PERMISSION TO ADD: MATH TEACHING LIMERICKS Marion Cohen PRE-CALC TYPES OF FUNCTIONS They come in all sorts of disguises in the various cool Exercises. Some ordered pairs some tables (not chairs) and some graphs with their xʼs and yʼses. DEFINITION OF DOMAIN OF A GIVEN FUNCTION Some xʼs the functions can play and others it canʼt, nay nay. Take all of the canʼs and include in the plans. Save the canʼts for a rainy day. FUNCTIONS AND THEIR GRAPHS, GRAPHS AND THEIR FUNCTIONS Functions have various drapes and graphs have their various shapes. We have a preponderance of correspondence from which nobody escapes. LINEAR FUNCTIONS The simplest graph is straight onto. And what does it correspond to? Why, ax + b. ʻTis easy to see unless, of course, we doesnʼt want to. SLOPE-INTERCEPT FORM: y = mx + b, m - slope, b =y-intercept Some statements that might help us to cope: Before x is written the slope and after the plus (without too much fuss) goes the y-intercept (so we hope). POINT-SLOPE FORM: y - y-one = m (x - x-two), has slope m and passes through (x- one, y-one) The m says how much it doth lean and as for the rest of the scene weʼve got, just for fun x-one and y-one with a comma in between. point-point: y - y-one = [ (y-two - y-one) / (x-two - x-one) ] (x - x-one) passes through both (x-one, y-one) and (x-two, y-two) Two points doth this straight line pass through (w,y) sub-one and sub-two. Conversely, these four (in this math so pure) determine whatʼs what and whoʼs who. PIECEWISE FUNCTIONS (“different strokes / for different folks” ) Thereʼs more than one rule, woe-betide. By which rule should we abide? That depends on x so stand back for two secʼs then do what that x is beside. QUADRATICS A fair maid from Indianapolis was expert on drawing parabolas. She extended their arms and showed off their charms and they looked absolutely fabulous. SIMPLIFYING COMPLEX FRACTIONS (FRACTIONS WITHIN FRACTIONS) Take the little denomʼs en masse then their lcm, with pizazz. Itʼll cancel all to get rid of the small but not of the big (alas). RATIONAL FUNCTIONS: DENOMINATOR ALERT (Are you a denominator-hater?) As fall months march on towards winter (and our evening squints become quint-er) watch out, this fine autumn that the x on the bottom wonʼt make it unfit for gozinta. TRANSLATING A LINEAR SYSTEM INTO A MATRIX Erase every x and each y anʼ each plus and each equal sign. Be sure to take care with each missing var in line after line after line. n EQUATIONS, n UNKNOWNS, UNIQUE SOLUTION The rref will be nice and tidy. On the left, an x-n I.D. With tons and tons of zeroes and ones and the answer on the righty. A ROW (OR TWO) OF 0ʼS (0 0 0 0 0...) This row is the ze-ro row. Itʼs a row which gives no info. And it will not budge. You be the judge -- it says no no no no no. NO SOLUTION, “BAD ROW” (0 0 0 0.... 1) This row says more is less. Canʼt make up its mind, I guess. It says right is wrong and short is long and no no no no yes. TELLING THE CALCULATOR TO PUT A GIVEN MATRIX INTO REDUCED ROW ECHELON FORM: A NON-LIMERICK Second x to minus one is how we get this thing begun. And then to get a further start two arrows right will do their part. Next, lest our poor T-8-3-plus get some impression wrong from us we must type in how many rows. How many columns also goes. And then the numbers, ends and center, separated by each ENTER. Now ʻtis time in this fine ode to take a pause with Second MODE. And then -- its second grand appearance -- sec x minʼ oneʼs interference. Arrows right and down (not left) enough to get us down to rref. Another ENTER -- Got that done?-- and one more sec x minus one. (I promise that was the last time. Itʼs honest and it makes it rhyme.) Now, one more ENTER -- you canʼt miss -- and then a close-parenthesis. Now one more keystroke from your mentor: big fat joyous final ENTER. Now weʼre done (unless one checks it): Not a keystroke but can EXIT. Functions are just like numbers. They get lugged around like lumber. They get add- and subtracted and more interacted making us dumb and dumber. (Cost Analysis) The cost of producing is mixed. Part of that cost is fixed. And the rest, never-ending on level depending (or maybe on politics). (Rational Functions) Said your favorite Zada or Tante “for an asymptote horizonta “you need to use both “of the leading coeffʼs “whether or not you wanta.” (Exponential Functions) The bottom is perfectly noble but x is now upwardly mobile so that constant, poor dear must downsize its career and itʼs feeling a bit claustrophob-al. These guys are as busy as beavers. Theyʼve got Monday morning fever. They just grow, grow, and grow. Theyʼre NEVER zero. And they donʼt remain one for long, either. DEFINITION OF LOG Do logs give you logarrhea? Is a logjam drawing near? Just use your credentials and take exponentials to make those olʼ logs disappear. TWO BASES FOR LOGS Two numbers have we to discuss -- e and 1-0, just for us. The first is a natch the second a catch and theyʼre both in the 83-plus. LOGS OF PRODUCTS; ln MN = ln M + ln N The product can change to a sum. But ʻtisnʼt entirely humdrum. Thereʼs an extra ln (one for M, one for N) and you know where that had to come from. LOGS OF QUOTIENTS: ln (M/N) = ln M - ln N From divide to subtract we can whittle. But again, thereʼs s smidgeon of diddle. lnʼs will appear in the front and the read with the minus in the middle. LOGS OF POWERS: ln (m^N) = N ln M That N is too up for our taste. Weʼll make it step down in great haste. But be sure, as you slide that that N lands outside the ln, or it must be erased. CHANGE OF BASE FORMULAS: LOTS OF LOGS Oh, how can we deal with this b? Itʼs neither a ten nor an e. Just spring into action by making a fraction with logs up and down, yesiree. SOLVING EQUATIONS LIKE 9^(2x-1) = 27^(x+1) Such a pile of glop and slop -- how can we make it all stop? Just play the game make the bases the same then compare what weʼve got on top. LOG-TRIGGERS: SOLVING EQUATIONS LIKE 3^(2x-1) = 5^(x+1) Hereʼs how to accomplish these missions: Take logs, change the powersʼ positions. Youʼll get rid of the ninny and acquire a linny with weird looking coefficients. SOLVING EQUATIONS LIKE ln (x+1) - ln x = ln 2 We want just one log for each side. So by Rule #1 weʼll abide. Weʼll clear up the fog by canceling log much to our supreme joy and pride. TRIG LIMERICKS: TWO NEW ANGLES ON ANGLES (1) Rinkity dinkity dink. More angles exist than we think. And some wrap around (without making a sound) as though they had too much to drink. (2) RADIANS Itʼs my supreme pleasure today to one other measure convey. The numbers are spare but they do get you there. 6 goes around most of the way. A lady name Katy OʼGrady was good at converting to radi- ans. “Simple”, said she “just take the degree “and times it pi over 180.” Her cousin named Mary Magee knew how to get back to degree. “Just multiply by “180 oʼer pi. “My cousin, Iʼm sure, will agree.” MORE TRIG Letʼs hereby consider our data: x, y, r, and our angle big-theta. It feels pretty blurred to the point of absurd but weʼll get it straight sooner or later. MNUEMONIC DEVICE/ sin theta = y/r, cos theta = x/r Y-inʼ rhymers with sine (at least if it keeps on tryinʼ). Does x rhyme with cos? Nope, not even close. (But maybe it will sometime.) INVERSE TRIG FUNCTIONS (Hark, hark, the arc!) We can go from x to sin x (or even to 9 sin 9x). But hey, can we play the opposite way with our calculators and Timex ? LINEAR REGRESSION: THE KEYSTROKES (START WITH STAT.): a non-limerick: STAT ENTER is the way to start. It gets us L1, if weʼre smart. Now list the xʼs, one by one. (There might be many, but itʼs fun.) Next arrow-right (get to L2) then list the yʼs, thatʼs what to do (as many yʼs as there are xʼes. -- takes us all the way to Texas) Donʼt forget, do what you mean and hit the ENTERʼs in between. Now, 2nd MODE is our next bet. It gets us where we want to get. STAT arrow-right and then a 4. ʻTwill get us what we need, and more. (Yup, a and b and r, r-square. The whole nine-yards, extraordinaire.) And now weʼre done -- Farewell, Godspeed! -- unless, perchance, we hap to need to view a dandy scatterplot or use the function we just got. To do the former in this sequel ENTER last, first sec- Y=. For the latter, I have reckoned the reverse, Y= 2nd followed by MODE VARS right-arrow ENTER ENTER straight and narrow. Next we hit the number that we want to find the function at. Then ENTER once again, thatʼs it to keep us satisfied and fit. And now weʼre done. Iʼm not pretending. Really truly ʻtis the ending. (Matrix multiplication) Just pair off each row with each column. Take the sums of those products so solemn. Yes, beat the odds with those scalar prods or whatever the devil you call ʻem. (Compound Interest) “Nothing succeeds like success,” as these formulas show with finesse. Thus again and again A gets bigger with n though not quite as big as youʼd guess. (Annuities) And now hereʼs another fine source (to tickle your brain in this course) of moʼ and moʼ dough. Itʼs dough a go-go. Oh, donʼt you just wish it was yours. LIMERICKS (AND OTHER CORNY POEMS) FOR MATHEMATICAL CONCEPTS I arranged by section numbers in the text, The Heart of Mathematics, Berger and Star- bird) (WELCOME) This first poem is just to say hi. Iʼm glad you had time to drop by. You might not like math but you do like to laugh and I promise, I wonʼt make you cry. (Chapter 1 -- HI DIDDLE DIDDLE, HOW ʻBOUT A RIDDLE?) Riddles are all fun and frolic and some are of cool math symbolic. Weʼve seen them before weʼll see them some more ʻtil we become riddle-a-holic. Just one stone is fancy and foreign. The eight others, weighing less, arenʼt. We have just two scales. After one use, each fails. Now, isnʼt that wholly abhorrent?