbed down. I think that's true. About 10 years ago, when I when back to school for that math degree, and then tutored it, I went to the Math Chairman's office and chewed his ass up one wall and down the other for the shit-f*ing dumbed-down Alg. I book they were using.

It proved to be a very amicable conversation. (Surprise!) He fully agreed with me, saying that they chose that book because it was the least dumbed-down they could find. We had a good long chat about how dumb math teaching has gotten, compared to how it was Back In The Day. I'm not at all sure that current math curricula are really good enough to prepare you for Calculus, unless that's dumbed down too (which it may be... see below).

Here's my point in all this: If you're really serious about it, hit up the used-book stores and find some really old math textbooks -- say, 1940's and 1950's era. And study those thoroughly too, along (or even instead of) your class text. If you are able to do that, you will definitely be ahead of the curve. (I once had a trig book from 1914!) And I still have a college algebra book from 1942, which is vastly better than the butt-wipe textbooks you can find now. Those old books always went into much greater depth.

Here's one more specific topic you should study on your own, because it's only covered superficially any more, but is great help to get your head around calculus: INEQUALITIES. If you can learn about that in detail, it will greatly help you learn to think about "epsilon-delta" definitions and proofs.

As for Calculus getting dumbed down too: I've been told that they don't even teach epsilon-delta any more in lower division -- it's now deferred to upper-division advanced classes. Still, if you know inequalities, that's a good conceptual thing to understand. My old 1942 college algebra book has a whole chapter on it, including proving unconditional inequalities (that is, given an inequality that is allegedly true for all values of all the variables, prove it). If you can find a textbook with that, it's a great help for wrapping your mind around epsilon-delta.

Since, as you say, you're teaching yourself, and if you are really wanting to get all into it, you should try to do all these things.

[Monty]

Does the college you intend to attend require a placement test prior to taking any Mathematics courses there? If so, you'll placed in the appropriate class to prepare for higher math.

[Nava]

Side question: you guys seem to have a very clear notion of what's the division between Algebra I and II, but we used descriptions. Basic Algebra, Linear Algebra, Multivariant Algebra, Matrix Algebra...

Is the division between I and II sort of homogeneous throughout the US/the Anglosphere, and if so, what is it? Or is each of you assuming it's going to be whatever it happened to be at his own Alma Mater?



I don't think it's important to know the formula for the volume of a sphere in order to be able to integrate it, but understanding the notions underlaying geometry is very useful in order to comprehend trig (as opposed to parroting it). Knowing what a tangent line is helps understand what the trigonometric tangent is.

[nivlac]

For calculus, you just need to know algebra and some trig. Plane geometry? Not that much. Of course you need to know basic mensuration formulas, but you can just look those up when necessary. As I recall, knowing right triangles and similar triangles will help you solve some problems during the course. You'll be introduced to analytic geometry in calculus, and your algebra skills will come in handy. One thing for sure ... if you're shaky in algebra you're going to have a hard time in calculus. You'll be expected to have a facility in dealing with equations and inequalities, and in manipulating and simplifying expressions. The last thing you want is to be caught in an algebraic fog when you're trying to absorb calculus concepts. Take this from someone who has taught calculus for a long time and have seen which students do well and which fall by the wayside.

[Relunctant Mathematics]

You mentioned never wanting to study much math, but you think you are ready to tackle it now. How many hours a week do you think you can stand studying math? Since you've already passed Algebra I, are you planning to start now with II, or take I again to brush up? You might very well need to take I again if it's been a while.I intend to work on this for one hour every night and a few more hours every weekend for as long as it takes. I'm starting from scratch, working the playlists at Kahn Academy including the very basic ones, if only for review. Hopefully I'll be able to get through it all by next fall. Thanks for the rest, good information.

Does the college you intend to attend require a placement test prior to taking any Mathematics courses there? If so, you'll placed in the appropriate class to prepare for higher math. Not that I know of. A student not ready for Calculus would be unlikely to get in here, and I've kind of side-stepped the requirements by being an employee. I have been given a second chance to get an excellent education after having squandered the first one, and a chance to overcome my past poor performance. I've earned this second chance with dedication to my craft and hard work and do not intend to let the opportunity pass me by. But it's going to require catching up on my own where necessary.

I'm holding a 3.8 GPA so far, having taken Multimedia, Philosophy, Literature and History classes, and I would like to keep it thereabouts. I'm convinced that I am capable of learning all of this, and unlike the first time, dedicated to doing so.

[Thudlow Boink]

Side question: you guys seem to have a very clear notion of what's the division between Algebra I and II, but we used descriptions. Basic Algebra, Linear Algebra, Multivariant Algebra, Matrix Algebra...In the US at least, the word "algebra" is used both for the basic mathematics courses, often taken in high school, as a prerequisite for things like trig and calculus, and for the branch of advanced mathematics that often called "abstract algebra" or "modern algebra" which includes things like linear algebra.

I don't know that there's a universal description of what "Algebra I" vs "Algebra II" would cover.

[BlinkingDuck]

I used to teach Calculus.

The below is what you need:

- As much Algebra as you can have...at least Algebra II from high school.
- Preferably a Senior prep course that conains even more Algebra and Trigonometry.

That's it...Algebra and Trig.

Trig is not a huge subject area. I used to teach a 2-week/10 day all day trig course during the summer and I didn't feel all that rushed. Trig is tiny compared to Algebra.

MOST IMPORTANT! - Have had no high school calculus.

The last one is snark, but students that came into Calc I that had high school calculus perfromed measurably worse than ones that didn't have any Calculus. I think this is because ones that had Calculus were more cocky/lax and, more importantly, had sacrificed that year of college prep math for Calculus (Calculus WAS their college prep math course) and so they didn't have as much background as students who didn't have any calc but still took college prep math.

If you are out of high school and asking as an 'adult'...then get as much algebra and Trig as possible and you'll be set.

[Thudlow Boink]

I particularly enjoyed learning of the sign, legend suggests hung outside of Plato's Academy, which read: 'Let none ignorant of Geometry enter". Plato valued geometry (in a form such as that presented by Euclid, in which everything is logically built up from "self-evident" axioms) as a prerequisite to philosophy, not calculus, because he thought it provided training in logical thinking and deductive reasoning.“In the course of my law reading I constantly came upon the word “demonstrate”. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?

I consulted Webster’s Dictionary. They told of ‘certain proof,’ ‘proof beyond the possibility of doubt’; but I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man.

At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies.”

[BlinkingDuck]

Each layer of mathematics is pretty much just doing the layer below it. So, first you learn your numbers. Then you learn arithmetic, which means making the numbers dance. Then you learn algebra, in which you use arithmetic to solve algebra problems. And then you have calculus, where you use algebra to solve calculus problems.

So if you're terrible at algebra, calculus is going to be a horrible slog, because everything you do in a calculus class is applied algebra and trig.

This!

Perfect answer :)

[brocks]

With your background, I'd say your best bet is to get a precalculus text like this one (http://www.amazon.com/Precalculus-Ron-Larson/dp/0618052852/) and work your way through it, using online resources like this (http://www.physicsforums.com/forumdisplay.php?f=152) to help you when you get stuck.

Almost all of the authors of the popular calculus texts, like Stewart, Anton, and Larson, also have precalculus texts. Those books pretty much start from scratch and cover algebra, trig, series, etc. And analytic geometry (think graphing), which is EXTREMELY important for calculus.

And DON'T pay $150 each for math books. Algebra, trig, and calculus haven't changed much in the last few decades, so get an older, used edition for no more than ten bucks. There are also a lot of free editions online.

Best of luck to you.

[BlinkingDuck]

A google search on 'Calculus prerequisites" suggests I should maybe not worry about geometry. Seems important, though, if Plato's sign is to be believed. I'm also going to have to take Physics at some point and the CS minor will require a bunch of algorithm business.

Any advice appreciated.

Do not worry about Geometry.

While something called 'Analytic Geometry' will appear in Calculus, don't let the Geometry fool you. You will not being doing Geometry proofs and SAS, ASA and stuff like that :P

You will just be using the formulas of Geometry to do Calculus which is much more Algebra-like than Geometry-like.

[mnemosyne]

Adding to Nava's post, I always find it interesting - and slightly disconcerting - that a lot of school systems present algebra, trigonometry and geometry as separate topics, as if anything useful can be done in one without knowing at least a little of the others. In my school system, I had Math classes, which covered everything in intertwined courses that covered a full year (my high school didn't have a semester set up).

Saying that you don't need to know geometry might be true in the sense that a lot of what you might have learned in a high school Geometry class isn't used, but of course you will need to be able to recognize the formulas for areas and volumes of circles and spheres and be able to work with similar triangles and understand how to find a given angle given other information. Some word problems won't give you the equations - you'll have to figure them out themselves for things like deriving the volume of a bowl (a half sphere) and such. There's not much point to trying to solve any Calculus problems if they have no relation to the physical world, and topics related to geometry and trigonometry are the means to do that.

I second the idea that you obtain a good pre-calculus textbook; looking at one now will tell you what sorts of topics you need to understand and help you narrow your focus on topics from the Khan Academy while building up the skills you need for calculus. It should be rather easy to find out what textbook your school is using in the Cal I, II courses and it might help you to get the pre-cal book by the same author. On the other hand, a different author might help reinforce a topic if he or she presents the material slightly differently.


I think it only came up in higher level calculus classes for me (or Ordinary/Partial differential equations classes) but an understanding of the basics of linear algebra came in handy - especially doing a simple elimination to solve a 3x3 system. It wasn't explicitly asked of me sometimes, but I'd occasionally be given problems where you end up with three solutions that need to be solved. Most students just start substituting equations/using general algebra to solve them at this point, but I tended to go the matrix route because it was much faster. It won't really be required for that you intend to study, but since you're in the computing/gaming fields it might prove to be really useful for you later on (or, you might find you've already seen this stuff in your programming courses!)

Good luck!

FWIW, I went back to school at 26 years old and relearned my math from Cal I through to the end of an engineering degree. You can learn anything, if you make the effort. :)

[engineer_comp_geek]

A lot depends on how the class is taught, and how you learn.

For me personally, I found calculus very easy to learn, but I often had to relate what we were doing to its geometric equivalent (i.e. an integral is just the area under a curve and the derivative is the slope of the curve). Some classes teach calculus with a bunch of proofs and theorems and lots and lots and lots of equations. That to me was a bit meaningless, but there are a lot of math weenies who enjoy that sort of thing and learn from it. If you "dread math" (as stated in the OP) you're probably not the math weenie type, and relating what you are doing to its practical geometric use may be helpful in allowing you to understand it.

This makes me think that you will likely need to focus on geometry. You'll also need to have a solid background in algebra just to be able to manipulate the equations around. If you take this approach, the proof type stuff you do in trig isn't so important, and the only bits you'll probably really need out of trig are things like the sin/cos/tan relationships and that sort of thing.

I've always had an easy time with math, but the key for me is to understand the purpose behind what I'm doing. Once I understand the "what" and "why", the "how" part becomes much easier. Learn why you are going what you are doing and it all will make more sense and will be much easier to remember (or at least it was for me).

Just my 2 cents.

[Relunctant Mathematics]

Great advice all. Thank you.

[Buck Godot]

Here are some questions for you, taken from memories my days as a Calc I TA:


Do you understand what a function is?
Do you understand what it means for a function to be invertable?
Do you understand the real line? The difference between the rational numbers and the reals?
Restricting the scope to functions from real numbers to real numbers, do you understand, in general, how to plot such a function on an x-y Cartesian plane?
Given a plot of an invertable function, can you sketch the plot of the inverse of that function?
Given the plot of a function y = f(x), can you sketch the plot of y = a*f(x), y=f(x) + b, y = f(cx), y=f(x+d), and y=1/f(x) for known quantities a, b, c, and d??
Can you sketch plots of the following functions, including labeling axis-intercepts and extrema: sin(x), cos(x), tan(x), exp(x), ln(x), x^n, 1/x^n?
Do you understand exponents? Polynomials?
Can you recognize and plot the equation of a line in it's various forms (e.g. y = mx+b, (y-y0)/(x-x0)=m, &c)? Can you identify the axis intercepts given such equations?
Given a quadratic polynomial, ax^2+bx+c, can you find its roots, minimum/maxiumum value and location of that value?


Those were the kinds of pre-calc activities we'd do before getting down to it. If you can answer "yes" to all of those, you're probably golden for Calc I.


This is a very good list.


A few other things I would add to the list is an understanding of how to combine 2 functions: f(g(x)). And an understanding of the laws of logarithms and exponents

The most important things that I think the OP might be lacking is a implicit understanding of how functions work. This is crucial for getting through calculus.

[Senegoid]

With your background, I'd say your best bet is to get a precalculus text like this one (http://www.amazon.com/Precalculus-Ron-Larson/dp/0618052852/) and work your way through it, using online resources like this (http://www.physicsforums.com/forumdisplay.php?f=152) to help you when you get stuck.

<snip>

And DON'T pay $150 each for math books. Algebra, trig, and calculus haven't changed much in the last few decades, so get an older, used edition for no more than ten bucks. There are also a lot of free editions online.

That pre-calc book he suggests is by Larson and Hostetler. I used various editions of their books in some of my Calc classes. I thought their texts were particularly good.

True, math hasn't changed all that much in decades. But the textbooks have, I think. As I wrote earlier, I think they are getting steadily more dumbed down. So that's another reason to hit up the used-book stores looking for older books, besides them being cheaper. The college textbook industry is a major racket, and everybody knows it.

[Nava]

Adding to Nava's post, I always find it interesting - and slightly disconcerting - that a lot of school systems present algebra, trigonometry and geometry as separate topics, as if anything useful can be done in one without knowing at least a little of the others. In my school system, I had Math classes, which covered everything in intertwined courses that covered a full year (my high school didn't have a semester set up).

That's how it works in Spain as well, we didn't start getting courses labeled after specific branches of Math until college. Previously, each year would focus on whatever aspects the national curriculum asked for, but the first couple of "themes" were always "review of [whatever was studied in previous years that is required in order to understand what comes next]".

For example, my 10th-grade Math involved trig, limits, series and derivatives, each building both from the material of previous years and from what had already been seen that year. Sometimes the curriculum would step back for a bit; for example, we'd done limits, then we'd do a new type of series, then we'd see how to do limits for that new type of series.

[Senegoid]

A few other things I would add to the list is an understanding of how to combine 2 functions: f(g(x)).

Yes! Good catch, Buck Godot! This combination of functions f(g(x)) is called composition and it comes in BIG in Calculus. And typically doesn't get much emphasis in Algebra, although it may be briefly mentioned. When you first see it in Algebra, be sure to get it engraved into your engrams! When I took Calc I, everybody (myself included) had a bear of a time with this for a while until we got it down.

[smiling bandit]

First off, Relunctant Mathematics, don't be intimidated. Calc is among the easiest math courses I ever took. I had the advantage of good teachers. Fortunately, it's one of the easier ones to teach yourself as well. I'm not sure how your school handles it, but I too went back to school after having no math in years. I got through with flying colors.

Calc is extremely logical. They may include some miscellaneous material under that heading, which isn't exactly Calc but related to interpreting what you find in it. That's why you might need some basic geometry. You could use it to compare two graphed functions, for instance.

But for a simpler method - just ask your school. Quite often, they can give you a rundown of basic needs and even give you a quick math test to show where you need work. I did this on the first day of class, got crushed, and still got an A+ by the time the first test rolled around. (Make sure you ask about natural log functions and a rundown of what various power notations mean.)

I have a theory: The grade you get in any level becomes the limit of what you can do thereafter. If you get a B in one class, you will have a hard time ever getting better than a B in any later class. If you get a C, you will be stuck a C math student. The things you don't know, that cause you to get anything less than A++++ will bite you ever after.

Complete tripe. I can personally vouch that it's not true, for several reasons. First, schools tend to give everyone the same instruction, regardless of personal need. Often enough, somebody won't get it at first. Given time, they practice with other things and the old difficult problem becomes trivial. Second, people may find a new apporoach they understand better. Third, they somtimes just need to digest the math a while. Courses might be taught at breakneck speed, jumping from one thing to the next; some people seem to suddenly improve their math skills weeks or months after first seeing (and failing) a problem.

[septimus]

This is a very good list.


A few other things I would add to the list is an understanding of how to combine 2 functions: f(g(x)). And an understanding of the laws of logarithms and exponents

The most important things that I think the OP might be lacking is a implicit understanding of how functions work. This is crucial for getting through calculus.

I agree with this.

My credentials in math education are flimsy to put it mildly, but let me opinionate anyway if only to stand correction. :D Is there not an Analysis I for Math Majors completely different from the Calculus I for Engineers and Technicians? Which such course we're preparing for makes a big difference. Either way, I doubt specialized knowledge in either geometry or trig would be of great importance, excepting simple formulae for area, angle, etc. As a mandatory prerequisite perhaps Alg I and, for practice, at least two or so of {Alg II, Geom, Trig}.

The comment about epsilon/delta doesn't sway me. From simple algebra, one can learn to express epsilon as a function of x and delta; ability to do this will stand one in good stead, e.g. should a formula be otherwise forgotten, or the intricacies of a calculus proof otherwise confound.

[brocks]

Is there not an Analysis I for Math Majors completely different from the Calculus I for Engineers and Technicians?

Yes, but it is normally at least a junior-level course, and two or three semesters of calculus is usually a prerequisite. Analysis is basically calculus redone in a very rigorous manner, with the emphasis on proofs, rather than calculation.

[Indistinguishable]

In my humble opinion: algebra, algebra, algebra. Everything else you can pick up on the way, as suits your needs. (But, one must note, much depends on the nature of the course you will be taking and your goals in taking it...)