Quadratic Modeling Exercises

Transcription

1 Quadratic Modeling Exercises Pages Problems 1,5-12,14,15,17,19,21 (for 19 and 21, you re only deciding between linear and quadratic; we ll get to exponential soon!) In class, we analyzed te function f(x) = x 2 to see wy te second differences were equal. We ten made te somewat bras claim tat, Well, since te second differences are constant for tis function, tey ll be tusly for all quadratics. (OK, I didn t actually say tat but we pretty muc bougt it, didn t we?) Let s be more matematical let s use te general quadratic function f(x) = ax 2 + bx + c, and sow tat te second differences are te same. E1. Wat does f(x+1) f(x) equal? E2. Wat does f(x+2) f(x+1) equal? E3. Wat does f(x+3) f(x+2) equal? Eac of tose represented te first differences in te quadratic function now, let s look at te second differences. E4. Subtract your result in E5 from your result in E6. E5. Subtract your result in E6 from your result in E7. E6. Wat do you notice about your answers in E8 and E9? Tat ougta do it, altoug it still isn t a formal proof. If you need (or want) more, I offer you a quiz below were you can! Now, refer back to exercise 5 in te text. E7. Find te average rate of cange (speed) of te football s eigt between 0 and 0.5 seconds. Be sure to include te unit! E8. Find te average rate of cange of te football s eigt between 0.5 and 1 seconds. E9. Find te average rate of cange of te football s eigt between 1 and 1.5 seconds. E10. Find te average rate of cange of te football s eigt between 1 and 2 seconds. E11. Find te average rate of cange of te football s eigt between 1.5 and 2 seconds. E12. Find te average rate of cange of te football s eigt between 1.75 and 2 seconds. E13. Find te instantaneous rate of cange of te football s eigt at 2 seconds.

2 E Answers. E1. f(x+1) = a(x+1) 2 + b(x+1) + c and f(x) = ax 2 + bx + c, so f(x+1) f(x) = a(x+1) 2 + b(x+1) + c (ax 2 + bx + c) = ax 2 + 2ax + a + bx + b + c ax 2 bx c = 2ax + a + b, or, if you like, a(2x+1) + b I tink you can andle E2 and E3! E4. I get 2a. You? E5. Hmmmmmm? E6. Interesting! Wic sows tat te second differences are equal. Altoug it doesn t prove it for all sequential values of (x + ), it s opefully enoug to convince you tat tere is a pattern (and inspire you to try te quiz, if you so coose). E7. (I m calling te function f(x) ere, were f is te eigt at x seconds) E9. 0 feet per second (see wy?) f(0.5) - f (0) E feet per second (wy s it negative?) feet second = feet second = 32 feet second You can also use te best fit curve to do tis! E13. OK, we need to do a little algebra ere. From te average rate of cange formula Average Rate Of Cange = f(x+) - f(x) We need to get to equal 0. If you remember, from class, we can t simply plug in = 0 (yet), since we get 0 0 (wic is, essentially, living in sin). So, let s finagle, algebraically: Average Rate Of Cange (for our football problem) = f(x+) - f(x) = -16(x+)2 + 40(x+)+6 - (-16x 2 +40x+6) = -16x2-32x x x 2-40x - 6

3 Ha! Notice tat everyting witout an term canceled. Onward! = - 32x = (-32x ) = 32x + 40 At tis point, we can let = 0, so we ave Instantaneous ( = 0) Rate Of Cange (for our football problem) = 32x + 40 So, if x = 2 (for 2 seconds into te experiment), we ave an instantaneous speed of -32(2) + 40, or -24 feet per second.

4 Quadratic Modeling Quizzes Quiz 1. (10 points) (w) Complete questions E8, E11, and E12 above. Make sure to sow me te division you did (you can use E7 as a guide).

5 Quiz 2. On eart, tings fall under te force of gravity. Tis is fun! We can do tings like ave a catc, go sky diving, or rappel all because of tis wonderful force! 1. (2 points) For starters, Google Wy is tere gravity and tell me, in a couple sentences, wat you find. Now even toug we re not entirely sure wy tere is gravity, we do know ow it works. For example, suppose you need to know ow ig a cliff is (remember above wen IO say it d be fun to rappel? Well, if you know ow long your rope is, you ll need to also know ow tall a cliff is to make sure tat it s long enoug to get down safely!). Ceck tis out: = -16t 2 + v o t + o Tat s a little equation tat governs te position of any object trown on eart. is your eigt after t seconds, v o is your initial velocity, and o is your initial eigt. Te -16 is a cool result tat s tied to te fact tat gravity accelerates objects at a predictable rate. Now suppose you re standing a te top of te cliff you want to rappel down. Tat cliff as a eigt and you don t know it. Let s let tat be te o tat we re trying to find. To find it, we re going to drop rock over te edge of a cliff, and time ow long it take for us to ear it it te ground 1. If we do tat, ten te eigt after t seconds ave passed will be 0 (since te rock is now on te ground). Te initial velocity (v o) will also be zero (since we re dropping te rock and not trowing it down). Tis means tat te equation above can be rewritten as 0 = -16t 2 + o 2. (2 points) Explain wy we can rewrite te equation in tat way! Use tis equation to figure out te eigt of te cliff you re standing on if te rock falls for 3. (2 points) 2 seconds. 4. (2 points) 3 seconds. 5. (2 points) 4 seconds. 1 We ll assume tat te cliff is fairly sort, by speed of sound standards. Wen you re climbing, you rarely carry a rope more tan 250 feet long, anyway wic means tat you wouldn t be rappelling more tan about 125 feet. Over a distance tat sort, we can ignore te effect of te speed of sound delay.

6 Quiz 3. In class, we spoke of lots of ways in wic quadratic relations sips appear in te world around us. I d like to explore anoter in tis quiz te Golden Ratio. 1. (2 points) Google Te Golden Ratio and tell me wat you find. Make sure to include its numerical value, in bot exact (wit te square root) and approximate form! You may ave learned (in your Googling) tat Te Golden Ratio gets a symbol:. Wy? Well, like and its lesser known cousin, e, tends to sow up quite a bit in te world (ence, it gets a cool name). Here are a couple of cool places sows up: In te Mona Lisa, te ratio of te subject s face (is er name Lisa?), lengt to widt, is approximately. In fact, tere are many occurrences of in te Mona Lisa tis is just one of tem. In te Partenon, te lengt to widt ratio, again, is approximately. Te nautilus sell s long side to sort side ratio is about, as well. (over, please!)

7 Here s one of my favorites: In any regular (all sides and angles equal) pentagon, if you draw in te diagonals, you get a five pointed regular star. Te lengts of some of tose segments formed can be combined into a ratio of for example, te ratio of DB to DC is. 2. (2 points) Find two oter lengts in tat figure tat ave, as teir ratio, a value of. You migt want to start by Googling golden ratio pentagon unless you love doing geometry! 3. (4 points 2 for eac) Find two oter places te golden ratio occurs one in nature (like te nautilus), and one in uman generated form (like te Mona Lisa or te Partenon). 4. (2 points) In te Fibonacci Series (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc.), eac successive element is formed by adding te previous two togeter. Wat does ave to do wit tis series? Again, Google is your friend! Wy = 2, you migt ask. Good for you! It as to do wit someting in geometry call similarity in a golden rectangle (tat is, a rectangle wose side are in te golden ratio), if you cut off a square, te smaller rectangle you leave beind is also golden (and, tereby, similar to te larger one). (extra 3 points) Te large rectangle sown at left is golden tat is, te ratio of its long side to its sort side (a + b to a) is. Te small rectangle is also golden, tat is te ratio of a to b is also. Give me a proof tat demonstrates from were te in te figure sown at rigt arises

8 Quiz 4. Number of Deats Year per 100K men Te data at left sow various years wort of data on men s cancer deats (per 100,000 men) in te United states (source: USA Today) For tese data, use your Excel Seet to create te best fit parabola (let x be te year and y be te number of men, per undred tousand, wo die from cancer). 1. (4 points 2 for te correct number, and 2 for te correct unit!) (w) Use your model to find te average rate of cange in cancer deats from 1990 to Remember tat you can use your TI to do tis; just fully explain wat you did. You ll ave to use per twice in your answer! 2. (1 point) According to tis model, ow many US men (per 100K) sould ave died from cancer in 2011? According to te American Cancer Society report Cancer Statistics (ttp://onlinelibrary.wiley.com/doi/ /caac.20121/pdf), in 2011, tere were a total number of 300,430 male deats from cancer in te US. 3. (1 point) How many US men are tere? You ll ave to do a little online researc to find tis number (I Googled population of US 2011 ; te second result was a Census Bureau quickfact table tat was perfect. You can assume te US is alf men and alf women. Also, assume all males are men ; te ACS document appears to make tis assumption) 4. (1 point) Time for some unit analysis move te decimal point in your answer to part c 5 places to te left. Tat makes it now te number of US men in undreds of tousands. Now, multiply tis number by your answer in b. Tat sould be te total number of US male deats from cancer in (1 point) Wat percent error is sown in your estimate? Use te formula Percent error = 6. (1 point) Wat could explain tis fairly large error? regression predicted result in ACS estimated value in 2011 ACS estimated value in 2011

9 Quiz 5. Years ago, someone told me If you drop a penny off te top of te Empire State Building in New York, and it its someone at ground level, tey ll die. #mindblown Of course, now tat I m older, I don t believe statements like tat at face value anymore. So, we ll model it! Go aead and open up te spreadseet tat accompanies tis quiz. Wen you do, you ll see tis! Hopefully, tis looks a tad familiar! It s based off of te seet we used for te in class quiz about my bike speed. In class, toug, we worked troug AROC (Average Rate Of Cange) until we arrived at my IROC (Instantaneous Rate Of Cange) at te moment I finised my 200 foot trip. Tis spreadseet s a tad different it looks at te IROC of te penny after a certain number of seconds tat it s fallen. In oter words, it s like te penny as a speedometer and you get to read it. So, just like we did in class, we ll ceck te IROC of tat penny after various points of time. And, just like in class, we ll measure tose points of time troug te reference of seconds until impact. 1. (1 point) How long does it take te penny to it te ground? You can round to te nearest undredt of a second if you like. (int: look at te table of values) 2. (1 point eac) Wat s te penny s IROC a. 8 seconds before impact? b. 5 seconds before impact? c. 2 seconds before impact? d. 1 second before impact? e. 0.1 seconds before impact? f seconds before impact? At tis point te penny disappears from te grap. It s, essentially, on te ground.

10 (please note: te penny, in tis situation, would actually fall straigt down, not out like a parabola. However, I tougt it elpful to grap it as eigt versus time, wic appears parabolic because it s accelerating as it falls) So now you know ow fast it s moving wen it its te ground (or, someone standing near te ground). And tat s pretty darned fast! However, speed s only part of te process of understanding if someone would die if tey were struck. Weigt, altoug it as noting to do wit ow fast te penny falls (everyting on eart falls at te same acceleration), as everyting to do wit ow te penny feels wen it its (imagine dropping an anvil and a crumpled piece of paper at te same time. Bot would it te ground at te same time, but well, you get te idea). So, let s do some pysics! Since te penny (used tis way) is a projectile, it s basically a bullet. 3. (1 point) Now, find te weigt of our bullet! Tat is, a penny. Ten convert tat weigt to grains. Apparently, grains is te preferred unit for ballistics. 4. (1 point) Convert your IROC speed in MPH to feet per second. Tat s te unit te calculator in te next step needs. 5. (1 point) Go ere: ttp:// How many foot pounds of energy does te bullet ave as it its te ground? Tat number migt seems a little cryptic to you (eck, it sure did to me). I did some Googling, and found tis: ttp:// Our little penny doesn t even come close to delivering te amount of energy (even after 1000 feet) tat any of tose rifles ave. Ten, I went back to tis: ttps://en.wikipedia.org/wiki/muzzle_energy. Tere, I discovered tat our little penny, basically, is like an undersized pellet fired from an AirSoft gun. It migt sting, but it sure as eck isn t gonna kill you. So tere you go! Myt busted! 6. (extra 2 points) I made an assumption about te acceleration of our penny tat well, it just isn t true. Wat was tat assumption (and wy does it make te dropped pennies are letal argument even less meaningful)? Hint: we mentioned it in class, but it didn t apply to te bike scenario.

11 Quiz 6. So, in class, we discussed ow linear equation ave constant first differences (a number tat s known as teir slope), wile quadratic equations ave constant second differences (a number tat s known as teir acceleration). And we illustrated tis wit a couple of examples in class, wic were fun and enligtening! But, examples do not te trut make! We need to sow tat tese results will old for all cases, for all linear and quadratic models! I ll start by sowing you ow, in te general quadratic model y = mx + b (tis is te form you all seem partial to), te first differences always come out to te same ting! So ere ya go! ttps:// (10 points) (w) Your job? Repeat tat sort of analysis for te general quadratic model y = ax 2 + bx + c, and sow tat te second differences are consistent. You can use te above metod as a template, and also te video below (were I get ya started). Make sure to sow me at least two more second differences (oter tan te one I sowed you in te video below); after tat, I figure we can believe tat te pattern will continue! ttps://