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).
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Tuesday, May 24, 2011
lsquared on what you must know to take calculus
(lsquared couldn't get this comment to post on Blogger):
I wonder if some basic complex analysis should be taught concurrently with trigonometry, so that formulas such as that for cos(2x) can be derived from Euler's formula,
ReplyDeleteexp(ix) = cos(x) + i*sin(x)
EPGY has placement tests for math from algebra through calculus at http://epgy.stanford.edu/applyandregister/upexam.html .
ReplyDeleteIn addition to Lsquared's suggestions, I'd add a bit of motivation: there are two ways to understand derivative calculus. Derivative calculus is understood by seeing how a certain function works in a given limit, or derivative calculus is understood in terms of how a certain trigonometric function works.
ReplyDeleteSo, to get that, you need to understand limits and you need to understand trig.
Demoivre's Theorem is usually part of a precalculus class. Understanding why it's true is usually not something students learn, but they do learn the formula. Ideally, you'd like a teacher who knew the formula and knew why it was true. In fact, if I were interviewing a potential precalc teacher, I'd ask them to tell me what Demoivre's theorem was and what it meant.
Before you get to sequences and series, you must be taught limits. Limits are really the bread and butter of understanding derivative and integral calculus. Sequences and series make sense only after limits make sense. Understanding convergence and divergence is critical to being prepared for calc.
Properly knowing how to find the limit of a polynomial or trig function requires mastery of the underlying algebra and trig. If you really don't understand polynomials, you will have a hard time with limits, sequences, and series.
Allison added "to get that, you need to understand limits and you need to understand trig."
ReplyDeleteAs a calculus teacher, I expect the students to come in knowing trig, but I don't expect them coming in knowing anything about limits. I agree that they make infinite series make more sense, but, again, I expect them to come in knowing how to work with finite series, but not infinite series. Teaching limits (and infinite series) is part of my job in teaching calculus. Thus, if you wish to learn limits in pre-calculus, that's fine (and if it is done in the pre-calculus C took, then you probably should--it may be part of the expectations for calculus at your school), but if a student turns up in my calculus class never having heard of a limit before, it's not a problem: I have 2-3 weeks built into my syllabus to teach them what they need to know about limits. If, on the other hand, they come in not knowing trig, they are up a creek without a paddle, because I don't have enough time to also bring them up to speed on the trig they don't know.