Yes, you need to be very good at algebra. Think factoring, equation solving and complex fractions.
You also need to be quite good at trigonometry: knowing right triangle trigonometry is sufficient for almost everything except calculus--for calculus you need to know how to solve equations that have trig functions in them, and you need to know your trig formulas (the double angle formulas are particularly useful). You don't actually do any of the "verify this trig formula" stuff in calculus, but those "verify the trig formula" problems are very useful for teaching the algebraic trigonometry you use in calculus. Oh--and radians. I'm morally convinced that radians were invented to make trig functions work for calculus. Degrees are better for everything else, but in calculus, you have to have radians.
You also need to know logs: solving exponential equations with logs, solving log equations with exponents, and manipulating logarithms.
Back to rational functions. Be able to solve rational equations, simplify rational expressions that are not equations, and graph (by hand, not by calculator) rational functions (using properties of the rational function to graph, not by plotting points) (I may be unusually picky about this).
Sequences and series, including finding sums using the formulas for arithmetic and geometric sequences. Also the binomial theorem.
You don't need matrices, though you do need to be able to solve simultaneous linear equations. If there's a section in the pre-calc book on find partial fraction decompositions, that's directly a calc algorithm.
I think that might be everything you need for pre-calc.
This was an excellent exercise for me. I’m going to have to keep a copy of this to use when counseling students about whether they are ready for calculus or not. Indeed, the homeschooled kid next door wants to get into my calc class in the fall, and I shall be handing this to him I expect. Good luck finding a good class--I think the community college might be a good choice (assuming that they have a pre-calc course).
Tuesday, May 24, 2011
lsquared on what you must know to take calculus
(lsquared couldn't get this comment to post on Blogger):