Most important things to be proficient in before Calculus 1?

Question

What are the main things one should be proficient in before taking Calculus 1? Please be specific.

Answer 1

Being comfortable and proficient with the fundamentals of algebra, functions, and graphs. These are more pervasive in Calculus I than even trigonometry (although that is important too).

More specifically, graphs of the standard functions (lines, power functions, abs value, exponentials, logs, basic rational functions, trig functions; translations and reflections; even, odd). Being comfortable manipulating abstract function expressions like $f(a+h)-f(a)$ and $f(g(x))$. Solving equations: linear, quadratic (factoring and quadratic formula), power functions (including fractional and/or negative powers), exponentials, logs.

After those, tack on the same skills for the 6 trig functions (certainly the big 3 at a minimum).

In my experience, weakness in these areas hinders the student moreso than doing the actual calculus.

PS Once those things are mastered, I'd encourage you---as much as possible---when you learn something new in calculus, always see if you can explain what you are learning in terms of a picture of some sort. Thinking geometrically will really help calculus to "make sense" and become more than pushing symbols around.

Answer 2

I would advise two things that took me a long time to realize. First, make sure algebra is very clear. Calculus is part of the study of functions and it is absolutely essential to be very clear about what a function is and what there is to be interested in about them before calculus. Second it is important to realize that the derivative of a function is a point property of the function, given by another function of the same variable, called the "derivative" of the original function, and has nothing to do with limits. Limits are useful for determining what a derivative is in a particular case but even that is academic since the properties of derivatives make it almost entirely unnecessary to differentiate using a limit. The properties of derivatives are especially nice and convenient. The antiderivative (integral) of a function is extremely curious. It is an interval property of a function and has the amazing property that for a given function, its value simply depends only on the location of the interval and not how the function behaves over the interval, contrary to what one might expect. (This gives rise to the so-called "definite" integral.) If you understand what calculus is about, it is very simple and ingenious.

Answer 3

Learn to manipulate polynomials:

• Add polynomials
• Multiply polynomials, FOIL technique and squaring a binomial especially
• Divide polynomials
• Partial Fractions of a polynomial quotient
• Binomial expansion
• Shifting up/down/left/right and scaling vertically and horizontally are helpful too
• Graphing polynomials (value at $\pm \infty$ and roots of small polynomials)

If you start calculus before you can do the above comfortably, you'll run into many roadblocks. If you want pointers on any of the topics leave a comment and I'll expand on them.