Starting Calculus with a weak foundation in Pre-Calculus

Question

I am struggling in Pre-Calc mathematics, and I want to know is it ok if I start Calculus I with a weak foundation in Pre-calculus mathematics? I understand the general gist of limits, function notation, geometry, surds, exponents, inequalities and so forth, but am not really comfortable with them. I want to know, will I struggle with Calculus.

The reason I ask this is because Calculus has so many literature online, and is easy to find help upon. So I want to start Calculus now and fill in any gaps as I go

Is their any test that checks how 'calculus' ready you are?

I really love mathematics and this site motivates me so much to continue, i Just feel dumb because I don't know how to do something if no one shows me how to do it first.

Answer 1

This is a difficult question to answer. You may find yourself able to understand the basic concepts of differentiation and integration but if you have a weak understanding of 'pre-calculus,' you may find yourself struggling with utilising them.

I suppose the most important aspect of 'pre-calculus' you need to be somewhat fluent in is the study of functions. You need to be familiar with polynomial, exponential, logarithmic (to an extent), rational, trigonometric etc functions. Knowing the graphs of these, as well translations (such as dilations in the $x$ or $y$ directions, things of that nature) is essential. You will also need to know how to solve linear, quadratic, and sometimes cubic functions (knowing the remainder and factor theorems and the quadratic formula), understand the gradient of straight lines (given $m$ is the gradient of the line $y=mx+c$).

Other aspects that are necessary are the trigonometric identites, such as $\cos^2x+\sin^2x=1$ and also the definition of the reciprocal trigonometric functions, such as $\sec x=1/\cos x$ etc.

It is worth checking out a comprehensive textbook of pre-calculus and having a browse through to see if you are ready.

Answer 2

There is a LOT amount of algebraic manipulation involved in typical calculus programs. So, YES you need to get that stuff down solid.

Buy a Schaum's and drill, drill, drill. (I like the 1958 Frank Ayres First Year College Mathematics as a simple precalc drillbook. But anything equivalent makes sense.)

Also, if you are learning calculus because you will be studying engineering or physics or similar realize that there will be a huge amount of algebra/trig in those topics. And chemistry, econ, geology, etc. are full of algebra (maybe not so much trig). So not only do you need the precalc for calc, you need it for ITSELF.

See this quote by Dick Feynman:

So, this guy comes into my office and asks me to try to make everything straight that I taught him, and this is the best I can do. The problem is to try to explain the stuff that was being taught. So I start, now, with the review.

I would tell this guy, “The first thing you must learn is the mathematics. And that involves, first, calculus. And in calculus, differentiation.”

Now, mathematics is a beautiful subject, and has its ins and outs, too, but we’re trying to figure out what the minimum amount we have to learn for physics purposes are. So the attitude that’s taken here is a “disrespectful” one towards the mathematics, for sheer efficiency only; I’m not trying to undo mathematics.

What we have to do is to learn to differentiate like we know how much is 3 and 5, or how much is 5 times 7, because that kind of work is involved so often that it’s good not to be confounded by it. When you write something down, you should be able to immediately differentiate it without even thinking about it, and without making any mistakes. You’ll find you need to do this operation all the time—not only in physics, but in all the sciences. Therefore differentiation is like the arithmetic you had to learn before you could learn algebra.

Incidentally, the same goes for algebra: there’s a lot of algebra. We are assuming that you can do algebra in your sleep, upside down, without making a mistake. We know it isn’t true, so you should also practice algebra: write yourself a lot of expressions, practice them, and don’t make any errors.

Errors in algebra, differentiation, and integration are only nonsense; they’re things that just annoy the physics, and annoy your mind while you’re trying to analyze something. You should be able to do calculations as quickly as possible, and with a minimum of errors. That requires nothing but rote practice—that’s the only way to do it. It’s like making yourself a multiplication table, like you did in elementary school: they’d put a bunch of numbers on the board, and you’d go: “This times that, this times that,” and so on—Bing! Bing! Bing!"