Math 148: Precalculus

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March 17, 2009 by vlorbik

It’s all over but the grading for me and my mind’s elsewhere. I’m doing a 103 and a 153 next quarter and have started work on those. Most of the blogging for the 148 classes now ending was in VME.

I’d like to thank my life-support system, Madonn’. Thanks for letting me hang on. Also all my students and anyone who commented here or even ever seriously considered it. Without you, no blog.

The world of mathblogs is expanding vigorously at all levels. It’s a great time to be alive.

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3.14

March 16, 2009 by vlorbik

JD’s Carnival of π.

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Let P_1 = (x_1, y_1) and P_2 = (x_2, y_2). The distance between P_1 and P_2 is then
$d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 +(y_2 - y_1)^2}$ .
Humans as opposed to computers would do well to understand that this is the square root of the sum of squares because of the Pythagorean Theorem; my habit in in-class computations is to write out the radical with the parens and superscripts first and fill in the differences after reminding everybody of this connection. Every now and then somebody gets it.
There’s a “formula” given in the same section for the midpoint of the line segment connecting P_1 to P_2: but I consider it foolishness to think of this as a formula. Average the x co-ordinates for the “new” x; average the y’s for the y.

One immediate consequence of the distance formula (or of the Pythagorean Theorem directly…) is that x^2 + y^2 = 1 is the equation of the unit circle; shifts-and-scalings lead to the circle equation
$(x - H)^2 + (y-K)^2 = R^2$
(for the circle with center (H,K) and radius R). One is prepared either to put an appropriate polynomial in x and y into this form (by completing the square) and “read off” the center and radius, or (what is much easier) to write out the appropriate equation for a given center and radius.

The financial section of the exp-and-log chapter is fraught with formulii. And typing ‘em out is no walk in the park, as I might have mentioned from time to time. But let’s see. There’s the old $A = P(1+{r\over n})^{nt}$ for compound interest; its limit as n grows without bound gives the continuous compounding formula $A = Pe^{rt}$ . Dividing on both sides of these gives the present value formulas (for “P”, when “A” is known); these should not be understood as separate formulas and I’m sure as heck not gonna write ‘em out here. I’ll gladly mention that the division I just mentioned is typically denoted in “negative exponent” style. The effective interest formulas are $(1+{r\over n})^n -1$ and $e^r - 1$ . As is usual, our hope in presenting this is that having thought it through, you can forget it and remember instead that one is here letting P and t be equal to unity (by selecting units appropriately); the “multiplier” on P, which is one, for one year, minus the original “one” (one hundred percent of P), is the total interest earned during that year (with its n compounding periods). But wait. Say something once; why say it again. This “formula sheet” project isn’t going to be the affair of an hour and I was crazy to think it would be. Anyway, as usual, there are typically real good summaries in our or any such book.

Better still, we’re on the net. Google “precalculus formula sheet” and look around. Finding the right ones from the best few you find sounds like a darn good exercise. I should have started here.

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More Review

March 12, 2009 by vlorbik

I posted some remarks during the review for Exam II (about rational functions and inverse functions) in the old blog. Then there’s this post about the “simplest” Rational Function, some discussion of the Factor Theorem, the review problems of 1/30, and a summary of Transformations.

“Quadratic Formula Lore” was one of my most popular posts. I don’t know whether my decision about Transformation notation counts as “review” (since these notations are “unofficial” for the course and won’t Be On The Test). I made up some hints about using the net and the library. Near the beginning, before I hit my stride, I posted this short rant about defining “relation”.

Let the record show that this was a very productive quarter for me (and that I’m proud of it). What the hell, I’m feeling generous… let the record even show that I had the time of my life doing it.

Posting about the Final will cease by Monday at 8:00 AM (so everybody gets the same info from here). Go do some math problems.

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Herculean Sense-Making

March 12, 2009 by vlorbik

Barry Garelick’s (PDF) “Discovery learning in math: Exercises versus problems” at the Third Education Group’s Review (I can’t quite read this site… there’s a sidebar covering up some text). Spotted in KTM.

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Forests And Trees

March 10, 2009 by vlorbik

Old fave Gilbert Highet noticed at KTM.

Dollars and sense at Social Maths.

Problem solved (probably), at Gowers’s.

Sez here I’m doing a 103 next quarter (8:00—9:50 AM MW) and a 153 (1:00—3:15). I’ve blogged about 153 before (badly) and probably will again real soon (it’s not for me to say how well).

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Remarks Toward The Dissolution Of This Blog

March 10, 2009 by vlorbik

The final’s soon. Somewhat to my surprise, the co-ordinators haven’t prepared a Practice Test (they recently took over the whole final exam; in the old regime one had to write half the Final for one’s own sections). Which, believe me, makes perfect sense: lots of good teachers consider this kind of thing a bad idea (the whole point of a final exam on their model being to cover the entire course).

So it’s on me and I don’t care if it’s a bad idea just now: my students have grown to expect such things and I hate to disappoint them. At the same time, I’ll preserve a little of the roll-your-own flavor: part of the point is to get students to read the book, not just to consider certain problem types. Anywhow, from top to bottom, we did, and will soon do again, problems like these.

Reflections. Given a graph, sketch its various reflections (origin, x-axis, y-axis); keep track of one or more points (given as ordered pairs); show algebraically that a function is “even” or “odd”.

Information From A Graph.Given the graph of a function, write down its Domain, its Range, its local Maxima or Minima (collectively Extrema; know the singular forms of the three plural nouns just mentioned… and in general, for heck sake, learn these words I keep slinging around… the point isn’t just to be able to do the calculations, but to talk about them…), the sets (subsets of the Reals, natch—specifically, of the domain… so we’re talking about x co-ordinates… behave appropriately here I beg of you) on which the given function is increasing or decreasing (or, I suppose, constant, though this strikes me as somewhat affected and I typically distain to give its definition), the sets on which it’s positive or negative, its various intercepts, and whatever I’ve forgotten to mention. This is stuff you mostly all knew before you even took 148 so just get the bloody notations right: what’s a point, what’s a set of x co-ordinates, what have you. Because if you can’t write this stuff down just by looking at the graph and be essentially correct about almost all of it, you aren’t ready to “talk about” graphs of functions…

Piecewise-defined Functions. Graph it if we give you the algebra; write down the code if we give you the graph.

Transformations. We did a lot of this and if I had my way we’d've done even more. Be able to name and calculate “shifts” (I actually prefer to call these “translations”), “shrinks” and “stretches” (“scalings”) as applied to algebraic “formulas” and to graphs (and their individual points).

Quadratic Functions. We looked in some detail at some Optimize-the-Area problems and some Optimize-the-Revenue problems, so prepare for these: in either case, the point is to set up the right quadratic and look at its vertex. We did more review in here than the syllabus called for… complete the square, QF (the Quadratic Formula), what have you. You can thank me later.

Zeros of a Polynomial. With real coefficients. In the easiest cases, one can read the factored form of a polynomial directly from its graph; in the more interesting cases we’ll have to “divide away” the factors—(x-R), say—associated to certain roots (R such that f(R)=0, also known as zeros and, loosely, as x intercepts [strictly speaking, the root is a number and the intercept is a point (i.e., an ordered pair of numbers); some students appear to feel that clinging to the belief that details like this are never important is some higher moral duty... as if thinking clearly is some sort of sacrilege...]); typically one finally cleans up by applying QF to get the last two roots. These can be real or complex. One should also be prepared to construct polynomials given certain roots (and their multiplicities): remember here that complex roots of real polynomials occur in conjugate pairs.

Rational Functions. A single problem of the form “graph y = R(x)“, where R is a rational function, involves lots of sub-problems. Find zeros and vertical asymptotes (factor numerator and denominator first if necessary; we reviewed this very little, alas); put appropriate features in the graph. Determine the “end behavior” (horizontal, oblique, or no asymptote); sketch. Find a “new” point & make the appropriate connections. There’s an inequality to solve under this heading too: I recommend solving these graphically even though the exam itself says “solve algebraically”. If they give you ruled paper, write the other way.

Composite Functions. Compute ‘em. But also, evidently because it’s slightly tricky, give their domains. The point here is to consider, not only the computed “formula”, but the domain of the right-hand “factor” in the composite (g in $f\circ g$ , for example).

Inverse Functions. Be able to compute the formula for the inverse of a function (given by a formula). First of all, in textbook fashion: to invert f, put x = f(y) and solve for y; the result is (a “formula” for) $f^{-1}(x)$ (so write down the appropriate equation). But, to thrill your instructor, use “shoes and socks” to calculate verbally (appropriate wigglings of chalk or pencil are optional here).