MATH 16 A-LECTURE. SEPTEMBER 2, PROFESSOR: WE'RE GOING TO START WITH A FEW SHORT ANNOUNCEMENTS.

Transcription

1 1 MATH 16 A-LECTURE. SEPTEMBER 2, PROFESSOR: WE'RE GOING TO START WITH A FEW SHORT ANNOUNCEMENTS. GSI: ANY NAME IS DAN-COHEN A CONE. I WORK FOR THE CAMPAIGNING. RUNNING FOR REELECTION THIS YEAR. THE NUMBER TWO RACE IN THE FACE AND WE'RE HIRING PAID INTERNS. IN YOU'RE INTERESTED IN GETTING INVOLVED OR WHATEVER I HECK THE LIKE FEEL FREE TO PICK UP A FLYER. THANKS VERY MUCH AND ENJOY THE CLASS. GSI: ANY NAME IS TIM I'M FROM THE A-BA. I LIKE TO TAKE THIS OPPORTUNITIES TO INVITE YOU TO OUR FIRST GENERAL MEETING WHICH IS BEING HELD WEDNESDAY, GOOD OPPORTUNITIES FOR YOU TO LEARN MORE ABOUT THE ORGANIZATION AND THE PROFESSIONAL, SOCIAL AND ASPECTS OF OUR ORGANIZATION. ALSO YOU CAN SIGN UP FOR MEETINGS AFTERWARDS. CHECK US OUFPLT IT WILL BE A LOT OF FUN. THIS KWERPBS, 7:30 P.M., 145 DWINELLE. PROFESSOR: SO FINALLY FOR AN ANNOUNCEMENT ABOUT THIS CLASS. GSI: HIGH ABOUT SCHEDULING. WITH ELECTROMAGNETIC IF A SISTER. SECTIONS WHICH ARE DONE ENTIRELY ON-LINE THIS SEMESTER. I THINK I HAVE SUCCEEDED IN SAYING THAT. IF YOU'RE IN SECTION FIVE CONFLICTS WITH THE REST OF YOUR SCHEDULE AND YOU WANT IT GET ON THE WAITLIST FOR ANOTHER SECTION THEN THE WAY TO DO THAT IS GO ON IT TELL BEAR, FIRST DROP THE COURSE AND ADD WAITLIST. AND THEN LOCATE IT FROM THERE. IF THAT SOUNDS A LITTLE RISKY. THAT'S BECAUSE IT IS. IF YOU COME TO MY OFFICE HOURS I CAN ASK YOU WHAT THE CONTACT IS AND FIGURE OUT IF IT'S A RISK WORK TAKING. OR MAKE THE RISK GO AWAY. YOU CAN'T HOLD A SPOT IN ONE SECTION 2 WHILE AT THE SAME TIME WAITING FOR SPOT IN NORTH SECTION. NOT THE MOST FRIENDLY WAY OF DOING THING. YOU CAN'T HAVE A BIG

2 CIRCLE OF PEOPLE WAITING FOR YOU TO MAKE A MOVE. ONE MORE THING IS THAT IF YOU'RE TRYING IT FIGURE OW, IF YOU'RE ON THE WAITLIST AND TRYING IT FIGURE OUT HOW YOU'RE DOING, IGNORE THE WAITLIST FOR THE COURSE AS A WHOLE. WHAT MATTERS IS THE LENGTH OF THE WAITLIST FOR YOUR INDIVIDUAL SECTION. EACH SECTION GET PROCESSED SEPARATELY. BY THE WAY OF AN UPDATE, WHEN I CHECKED THIS MORNING ON-LINE SCHEDULE OF CLASSES THAT THERE WERE OPENINGS IN SECTIONS ONE OWE ONE, ONE OWE TWO, ONE OWE THREE AND 104. SO IF YOU'RE ON THE WAITLIST FOR A DIFFERENT SECTION AND ANY OF THOSE TIMES IS COMPATIBLE WITH YOUR SCHEDULE, TRY TO GET STRAIGHT NOT COURSE IN ONE OF THOSE SECTIONS. AND YOU CAN ALSO TRY IT GET ON TO A SHORTER WAITLIST. OKAY, THANKS. PROFESSOR: THANK YOU. I JUST HOPE HE DIDN'T SEE PLACES IN SECTIONS SHOWING UP ON E-BAY. OKAY, SO I SEE THERE ARE SOME PEOPLE STILL SITTING ON THE SIDE AND I DO SEE EMPTY SEATS IN THE MIDDLE. THERE ARE A FEW HE SEATS IN THE BACK AND MIDDLE THERE. SO, DO WE HAVE ANY QUESTIONS FROM LAST TIME. I WOULD LIKE TO JUST CONTINUE. SO SEVERAL PEOPLE ASKED ME ABOUT THE WEB PAGE AND IN PARTICULAR WHEN I SAID HOMEWORK FOR SECTIONS 0.1 TO 0.6, WILL BE DUE NEXT TUESDAY, THEY ASKED DO I HAVE TO DO ALL THE QUESTIONS IN SECTIONS, THE ANSWER IS NO. AND EDIT THE WEB PAGE IT MAKE THAT CLEAR AM ILL POST THE SUBSETS OF QUESTIONS FROM THE SECTIONS THAT YOU'RE EXPECTED TO DO. IF YOU DO THEM ALL, IT'S 3 GREAT PRACTICE SO I WOULDN'T DISCOURAGE THAT. I'LL LEAVE THIS UP FOR A LITTLE WHILE. THIS IS BETTER THAN THE LAST POLITICAL ANNOUNCEMENT I HAD IN THE CLASS. SOMEONE CAME IN AND ASKED FOR ONE MINUTE. HELD UP THE TEXTBOOK AND TOLD THE STUDENTS THEY'RE BEING RIPPED OFF BY THE TEXTBOOK. SO THIS IS MUCH ALREADY. THE MATERIAL IN THIS CLASS HAS NOT CHANGED FOR

3 50 YEARS. WHY SHOULD YOU HAVE TO BUY A NEXT TEXTBOOK. I THOUGHT THAT WAS AMUSING. SOME OF THE RESULTS IN THE BOOK WERE FROM PEOPLE WHO AREN'T 50 YEAR OLD YET. IT'S A -- AND IF YOU LOOK AT MY WEB PAGE YOU'LL SEE A LOT OF THING HAVE EXCHANGED IN THE LAST 50 YEAR TOO. I'M GOING TO TALK ABOUT THAT. SO LAST TIME WESTERN TAKE ABOUT GRAPHS OF FUNCTIONS. TALK ABOUT THAT BEFORE WE GO ON TO GRAPH OF FUNCTIONS. SO THE GRAPH OF A FUNCTION, Y-EQUALS F-OF X-WAS THE SET OF ALL POINTS IN THE X-Y-PLAIN OF THE FORM X-COMMA F-OF X. AND SO FOR EXAMPLE THERE'S PARABOLA YOU MIGHT GET THAT IF F-OF X-WAS SOME SORT OF POLYNOMIAL. SO THAT'S SORT OF GRAPHS WE TALK PERIODS LAST TIME. BUT NOT EVERY INTERESTING GRAPH COMES FROM A FUNCTION. SO A FUNCTION OF THE FORM Y-EQUALS N-F-OF X-. WE'RE INTERESTED IN PLOTTING THE GRAPH, SO HERE'S AN EXAMPLE. TWO X-PLUS THREE Y-EQUALS TPAOEUFRZ. THAT IS THE GRAPH OF, THIS SHOULD BE FAMILIAR, A STRAIGHT LINE. AND THAT'S NOT A PROBLEM TO ACTUALLY SOLVE IT SO IT WOULD LOOK LIKE THIS. SUBTRACT -- THEN DIVIDE BY THREE, AND FIRST YOU COULD MAKE A PARABOLA LOOKS LIKE THIS. WHO 4 CAN TELL ME WHAT THIS POINTS IS HERE. THE X-INTERCEPT. THAT'S THE NAME OF THAT POINTS. SO IT'S THE POINT WHERE Y-IS ZERO. SO I CAN JUST GO AHEAD AND FILL THAT IN. WHAT IS X-HAD THAT HAPPENED? JUST SORT OF, CAN YOU SOLVE THAT FOR Y-EQUALS ZERO? SO (ON BOARD) AND THIS POINTS THERE THAT'S WHY THAT INTERCEPT THE Y-AXIS. THAT'S CALLED Y-INTERCEPT. SO THAT'S THE POINTS X-EQUALS DISTORTION FIVE THIRDS. SO SOMETIMES YOU'RE EQUATIONS CAN BE TURNED INTO A FUNCTION OF THAT'S NO PROBLEM AND BUT THERE ARE ONLY MORE INTERESTING ONES THAT CANNOT BE. SO HOW ABOUT A GRAPH OF X-EQUALS THREE? WHAT DOES THAT LOOK LIKE? HERE'S THE

4 X-Y-PLAIN. IT'S A STRAIGHT LINE BUT WHAT KINDS OF STRAIGHT LINE? IT'S A VERTICAL LINE. X-EQUALS THREE AND IT'S NOT A FUNCTION, WHAT WAS OUR TEST FOR WHETHER A GRAPH WAS A PHOTOGRAPHER OF A FUNCTION OR NOT? YOU DRAW A VERTICAL LINE. IF IT HITS MORE THAN ONE POINTS IT'S NOT A FUNCTION. HERE'S ANOTHER ONE I HOEPDZ IS FAMILIAR. X-SQUARED BLOUSE Y-SQUARED IS ONE, THAT'S A CIRCLE? AND IT'S CENTERED TER ORIGIN. ZERO, ZERO, AND IT'S RADIUS IS ONE. SO THAT IS NOT A FUNCTION BECAUSE THAT VERTICAL LINE INTERSECTS TWO POINTS. BUT I CAN BREAK IT UP INTO TWO FUNCTIONS. THIS IS ACTUALLY THE GRAPH OF TWO FUNCTIONS ON TOP OF ONE ANOTHER. SO WHAT ARE THOSE TWO FUNCTIONS IF YOU WANTED TO DO IT? I WOULD. SO THIS FOR Y, SO WRITE Y-SQUARED EQUALS ONE MINUS X-SQUARED AND AND THEN I NEED THE SQUARE ROOTS. BUT WHETHER I GET THE SKRAORBGTS IS THERE ONLY ONE ANSWER? NO. SO THERE'S ONE FUNCTION. THIS ONE GIVES ME THE TOP HALF. AND 5 THIS ONE GIVES ME THE BOTTOM HALF. SO SOMETIMES YOU CAN BREAK IT UP INTO, A GRAPH INTO TWO FUNCTIONS THAT WAY OR THREE OR WHATEVER IT TAKE. THIS ONE YOU CAN'T DO THAT. (ON BOARD). THIS IS YOUR LAST CHANCE TO WRITE THIS DOWN IF YOU'RE INTERESTED. I'M ABOUT TO ERASE IT. I GUESS I'M NOT GOING TO ERASE IT. I FORGOT ABOUT THIS CHALK, IT'S TRICKY. ALL RIGHT. I'LL DO MY BEST. SO LET'S SEE HERE. I HOPE IT'S, I HOPE THIS IS, MAKE THIS AS INDELLABLE IMPRESSION IN YOUR BRAIN AS THIS MAKES ON THE BOARD. SECTIONS 0.2, NOW WE'RE GOING TO TALK ABOUT SOME IMPORTANT FUNCTIONS THAT COME UP ALL THE TIME AND YOU SHOULD RECOGNIZE. AND I HAVE TO REMIND MYSELF INDEED I NEED IT LOOK UP TO SEE IF THERE'S A HAN BACK THERE. SO IF I DON'T, YELL. SO LET'S WALK THROUGH THE SIMPLEST ONE. AND THIS ONE WE'VE DONE BEFORE. JUST ANOTHER

5 NAME FOR A STRAIGHT LINE. AND I TOLL YOU THAT ONE OF THEMES IN THIS COURSE IS THERE'S A BUNCH OF ALGEBRA DEALING WITH FORMULAS AND THERE'S ALSO THE GEOMETRY, WITH PICTURE. I WANT IT USE BOTH LANGUAGES ALL THE TIME. A LINEAR FUNCTION GETS TKPWRAFD AS WE'VE SEEN MANY TIMES AS A STRAIGHT HRAOEUPBL. SO NO BIG DEAL THERE. BUT LET ME JUST REMINDS YOU OF SOME TERMINOLOGY. SO WRITE DOWN ONE VERY SIMPLE LINEAR FUNCTION. AND SO THE FIRST POINTS I WANT IT MAKE IS THAT THIS HAS A NAME. I CALLED IT, I USED THIS NAME BEFORE, IT'S CALLED THE Y-INTERCEPT BECAUSE WHEN YOU PLUG IN X-EQUALS ZERO YOU KNOW THAT THE LINE HAS TO GO THROUGH THAT POINTS. AND THE OTHER TERM HERE HAS AN IMPORTANT NAME WE'RE 6 GOING TO USE A LOT. AND IT'S CALLED THE SLOPE. AND IT TELLS YOU HOW STEEP THE LINE IS. AND SO FOR EXAMPLE IF YOU WERE TO DRAW A LIMB TRIANGLE HERE WHERE THE BASE IS ONE. A LITTLE RIGHT TRIANGLE THEN IT WOULD BE AN M BY ONE TRIANGLE THAT WOULD FIT RIGHT ON THAT LINE. SO THAT'S WHAT THE SLOPE S. USE THIS TERMS AS WE GO FORM. I HOPE THEY'RE FAMILIAR. SO I WANT TO MOVE ON, LET'S SEE, ONE OTHER FACT I WANT TO SAY THAT'S ON THE GEOMETRIC SIDE. WHICH IS FAMILIAR. BUT I'M GOING DO GENERALIZE IT TO OTHER FUNCTIONS. TWO POINTS DETERMINE A STRAIGHT LINE. THIS IS I HOPE REALLY REALLY FAMILIAR, IF I HAPPEN TO PICK TWO-POINT THERE'S OBVIOUSLY ONLY ONE STRAIGHT LINE I CAN DRAW. THIS TWO PARTICULAR POINTS HAPPEN TO BE THE Y-AND X-INTERCEPT. OTHER FUNCTIONS ARE ALSO GOING TO BE INTERESTING PROPERTY AM MIGHT BE DETERMINED NOT BY TWO-POINT BUT BY THREE POINTS. SO I'LL POINT THAT OUT SHORTLY. SO LET ME DO A ECONOMIC STKAFPLT A FUNCTION THAT WOULD COME UP IN A A BUSINESS CLASS. ECONOMIC EXAMPLE. SO THIS SHOULD BE, THE MATHEMATICS HERE IS JUST WHAT I WROTE DOWN THERE. THIS IS

6 NOTHING KNEW. JUST AN EXERCISE IN TRANSLATING FROM ENGLISH. SO TO PRODUCE A PRODUCT YOU MIGHT KNOW AS A BUSINESS PERSON THAT THERE'S SOME COSTS ASSOCIATED WITH IT. IT COSTS LET'S SAY $10,000 IN FIXED COSTS, LIKE THE RENTS ON THE BUILDING AND WHATEVER MACHINES YOU MAY TPHAOEFPLD YOU BOUGHT THEM. THEY COST MONEY. IT COST LET'S SAY $10 PER ITEM. TO MAKE ONE OF THESE, MAKE A WIDTH IT. W-I-G-E-T. MAKE AN ITEM. SO THAT'S WHAT IT 7 COSTS IN RAW MATERIAL. AND BECAUSE YOU'RE IN BUSINESS IT COSTS $15 PER ITEM TO SELL. THAT'S THE COST. SO THE QUESTION IS HOW MANY ITEMS DO I NEED TO SELL TO TURN A PROFIT? SO THAT'S A PERFECTLY REASONABLE QUESTION YOU MIGHT ASK. SO LET'S ME WRITE DOWN THE PROFIT ON X-ITEMS AS A FUNCTION. SO I CALL THE FUNCTION F-OF X, MAYBE I'LL YOU CALL IT P-OF X-FOR PROFITS. SO WHAT THE THE PROFIT GOING TO BE? IT'S GOING TO BE HOW MUCH I SELL THEM FOR, SO IF I SELL KP-FRBGS THEM IT'S GOING TO BE 15 X. THAT COMES IN BUT I HAVE TO SUBTRACT OUT MY COSTS. SO THAT'S AMOUNT I HAVE TO SPENDS. AND THE COSTS ARE GOING TO BE 10,000 PLUS TEN X. SO INCOME MINUS COSTS. AND WHEN I PUT THAT \ALTOGETHER\ALL TOGETHER I GET, CANCEL. I GET FIVE X MINUS 10,000. SO THAT IS MY TOTAL, WHAT HAPPENS TO MY BANK ACCOUNT. AND SO HOW DO I DECIDE WHAT X-WILL GIVE ME A PROFIT. WHAT DO I HAVE TO DO WITH THIS FUNCTION? I HAVE TO ASK WHICH VALUES OF X-IS THIS FUNCTION POSITIVE? WHICH, X-IS F-OF X-GREATER THAN OR EQUAL TO ZERO SO THAT'S THE PROFIT QUESTION. AND SO WHAT WE NEED TO DO THEN IS SOLVE, THIS IS, SO WE'RE GOING TO SOLVE FOR F-OF X-EQUALS ZERO AND WHILE I'M AT IT LET ME ALSO PLOTS OF THING. SO P-OF X. AND SO THIS IS, THIS FUNCTION, SO WHAT HAPPENS WHEN X-EQUALS ZERO IF I HAVEN'T SOLD ANYTHING? WHAT'S MY PROFIT? IT'S MINUS 10,000. WHAT IS NEGATIVE PROFIT? WHAT'S THE SLOPE OF THAT CURVE? FIVE.

7 THAT'S GOING TO GO UP AT A PRETTY STEEP SLOPE. I'LL MAKE IT UP THERE. AND SO ON THIS SIDE I WIN AND ON THIS SIDE I LOSE OR POSITIVE, NEGATIVE, WHATEVER WORPBDZ UNIT TO USE THIS. SO I 8 NEEDS IT FIND THAT POINTS SO I NEED TO SOLVE ZERO EQUALS FIVE X-MINUS 10,000. AND FIVE X-EQUALS 10,000. AND SO I BETTER SELL AT LEAST 2000 OF THESE OBJECTS TO TURN A PROFIT. SO THIS POINTS RIGHT HERE IS 2000 COMMA ZERO RIGHT FROM. AT 2000 I MAKE ZERO PROFIT. SO HOPEFULLY I'LL BE ON THIS SIDE OF SO JUST A WORD PROBLEM TO TRANSLATE THE MATH TO THE, BUSINESS TO MATH. SO LET ME DO A SLIGHT EXTENSION OF THIS FUNCTION THAT ALSO COMES UP WHICH IS CALLED A PIECE WISE, SO THAT WAS A LINEAR FUNCTION. NOW I'M GOING TO TALK ABOUT A PIECE WISE LINEAR FUNCTION WHICH IS ALSO RELATED. SO HERE IS AN EXAMPLE I THINK I USED LAST TIME. IF THE BUSINESS, VOLUME DISCOUNTS, HOW MUCH SOMETHING COSTS. SO IT'S GOING TO BE THE COST IS A FUNCTION OF X-EQUALS NUMBER PURCHASED. AND SO A PIECE WISE LINEAR FUNCTION SAYS I'M GOING TO HAVE A DIFFERENT FORMULA FOR THIS FOR DIFFERENT VALUES OF X-BUT IN BOTH INDICATIONS ITS GOING TO BE A STHRAEUT LINE. SO THIS SAYS AT THAT UP TO X-IT'S GOING TO COST 200 EACH. BUT IF X-IS BIGGER THAN 200 I'M ONLY GOING TO EXCHANGE YOU EIGHT BUCKS EACH BUT I BETTER KNOTS DO THIS BECAUSE ALL OF A SUDDEN OF PRICE TKROFPLZ I KNOW IT PRICE TO KEEP GOING OCCUPY BECAUSE I'M IN BUSINESS. SO TO ADD SOMETHING TO THIS SO THESE THINGS MATCH AT X-EQUALS 200. SO WHAT SHOULD I ADD TO IT. LET ME DRAW THE FUNCTION. SO HERE I HAVE SOMETHING QUITE STEEP. HERE'S THE FUNCTION, TEN X-. THAT GOES ALL THE UP TO 200. AND HERE THE VALUE IS BECAUSE TEN TAOEUPBLZ 200 IT HERE I WANT TO HAVE A SLOPE THE EIGHT. THAT'S GOING TO BE MY PRICE AS A 9

8 FUNCTION OF X. THERE'S X. HERE'S F OF X. SO THIS IS GOING TO HAVE A SLOPE EIGHT AND IT'S GOING TO BE EIGHT X-PLUS, WELL I'LL JUST WRITE DOWN 400. THAT MAKES THESE TWO LINES MAX. THE OTHER WAY I COULD HAVE SAID IT IS FOR EVERYTHING YOU BUY PAST 200 THAT COSTS YOU EIGHT, PLUS YOU HAVE TO BUY THE FIRST 200 OF AT THAT PRICE SO IT'S 2000 FOR THAT AND YOU GET THE SAME ANSWER THAT WAY. SO THIS IS CALLED A PIECE WISE LINEAR FUNCTION BECAUSE IT'S TWO STRAITS LINES STUCK TOGETHER. AND IT COMES UP NATURALLY. SO LET'S ME DO ANOTHER FUNCTION. WELL, LET ME DRAW A DIFFERENT GRAPH AND YOU CAN TELL ME THE FUNCTION. SO I'LL DO IT THIS WAY. SO HERE'S ANOTHER EXAMPLE. CALL IT G-OF X-. IT WILL LOOK THE SAME HERE. TEN X-FOR X-LESS THAN OR EQUAL TO 200. AND THEN MAKE IT EIGHT X-PLUS 500 FOR X-BIGGER THAN 200. SO WHAT DOES THAT LOOK LIKE? IT LOOKS LIKE THIS. RIGHT. SO HERE IS ON THIS SIDE JUST A LITTLE BIT PAST IT IT'S GOING TO BE I WROTE THIS DOWN I DREW A PICTURE. THIS IS A FUNCTION? THIS IS A FUNCTION OF THIS IS A FUNCTION, AND IT'S BECAUSE IT PASSES THE TEST IF YOU DRAW A VERTICAL LINE IT ON THE INTERS-BGTS ONCE BUT IT SHOWS THIS FUNCTIONS DO NOT HAVE TO BE CONTINUOUS. THEY CAN HAVE JUMPS IN THEM. THAT'S PERFECTLY FINE. CONTINUOUS MEANS I CAN DRAW WITHOUT LIFTING MY CHALK OFF THE BOARD. FUNCTION. DON'T HAVE TO DO THAT. THAT'S A PERFECTLY FINE SOMETIMES WE'LL HAVE THOSE TOO. OKAY. ENOUGH OF STRAIGHT LINES. LET ME GO ON TO SOMETHING, THE NEXT STEP MORE INTERESTING. STRAIGHT LINES ARE INTERESTING. 10 WE'LL KEEP USING THEM ALL SEMESTER FOR LOTS OF REASONS BUT STILL LIFE IS MORE INTERESTING IF YOU ALLOW YOURSELF SOMETHING ELSE. SO MY NUMBER, LET ME GO ON NOW TO QUADRATIC FUNCTIONS.

9 SO THEY'RE OF THE FORM, SOME CONSTANT, CALL IT A-TIMES X-SQUARED PLUS SOME CONSTANT CALL IT B-PLUS ANOTHER CONSTANTS. SO A-B-AND C-ARE CONSTANT. AND I HOPE YOU'VE SEEN THIS BEFORE. WHAT IS THE GRAPH OF THIS? THERE'S THE NAME FOR THE KINDS OF GRAPH WE GET. IT'S CALLED A PARABOLA. SO THEY ALWAYS LIKE EITHER LIKE THIS OR MAYBE THEY LOOK LIKE THIS. HOW DO YOU TELL WHICH IS WHICH? JUST BY LOOKING AT THIS, CAN YOU TELL WHETHER THE GRAPH IS GOING TO GO LIKE THIS OR THIS? IF A-IS POSITIVE, IT'S GOING TO GO UP. WHEN X-IS REALLY BIG IT'S GOING TO GO UP. AND IF A-IS NEGATIVE, THEN MAYBE IT'S GOING TO GO DOWN. SO IT JUST TELLS YOU THAT. WHAT IF A-IS ZERO? WHAT HAPPENS TO THIS FUNCTION? I GET A STRAIGHT LINE. SO IT'S A PARABOLA UNLESS A-EQUALS ZERO IN WHICH CASE YOU'RE BACK TO A STRAIGHT LINE. SO LET ME JUST DRAW AN ANALOGY HERE. TWO POINTS DETERMINE THE STRAIGHT LINE WHICH IS THE GRAPH OF A FUNCTION AND THAT FUNCTION IS, I'M JUST WRITE IT DOWN, SOME CONSTANT TIMES X-PLUS ANOTHER CONSTANT. A-X-PLUS B. SO NOW LET ME ASK, HOW MANY POINTS DO YOU THINK DETERMINE A PARABOLA WHOSE GRAPH IS, FOR THIS A-X-SQUARED PLUS B-X-PLUS C? WHICH HAS THREE PARAMETERS. SO HERE I HAVE TWO UNKNOWNS. TWO POINTS, THAT'S ENOUGH INFORMATION TO TIE DOWN THESE TWO CONSTANTS SO. IF I HAVE THREE I DON'T KNOW IT TURNS OUT THREE POINTS ARE ENOUGH TO DETERMINE A PARABOLA. SO THAT 11 MEANS IF I DRAW ANY OLD THREE POINTS NOTICE PLAIN I CAN, THERE'S EXACTLY ONE PARABOLA THAT GOES THROUGH THEM AND WE'LL TAKE A WILD GUESS WHAT THAT LOOKS LIKE. IT'S SOMETHING LIKE THIS. EXACTLY ONE OF THOSE. I WON'T USE THAT FACT BUT IT'S ONE OF THOSE, AGAIN THERE'S ALGEBRA -RPBG COUNT HOW MANY UNKNOWNS THERE ARE AND THERE'S A PICTURE WHICH IS THREE POINTS AND THERE'S EXACTLY ONE PARABOLA. SO IT'S NICE TO HAVE BOTH IN YOUR HEAD.

10 SO PARABOLAS ARE NICE. LET ME GO ON. NEXT CATEGORY AM BOTH STRAIGHT LINES AND QUADRATIC FUNCTIONS ARE SPECIAL CASES BEING SOMETHING CALL POLYNOMIALS. AND IN THAT CASE, MY FUNCTION, WELL I MAY NOT JUST V LET ME GIVE YOU A GENERAL EXAMPLE I MAY HAVE A-X-CUBED PLUS B-X-SQUARED AM SO I HAVE LOTS OF POWERS OF A. AND A-B-C-D-ARE CONSTANTS SO. THAT'S AN EXAMPLE. BUT FOR POLYNOMIAL I CAN HAVE AS MANY POWERS AS I WANT. I DON'T HAVE TO STOP AT THREE. SO THE MOST GENERAL CASE WOULD BE IF THE HIGHEST POWER IS N. JUST MAKE UP A NAME FOR TA N, I'M GOING TO HAVE SOME CONSTANTS, A-N-X-TO THE N. AND KEEPING THIS EXAMPLE, AND THERE'S THREE PLUS YOU NEED ANOTHER CONSTANTS. I'M GIVE IT A NUMBER. A-SUB N-MINUS ONE TIMES X-T*T N-MINUS ONE AND JUST KEEP GOING UNTIL I GET THEM ALL. PLUS SINCE I'M TIRED OF WRITING THIS I WRITE DO THE DOT DOT AND COVERS ALL THE THE ONES I'M LEAVING OUT. AND A-ONE TIMES X-TO THE ONE AND THAT'S JUST X-AND FINALLY I GET THE CONSTANT TERM. THAT'S THE MOST GENERAL CASE OF A POLYNOMIAL. I HAVE HAVE AN N-IS CALL THE DEGREE OF POLYNOMIAL. AND THAT CAN BE ANY POSITIVE INTEGER YOU 12 LIKE. SO THAT'S THE MOST GENERAL CASE. SO N-EQUALS ONE, STRAIGHT LINES. N-EQUALS TWO GAVE US QUADRATIC. N-EQUALS THREE, THERE'S A WORD FOR IT BECAUSE IT'S PRETTY COMMON. THAT'S CALLED CUBIC. BUT AFTER AWHILE WE RUN OUT OF EITHER GREEK OR LATIN WORDS AND JUST CALL THEM POLYNOMIAL. SO TO WRITE DOWN ALL THESE FUNCTIONS I'VE BEEN ADDING AND MULTIPLYING STUFF. WHAT HAVEN'T I DONE YET? I HAVEN'T DIVIDED ANYTHING ELSE. SO LET'S GO ON AND TRY DIVISION. I GOT IT. THIS IS DANGER WHEN I WEAR DARK PANTS AND I COME OUT WEAR WHITE AFTER LECTURE. IS THE LAST CATEGORY TALK ABOUT HERE, CALLED RATIONAL

11 FUNCTIONS. AND THAT'S THE SIMPLY QUOTIENTS OF ANY TWO POLYNOMIALS. SO HERE'S A NICE SIMPLE EXAMPLE F-OF X-EQUALS THREE X-MINUS TWO DIVIDED BY EIGHT X-PLUS SEFPLT YOU CAN TAKE ANY TWO PAIRS OF POLYNOMIALS UNIT AND DIVIDE THEM. MAKE THE MOST GENERAL CLASS OF FUNCTIONS THAT WE CARE ABOUT. I DON'T HAVE TO TAKE ABOUT MUCH MORE. HERE'S WHY, ANY FUNCTION THAT YOU CAN WRITE DOWN USING, WELL, X-OF COURSE, AND CONSTANTS, ANY CONSTANT. AND YOU'RE ALLOWED TO DO ADDITION. SUBTRACTION, MULTIPLICATION, DIVISION, AND YOU'RE ALLOWED TO USE PARENTHESES. ANYTHING YOU WRITE, YOU CAN POSSIBLY WRITE DOWN THAT'S CORRECT. ANYTHING YOU CAN WRITE DOWN IS A RATIONAL FUNCTION. YOU CAN IF I HAD HE WILL WITH IT UNLESS IT LOOKS LIKE ONE POLYNOMIAL DIVIDED BY ANOTHER POLYNOMIAL NO MATTERS WHAT WEIRD STUFF IT IS, IS A RATIONAL FUNCTION, OOPS, AS LONG AS YOU DON'T DO SOMETHING BAD. DON'T 13 WHAT? DIVIDE BY ZERO. OKAY. AND THAT'S A PROBLEM. NOW IF YOU LOOK AT THIS THING, THERE'S A FUNCTION THAT'S NOT HERE. EIGHT X-PLUS SEVEN, THAT CAN BE ZERO SOMETIME WHEN EIGHT, YOU JUST KAEL EQUALITY THIS FUNCTION WHEN ZERO. MOST TIME YOU'RE NOT DIVIDING BY ZERO THAT'S FINE. BUT THIS IS NOT A GOOD RATIONAL NUMBER. SO YOU WON'T DO THAT. LET ME JUST DO AN EXAMPLE THERE. I'M NOT GOING TO PROVE THIS. I'M JUST GOING TO DO ONE EXAMPLE TO SHOW YOU WHY YOU SHOULD BELIEVE IT. SO LET ME TAKE RAY NOT TOO UNCOMPLICATED FUNCTION OF ADD TWO RATIONAL FUNCTIONS. AND I'LL GO A LITTLE BIT OF ALGEBRA TO CONVINCE YOU THAT I CAN MAKE IT LOOK LIKE THIS WITH ONE POLYNOMIAL DIVIDED BY ONE POLYNOMIAL. AND THE IDEA IS NOT HARD. PUT OVER COMMON DENOMINATOR. SO WHAT DO I HAVE TO MULTIPLY THIS BY TO GIVE DID A COMMON DENOMINATOR. SO I'M GOING TO MULTIPLY THIS BY X-PLUS FOUR. SO X-PLUS FOUR DIVIDE BY X-PLUS

12 FOUR. THIS IS OKAY AS LONG AS X-ACCIDENTS EQUAL MINUS FOUR AM I'M ALLOWED TO DO THAT. THAT'S PERFECTLY FINE. AND THIS GUY I'M GOING TO MULTIPLY BY X-PLUS TWO OVER X-PLUS TWO. AND FINALLY PUT IT ALL TOGETHER, AND YOU SHOULD BE VERY GLAD THAT I DIDN'T DO A COMPLICATED EXAMPLE. OKAY AND FINALLY YOU CAN MULTIPLY IT OUT BECAUSE THIS IS INDEED A POLYNOMIAL. MULTIPLY IT OUT, COLLECT ALL THE TERM AND YOU TPAOEUPLTD WILL I GET TWO X-SQUARED PLUS MINUS X-PLUS SEVEN, THAT'S WHAT THE NUMERATOR TURNS IN IT IF MULTIPLY IT OUT AND ADD IT UP. AND THE DENOMINATOR TURNS INTO X-SQUARED PLUS SIX X-PLUS ONE. AND THAT'S ONE POLYNOMIAL 14 DIVIDED BY ANOTHER POLYNOMIAL. SO IF YOU WANT ME TO I CAN DO ALL THE ALGEBRA BUT I DIVIDED TO LET I BELIEVE ME IF I MULTIPLY THOSE TWO AND MULTIPLY THOSE TWO AND ADDS THEM UP YOU GET THAT. YOU'LL GET SOME PRACTICE. BUT I'M HAPPY TOO IF YOU WANT ME TO DO IT. OKAY FINE. SO THAT'S WHY THIS STATEMENT IS EASY. IT SAYS NO MATTER WHAT YOU DO YOU GET A RATIONAL FUNCTION. IT MAYBE A LOT OF HANDWRITING BEFORE YOU FINALLY GET THERE BUT... OKAY LET'S SEE. I'M ALMOST DONE WITH EXAMPLES OF FUNCTIONS. NUMBER FIVE. POWER FUNCTIONS. AND THIS LOOKS LIKE A SPECIAL CASE OF POLE MOMENT KWRAL. X-TO SOME CONSTANT POWER. SO THIS CERTAINLY INCLUDES SIMPLE POLYNOMIAL. BUT IT'S ALSO MEANT IT INCLUDE THING LIKE X-TO THE ONE-HALF. OKAY. AND WHAT IS THAT? THAT'S A SKRAOFTS X-. THAT'S ANOTHER WORD FOR IT. AND IN FACT ONE THIRD, THAT'S AN EXPONENT THERE. THAT WOULD BE THE CUBED ROOT OF X. AND IN GENERAL IF R, I'M GOING IT DEFINE THIS MORE CAREFULLY LATER, IF R-IS A RATIONAL NUMBER LIKE SOME INTEGER M DIVIDED BY SOME OTHER INTEGER M THEN YOU CAN MAKE SENSE OUT OF IT, BY, FIRST YOU TAKE THE N-TH ROOT. AND THEN YOU MULTIPLY THAT BY TIME AND THAT'S A PERFECTLY GOOD FUNCTION TOO. SO FOR, AS

13 LONG AS R-IS RATIONAL WE UNDERSTAND WHAT IT MEANS. AND LATER, I'M GOING DID EXTEND THIS FUNCTION, POWER FUNCTION TO ALL REAL NUMBERS, RATIONAL OR TPHOFPLT BUT I'LL GOING TO HAVE TO WAIT UNTIL WE LEARN ABOUT LOGARITHM UPLZ LATER. IN WHATEVER CHAPTER THAT IS, CHAPTER FOUR HRAURTD. BUT THE POWER FUNCTION? A THAT COULD BE BASICALLY ANY REAL NUMBER. 15 AND IT HAS NO SQUARE ROOTS,... I LIED. ONE MORE FUNCTION THAT WE'RE GOING TO USE FREQUENTLY, THE ABSOLUTE VALUE. AND IT HAS DRAW TWO VERTICAL LINES AROUND IT. AND THERE ARE TWO CASES. X-X-IS GREATER THAN OR EQUAL TO ZERO, MINUS X-IS X-IS LESS THAN ZERO. AND IT HAS A NICE SIMPLE GRAPH. WHEN X-IS POSITIVE IT'S JUST A STRAIGHT LINE. AND IT'S JUST X. BUT WHEN IT'S NEGATIVE, IT LOOKS LIKE THIS. OKAY. TWO STRAIGHT LINES THAT HAS SLOPE PLUS ONE AND SLOPE, TWO FUNCTIONS WE'RE GOING TO CARE ABOUT. OKAY. NOW DOES ANYBODY, THIS IS A BREAK POINT, DOES ANYBODY HAVE ANY QUESTIONS ABOUT ALL THESE FUNCTIONS BEFORE I GOOD TO A SLIGHTLY DIFFERENT TOPIC. STUDENT: HOW DID YOU GET TWO X-SQUARED PLUS MINUS (INAUDIBLE). PROFESSOR: FINE. LET ME, THIS IS A TEST. DID I DO THE ALGEBRA CORRECTLY? WE'LL TRY. I'LL DO IT OVER HERE. X-PLUS ONE TIMES X-PLUS FOUR, I MULTIPLY, TAKE ALL THESE PAIRS. ALL THE COMBINATIONS AND MULTIPLY THEM. I GET X-SQUARED PLUS X-PLUS FOUR X-PLUS FOUR, THAT'S WHEN I GET FROM THIS ONE. I KEEP ONGOING. I TAKE ALL THE POSSIBLE COMBINATIONS I GET X-SQUARED PLUS THREE X-PLUS TWO X-PLUS THREE TIME TWO IS SIX, MAYBE I DID IT WRONG? LET'S SEE HERE. WROTE IT TOO FAST. SORRY. HOW ABOUT

14 THAT. THANK YOU FOR CHECK. YOUR HOMEWORK WILL BE TO CORRECT THE PROFESSOR. I DID IT TOO FAST. OOPS. 16 OKAY. THAT WAS A GOOD STOPPING POINTS. THAT WAS A TEST TO SEE IF YOU WERE PAYING ATTENTION. STUDENT: ON THAT ONE, THE SECOND ONE WITH THE CAN YOU EXPLAIN THAT ONE MORE TIME. PROFESSOR: SO WHAT, SO THIS FUNCTION YOU MADE UP BECAUSE IT, SOMETHING CALLED THE VOLUME STKOUFRPBT. IF YOU DON'T BUY VERY MUCH THEM. BUT THE POINTS IS IF YOU BUY A LITTLE BIT LESS OR A LITTLE BIT MORE THE PRICE IS ABOUT THE SAME. THIS FUNCTION IS CONTINUOUS. THOSE TWO STRAIGHT LINES MEET AT THAT POINT. AND THAT MAKE CERTAIN SENSE FROM WHERE THIS PROBLEM CAME FROM. BUT I COULD WRITE DOWN ANY OLD FUNCTION I WANT. IT STK-FRPBTS HAVE TO COMPLETE THERE. IT DOESN'T HAVE TO, YOU KNOW, BE CONTINUOUS. THERE WOULD BE A JUMP. THAT'S A PERFECTLY GOOD FUNCTION. STUDENT: YOU JUST SAYING SOMETIME IF YOU BUY, SAY I BUY MORE THAN 200 THE PRICE DIFFERENCE WOULD BE STPAERBL. PROFESSOR: THIS WOULD BE A RIP OFF. THIS IS NOT A BUSINESS FUNCTION BUT IT'S A OMEGA O* MATHEMATICAL FUNCTION. THIS MAKE SENSE FROM THE BUSINESS POINTS OF VIEW. I DOES THIS JUST TO MAKE SURE YOU REALIZE THAT NOT ALL FUNCTIONS ARE CONTINUOUS. NOT ALL FUNCTIONS YOU CAN DRAW WITHOUT LIFTING YOUR CHALK OFF THE BOARD AM LET ME WRITE DOWN ONE OTHER SIMPLE FUNCTION. SIGN, S-I-G-N. IT'S PLUS ONE IF X-IS POSITIVE. JUST CALL THE ZERO IF X-IS ZERO. AND MINUS ONE IF X IS LESS THAN ZERO WHAT'S ITS GRAPH LOOK LIKE? THIS IS REALLY SIMPLE FUNCTION. IT ONLY HAS THREE VALUES. OVER HERE IT'S EXACTLY THE ONE, UNTIL YOU GET TO THAT 17

15 POINTS, THEN IT'S EXACTLY ZERO THERE. AND THEN OVER HERE, IT'S EQUAL TO NEGATIVE ONE. THAT'S A FUNCTION. BUT OBVIOUSLY IT JUMPS THERE IN THE MIDDLE. THAT'S OKAY. FUNCTIONS STILL THERE. OKAY. SO I WANT TO SAY A FEW WORDS ABOUT SOMETHING THAT PEOPLE HAD ASKED ME BEFORE IN . WHICH IS THE USE OF CALCULATORS. SO IT'S OKAY, YOU CAN USE THEM AS MUCH AS YOU LIKE ON HOMEWORK. AND THEY ARE FORBIDDEN ON EXAMS. OKAY. AND THERE ARE SEVERAL REASONS FOR THIS. BUT I WANT TO TELL YOU THE MATHEMATICAL REASON? A MY PERSONAL RESEARCH IS HOW TO MAKE COMPUTERS EVALUATE FUNCTIONS FASTER AND BETTER. SO ONE OF THE RESULTS OF BEING AN EXPERT MANY THIS AREA IS THAT I KNOW THAT COMPUTERS CAN LIE. AND IN FACT THEY HAVE TO LIE. IN FACT ALL OF THEM HAVE TO. SO MOST OF THEIR LIES ARE LITTLE WHITE LIES. SO FOR EXAMPLE IF YOU TYPE ONE THIRD OF YOUR CALCULATOR, WHAT DO YOU THINK YOU'RE GOING TO GET? YOU'RE GOING IT GET.33 I WHOLE STRING OF THREES BUT EVENTUALLY IT HAS TO STOP. THE CALCULATOR OHM HAS ROOM FOR A FINITE NUMBER OF THREES. SO WE HAVE ONLY, MAYBE IF YOU'RE LUCKY, 16 DIGITS IN YOUR CALCULATOR. SO I WILL CALL THAT IF YOU LIKE, A LITTLE WHITE LIE BECAUSE MOST OF THE TIME THAT'S GOOD ENOUGH. BECAUSE IT USUALLY DOESN'T MATTER. OKAY. ON THE OTHER HAND, BUT SOMETIMES 18 THESE LITTLE WHITE LIES CAN ADD UP TO ONE REALLY BIG ONE. SO LET ME GIVE YOU AN EXAMPLE LET'S TRY PLOTTING A FUNCTION WHICH I THINK WORK FOR 24 PURPOSE AM Y-EQUALS X-MINUS TWO TO THE POWER OF 24.

16 LET JUST THINK ABOUT WHAT THAT FUNCTION SHOULD LOOK LIKE. OKAY. AND I'M ONLY GOING TO PLOT IT BETWEEN, ONE AND THREE. SO THINK ABOUT WHAT THAT SHOULD LOOK LIKE. SO X-MINUS TWO, SO X-IS GOING TO BE BETWEEN, ONE AND THREE. EVALUATE THIS FUNCTION. THAT'S WHERE YOU WANT TO SO WHERE IS X-MINUS TWO GOING TO BE? X-MINUS TWO IS GOING TO BE BETWEEN WHAT AND WHAT? SO IF X-NO BIG AND THAN THREE X-MINUS TWO CAN'T BE ANYTHING BIGGER THAN ONE I CAN SUBTRACT TWO FROM THIS INEQUALITY. SO X-MINUS TWO IS GOING TO BE BETWEEN PLUS WITHIN AND MINUS ONE. SO NOW I'M GOING IT TAKE THAT NUMBER BETWEEN MINUS ONE AND PLUS ONE, MULTIPLY IT BY ITSELF BETWEEN FOUR TIMES. SO HOW BIG CAN THAT BE? WHAT IS THE BIGGEST IT COULD POSSIBLY BE? I THINK I HEARD IT. ONE. CAN'T BE ANY BIGGER THAN ONE. HOW SMALL CAN IT BE? COULD IT BE NEGATIVE ONE? WHY NOT? BECAUSE I'M MULTIPLYING THIS THING AN EVEN NUMBER OF TIME SO ALL THE MINUS SIGNS CANCEL OUT. AND WHERE IS IT ZERO? WHEN X-EQUALS TWO. SO THAT'S OUR INTUITION AND WITH DID THAT WITHOUT USING A CALCULATOR. SO 19 LET ME DO IT HERE AM DO IT 21 AND THREE. SO THE FUNCTION IS GOING TO LIVE IN THIS LITTLE BOX. HERE'S TWO, IT'S GOING TO BE ZERO THERE. WHEN X-EQUALS ONE WHAT DOES THIS EQUAL? WHETHER I PLUG IN X-EQUALS ONE WHAT DO I GET? ONE. SO THE FUNCTION'S GOING TO BE UP HERE AM WHEN I PLUG IN X-EQUALS THREE WHETHER DO I GET? ONE. SO I THINK THE FUNCTION IS GOING TO BE TO THOSE THREE POINTS.

17 SO WHAT HAPPENS IF I START AT X-EQUALS ONE AND START MAKING X-BIGGER AND BIGGER, CLOSER TO TWO? WHAT'S A FUNCTION GOING TO DO? IS IT GOING TO GO UP OR DOWN? IT'S GOING TO GO DOWN. BECAUSE IF X-IS GETTING CLOSER TO TWO, -- SO THE FUNCTIONS GOING TO LOOK LIKE THIS. IT SHOULD BE PRETTY TINY IN THERE. SOMETHING LIKE THIS. SHOULD BE PERFECTLY SYMMETRIC SO. WE FIGURED THAT OUT WITHOUT USING YOUR CALCULATOR. OKAY. NOW YOU MIGHT SAY TO YOURSELF, I WOULD MUCH RATHER USE MY CALCULATOR TO FIGURE THAT OUT. SO LET'S DO THAT EXPERIMENT. I ACTUALLY DID IT AND THE PICTURES ON THE WEB PAGE. I AM TRY IT DRAW IT CAREFULLY HERE AM SO HERE'S HOW I'M GOING TO DO IT. BUT I'M GOING TO DO IT SLIGHTLY DIFFERENTLY. BECAUSE COMPUTER MIGHT DO IT. THIS IS A POLYNOMIAL. THAT I COULD MULTIPLY OUT. SO JUST LIKE I DID THE MULTIPLY OUT OF THIS AND GOT THIS I COULD MULTIPLY 20 THIS OUT. SO IF I DID THAT I'M NOT EXPECTINGING YOU TO DO THIS, THIS IS X-TO THE 24TH IN THERE. THEN THERE'S THE NEXT TURN HAPPENS TO BE 48 MINUS (ON BOARD). AND FIND THE ANSWER. SO I COULD ASK, THIS IS MATHEMATICALLY THE SAME PLOT. SO WHEN I DID WAS I WENT TO THE COMPUTER AND SAID PLEASE PLOT SO. LET ME DO IT. AND I'M GOING TO TRY TO DRAW IT AS CAREFULLY AS POSSIBLE. IT'S ON THE WEB PAGE TO MATCH WHAT THE COMPUTER DID. SO HERE'S THE PLOT OF WHAT THE COMPUTER DID. (ON BOARD). YEAH THAT'S ABOUT RIGHT. THAT'S LITERALLY THE PLOT THAT THE COMPUTER MADE AND THAT'S BECAUSE ALL THE LITTLE

18 WHITE LIES ADDED UP. AND SO YOU KNOW BECAUSE YOU KNOW SOME CALCULUS THAT THIS IS NONSENSE. IT SHOULD NOT LOOK LIKE THAT. AND IT'S ONLY BECAUSE AOUPBLGDZ WHAT IT OUGHT TO LOOK LIKE. AND SAY THIS IS WRONG AND YOU WOULD NOT TRUST YOUR KHRAEFT KHRAEFT MUCH I DO THIS EXAMPLE BECAUSE IT'S EASY TO DRAMA I WROTE OUT THE WHOLE POLYNOMIAL ON THE WEB PAGE. BUT IT'S A PERFECTLY REASONABLE THING TO ASK NOW, WHO CARES? SO I WANT TO EXPLAIN WHY YOU MIGHT ACTUALLY CARE. IS THIS IMPORTANT? AND I ACTUALLY GAVE YOU THE REASON LAST TIME. I'LL WRITE IT DOWN AGAIN. YES, BECAUSE YOUR LIFE DEPENDS ON IT. OKAY. 21 LET'S ME JUST GIVE THREE EXAMPLES. TWO OF THEM I GAVE LAST TIME. WILL THIS BUILDING FALL DOWN IN AN EARTHQUAKE? PEOPLE SOLVED EQUATIONS AND WE'RE BETTING OUR LIVERS THAT THE ANSWERS DO NOT LOOK LIKE THIS. SO THERE'S ONE EXAMPLE. THE NEXT ONE IS ALSO FROM LAST TIME. IS CLIMATE CHANGE REALLY CAUSED BY WHAT PEOPLE DO? THIS IS A BIG QUESTION. SO PEOPLE RUN THESE HUGE COMPUTER PROGRAMS AND THEY SAY IT'S BECAUSE WE BURN TOO MUCH ENERGY. DO YOU BELIEVE THE ANSWER? SO THIS IS A, SO THESE ARE CAUSED BY BURNING TOO MUCH FOSSIL FUEL FOR EXAMPLE. SO SINCE THIS IS SUCH AN IMPORTANT QUESTION THERE ARE TEAM OF PEOPLE AROUND THE WORLD TRYING IT MAKE SURE THESE PROGRAMS ARE RIGHT. THERE ARE A LOT OF REASONS, THERE ARE A LOT OF POSSIBLE ERRORS. NOT JUST THAT THE PICTURE I DREW OFTEN THERE. BUT THERE ARE

19 TEAMS OF PEOPLE TRYING IT MAKE SURE YOU BELIEVE THESE BECAUSE WE'RE BETTING THE PLANET ON IT AM TEAMS OF PEOPLE WORK ON THIS WORLDWIDE. I HAD THESE TWO EXAMPLES LAST TIME. I HAVE ONE MORE. THERE ARE A LOT OF DIFFERENT EXAMPLES. I CHOOSE THIS ONE BECAUSE IT HAS A BERKELEY CONNECTION. THREE, WILL OUR NUCLEAR BOMBS BELOW UP BY MISTAKE? WHATEVER YOUR OPINION ABOUT NUCLEAR WEAPONS IS PROBABLY DON'T WANT THEM TO BLOW 22 UP BY MISTAKE. SO WHY IS THIS A CALCULUS PROBLEM? IT'S BECAUSE SINCE I THINK IT WAS 1992, U.S. HAS OBSERVED THE INTERNATIONAL TEST BAN TREATY. OKAY WE DON'T BLOW UP BOMBS ANYMORE TO SEE IF THEY WORK. INSTEAD WE RELY ON AMONG OTHER THINGS, BIG COMPUTER PROGRAMS SOLVING, DOING CALCULUS PROBLEMS, SOLVE DIFFERENTIAL EQUATIONS TO PREDICT HOW THEY WILL WORK. SO ONE FOR EXAMPLE IS YOU'RE IN AN AIRPLANE AND THE AIRPLANE CRASHES, WILL THEY BLOW UP. THIS IS ONE EXAMPLE. SO WHY. AND THERE'S A, I HAVE A LINK ON THE WENT PAGE TO WHAT PEOPLE DO TO SOLVE THESE PROBLEMS. SO WHAT'S THE BERKELEY CONNECTION. AND HERE'S SOME HISTORY, BOTH PAST AND CURRENT THAT YOU MAY OR MAY NOT KNOW. SO THE NAME OF THE PROJECT, BUILT THE FIRST ATOMIC BOMB A LONG TIME AGO WAS THE MANHATTAN PROJECT. AND THE NAME OF THE PERSON WHO IS IN CHARGE OF IT, THE SCIENCE PART IT, WAS ROBERT OPPENHEIMER. AND HIS OFFICE WAS NEXT DOOR. USED TO BE NEXT DOOR. HE WAS A PHYSICS PROFESSOR HERE. SO THAT WAS AWHILE AGO. BUT THE

20 CONNECTION CONTINUES. AND I HAVE LINKS TO SHORT BIOGRAPHYS OF ALL THESE PEOPLE ON THE WEB PAGE. AND THE UNIVERSITY OF CALIFORNIA STILL, AND THEY HAVE FOR HALF A CENTURY, STILL MANAGES ALL THE WEAPONS LABS. THAT'S ONE OF THE JOBS OF THE UNIVERSITY 23 BESIDES ORGANIZING CLASSES LIKE THIS. AND I HAVE LINKS THERE. WHAT ELSE DID I WANT TO SAME SO ARE THERE ANY QUESTIONS? THAT'S ONE OF THE LITTLE DIGRESSIONS, JUST TO CONVINCE YOU THIS IS IMPORTANT. ONE MORE CONNECTION I WANT DOLLARS TO MENTION. SO THE QUESTION OF HOW COMPUTERS SHOULD DO ARITHMETIC TO MAKE SURE THIS KINDS OF STUFF DOESN'T HAPPEN WAS PREF CONTROVERSIAL FOR AWHILE UNTIL SOMEBODY FINALLY CAME UP WITH A 70 RULES. EVERYONE SHOULD DO IT THIS WAY AM AND I CONVINCED EVERYONE IN THE SILICON VALLEY. AND HE WRONG A NOBEL PRIZE IN COMPUTER SINCE FOR CONVINCING EVERYONE IN THE SILICON VALLEY TO ROUND OFF NUMBERS THIS WAY. HIS NAME IS, HE'S STILL HERE, THE TURG AWARD. T-U-R-I-N-G-AWARD, NOBEL PRIZE IN COMPUTER SCIENCE FOR FIGURING OUT HOW COMPUTERS SHOULD DO ARITHMETIC TO AT LEAST MAKE THE CHANCE OF THAT PICTURE AS SMALL AS POSSIBLE. CAN'T MAKE IT GO AWAY BUT AT LEAST AS SMALL AS POSSIBLE. WE'RE ALL THE WAY UP TO SECTION 0.3. ALGEBRA FUNCTIONS. OKAY. SO WE'RE GOING TO SPENDS DURING THE COURSE A LOT OF TIME COMBINING OLD FUNCTIONS TO GET NEW FUNCTIONS. NOW I'VE BEEN DOING THAT FOR THE LAST TWO LECTURES SO IT SHOULDN'T COME AS A SURPRISE. THE EXAMPLE I'VE BEEN USING OVER AND

21 OVER AGAIN IS FROM BUSINESS. X-IS THE NUMBER OF 24 ITEMS THAT YOU WANT IT MANUFACTURE AND SELLING. ONE FUNCTION THIS I'VE WRITTEN DOWN MANY TIMES IS THE COST TO MANUFACTURE THEM., SO C-FOR COST. R-FOR REVENUE. THAT'S THE REVENUE FROM SELLING THEM. BUT WHAT REALLY MATTERS IS OF COURSE THE PROFIT. AND THE WAY I'M GOING DO GET THE PROFIT IS BY COMBINATION THESE TWO FUNCTIONS. WHAT DO I DO WITH THE TWO FUNCTIONS TO GET THE PROFIT. I TAKE THE REVENUE AND I SUBTRACT THE COSTS. SO I TAKE TWO FUNCTIONS, SUBTRACT THEM AND GET A NEW FUNCTION. NO SURPRISE. JUST WRITING DOWN WHAT WE KNEW ALL ALONG. SO LET JUST PRACTICE A LITTLE BIT HERE. TWO FUNCTION F-OF X-AND G-OF X. AND SO IF I ADDS THEM, ANYBODY WHAT DO I GET? THREE X-MINUS FOUR. I CAN SUBTRACT THESE TWO FUNCTIONS. AND I GET X-PLUS TEN. I CAN MULTIPLY THEM. AND I CAN JUST LEAVE IT THIS WAY IF I WANT OR I CAN MULTIPLY THEM OUT. I GET TWO X-SQUARED PLUS THREE X-MINUS 14 X-. THAT'S MINUS 11 X-I HOPE MINUS 21. SO I CAN DO IT THAT WAY. AND I CAN DIVIDE THEM AND DO ALL THAT GOOD STUFF. SO I'M GETTING TIRED OF THIS. LET ME TRY SOMETHING ELSE. HOW ABOUT F-OF T-EQUALS THREE OVER T-AND G-OF Y-EQUALS Y-MINUS ONE OVER Y-PLUS ONE. AND I WANT TO KNOW WHAT IS F-OF X-PLUS 25 G-OF X. SO T-IS, I CAN DEFINE A FUNCTION WITH ANY

22 OLD NAME I WANT. SO NOW WHAT IS F-OF X. IT'S THREE DIVIDED BY X. AND WHAT IS G-OF X? I'M SORRY, IT'S Y-MINUS ONE OVER Y-PLUS ONE. EXCUSE ME. SO NOW G-OF X-IS X-MINUS ONE OVER X-PLUS ONE. AND I CAN GO AHEAD AND ADD THESE UP. I CAN LEAVE IT THERE IF I WANTED. ALL RIGHT. (ON BOARD). AND TO BE SIMPLIFIED. SO MY POINT HERE WAS SIMPLY WHEN I DEFINE A FUNCTION I CAN HAVE T-THERE OR X-THERE OR Y-THERE. THAT'S OKAY. I CAN CHANGE IT LATER. SO LAST TIME JUST TO REVIEW A BIT WE TALKED ABOUT DOMAINSS OF FUNCTIONS WHICH IS WHERE IT WAS OKAY TO EVALUATE THEM. SO LET JUST ASK OURSELVES THAT QUESTION HERE. SO THIS EXAMPLE, WHAT'S THE DOMAIN OF THAT FUNCTION? WITH RESPECT I ALLOWED TO EVALUATE, LET ME CALL IT F-OF X, WHERE IS IT OKAY TO VALUE F-OF X? AS LONG AS X-IS NOT EQUAL TO ZERO BECAUSE THAT'S THE ONLY THING THAT CAN GO WRONG THERE WHAT IS THE DOMAIN OF G-OF X. WITH RESPECT I NOT ALLOWED TO EVALUATE IT. AS LONG AS X-IS NOT EQUAL TO SOMETHING. WHAT CAN GO WRONG HERE? IT'S WHEN I WOULD DIVIDE BY ZERO SO X-PLUS ONE EQUALS ZERO. SO AS LONG AS X-IS NOT EQUAL TO MINUS ONE I'M OKAY. STUDENT: IF F OF X-IS TWO X-PLUS THREE. 26 PROFESSOR: I'M SORRY. I'M TRYING TO, THIS ALL BELONGS TOGETHER HERE. SO I'M DOING THIS F-AND THAT G-. TRYING TO SAVE BOARD SPACE. REAL ESTATE IS EXPENSIVE IN BERKELEY. SO TRYING TO SAVE BOARD SPACE. IS IT CLEAR NOW? AND THEN FINALLY I'VE ADDED THEM, WHAT IS THE DOMAIN OF F-OF X-PLUS G-OF X? SO NOW THIS GUY?

23 WHERE AM I NOT ALLOWED TO EVALUATE? EVERYWHERE, X-DOES NOT EQUAL ZERO AND X-DOES NOT EQUAL MINUS ONE. SO WHEN YOU ADD, WHEN YOU COMBINING FUNCTIONS YOU HAVE TO THINK THEY'RE DOMAINS. YOU HAVE TO WATCH THAT. OKAY. SO THERE'S ONE OTHER IMPORTANT WAY TO COMBINE FUNCTIONS TO DO CREATE NEW FUNCTIONS OUT OF OLD FUNCTIONS. AND THAT'S CALLED COMPOSITION. COMPOSITION OF FUNCTIONS. WHICH IS SUBSTITUTING, (ON BOARD). SO WAY WE WOULD WRITE THAT IS COMPOSITION OF G-WITH F-WRITTEN F-OF G-OF X. SO TAKE G-EVALUATE IT, PLUG IT IN THERE. SO HERE'S A LITTLE EXAMPLE. F-OF X-EQUALS X-SQUARED PLUS THREE X-PLUS ONE, AND G-OF X-EQUALS X-MINUS SIX. AND SO I WANT TO COMPUTE F-OF G-OF X, SO WHAT, THAT SO THAT'S GOING TO BE THE SAME OF F-OF X-MINUS SIX BECAUSE THAT'S WHAT G-OF X-IS. AND THAT'S GOING TO BE X-MINUS SIX SQUARED PLUS THREE TIMES X-MINUS SIX PLUS ONE. (ON BOARD). X-SQUARED IS GOING TO WRAOEUPB WRAO*EUT THIS AS (ON BOARD). IF YOU MULTIPLY IT OUT. SO THAT'S WHAT COMPOSITION IS ABOUT. LET ME DO -- TIRED OF MONEY. ONLY BRIEFLY. SO HERE'S THE EXAMPLE. IN A LAKE, WE HAVE BUNCH OF FISH CALL BASS. THE BASS ARE HUNGRY. THEY EAT THE MINUTE OTHERWISE. THE M I-N-N-O 27 W-S-THE THE M I-N-N-O W-S, ARE HUNGRY, THEY EAT THE PLANK TON. SUPPOSE I KNOW HOW MANY MINUTE OZICK HAVE DEPENDING ON HOW MUCH PLANNING TON I HAVE. SO LET LET M OF P-BEING HOW MANY M I-N-N-O W-S-IF I KNOW HOW MUCH PLANK TON I HAVE. AND I'LL JUST MAKE THIS UP. SO THIS IS NUMBER OF M I-N-N-O W-S-AS A FUNCTION OF THE NUMBER OF PLANK TON. AND LET'S SUPPOSE I ALSO KNOW HOW MUCH THE BASS NEED TO EAT. SO SUPPOSE I ALSO KNOW THE NUMBER OF BASS, B, AS A FUNCTION OF HOW MANY M I-N-N-O W-S-THERE ARE. AND I'LL JUST MAKE THIS UP. SO THAT'S NUMBER OF BASS AS A FUNCTION OF THE

24 NUMBER OF M I-N-N-O W-S-. SO THE NATURAL BIOLOGICAL QUESTION IS I KNOW HOW MUCH PLANK TON THERE IS. HOW MANY BASS ARE THERE? SO GIVEN P-PLANK TON, HOW MANY BASS ARE THERE? OKAY. SO AND THE ANSWER IS GOING TO BE BY COMPOSING TWO FUNCTIONS THAT ARE ON THE BOARD. WHAT WOULD I DO? SO I HAVE ONE FORMULA THAT TELLS ME HOW MANY PLANK TON AND HOW MANY M O N-N-O W-S. AND ANOTHER ONE THAT SAYS HOW MANY M O N-N-O W-S- A HOW MANY BASS. STUDENT: THE FUNCTION OF BASS INSIDE OF THE. PROFESSOR: BASS OF SOMETHING. STUDENT: INSIDE OF N- PROFESSOR: BASS OF M O N-N-O W-S-OF PLANK TON, WHICH IS 50 BLOUSE THE SQUARE ROOT OF FOUR P-PLUS THREE OVER 150. SO THAT'S WHAT THAT IS. I WOULDN'T GO OUT AND BUY BAIT BASED ON THIS COMPUTATION. BUT THAT'S THE IDEA. OKAY. ARE THERE ANY QUESTIONS? THAT SECTION, I'M GOING TO START ON SECTION 0.4, 28 BECAUSE I HOPE TO GET TO 0.6 TODAY WHICH I'M NOT GOING TO DO. BUT LET'S GO ON. LET ME SAY A FEW WORDS. I HOPE IN REVIEW ABOUT FINDING ZEROS. IF YOU HAVE SOME NUMBER AND IT HAPPENS TO SATISFY F-OF X-EQUALS ZERO, THEN X-IS CALLED A ZERO OF THE FUNCTION F. SO PERFECTLY NATURAL THING. SO IF I DRAW A PICTURES OF A GRAPH, SO THERE'S A FUNCTION WHICH I JUST MADE UP. AND BASED ON THIS FUNCTION IT HAS EXACTLY THREE ZEROS. THERE'S ONE, THERE'S ONE, AND THERE'S ONE. SO THERE'S THE GEOMETRY, THE PICTURE THAT GOES WITH THE ALGEBRA. SO LET JUST REVIEW (ON BOARD). FOR THE SIMPLEST FUNCTIONS. SO THERE IS ANY LINEAR FUNCTION OF A STRAIGHT LINE. IT HAS EXACTLY TEN IF IT HAS ANY AT ALL. SO WHERE IS THAT ZERO? YOU GET IT BY PLUGGING IN. AND SO I SOLVE

25 FOR X. AND THERE I AM. (ON BOARD). IS THAT ALWAYS TRUE? DID I MAKE ANY ASSUMPTIONS ON MY WAY TO FINDING THIS POINT THERE? STUDENT: (INAUDIBLE). PROFESSOR: SO WHAT IF MY STRAIGHT LINE LOOKS LIKE THIS? HOW WOULD I TELL BY JUST DOING THE ALGEBRA. HOW WOULD I TELL THE ALGEBRA IF THERE'S NO ZERO. WHAT PROPERTIES WOULD M AND E-HAVE? STUDENT: M WOULD BE A ZERO AND -- PROFESSOR: M WOULD BE ZERO, THAT'S FLAT LINE, SLOPE ZERO AND -- WHAT IF N-WERE ZERO AND B-WERE ZERO? EVERYTHING'S A ZERO. SO NOT SUCH AN INTERESTING FUNCTION BUT A POSSIBLE. SO LET'S GO ON TO QUADRATICS. QUAD RAT EUBGZ. Y-EQUALS A-X SQUARED PLUS B-X-PLUS C. SO A FORMULA THAT SHOULD BE BURN IN YOUR BRAIN FROM 29 HIGH SCHOOL IS THE QUADRATIC FORMULA. SO WHERE ARE THE ZEROS? WHO REMEMBERS? NEGATIVE B-PLUS OR MINUS THE SQUARE ROOT OF B-SQUARED MINUS (ON BOARD). MUSIC TO MY EARS AM GOOD TO REMEMBER THAT. SO LET ME JUST DRAW THE ALGEBRA. SO THE ALGEBRA AND THE GEOMETRY. SO FIRST START OFF WITH, THIS IS REALLY QUADRATIC. SO A-BETTER NOT BE ZERO OTHERWISE THIS DOESN'T MAKE SENSE. SO WHAT IF B-SQUARED MINUS FOUR A-C-IS GREATER THAN ZERO? HOW MANY ROOTS DO I HAVE? ZEROS DO I HAVE? WHAT DOES THE GRAPH OF THE FUNCTION LOOK LIKE? IN THIS CASE? SO IF THIS IS A POSITIVE NUMBER I TAKE THE SQUARE ROOTS AND DO PLUS AND MINUS IT HOW MANY DIFFERENT VALUES DO I GET? TWO. SO THAT MEANS THAT THE GRAPH OF THE QUADRATIC FUNCTIONS IS A PARABOLA AND THE PARABOLA LOOK LIKE THIS. AND THERE ARE THE TWO ZEROS. IS THERE ANY OTHER WAY THE PARABOLA COULD LIKE. IT COULD LOOK LIKE THIS. HOW DO YOU TELL THE DIFFERENCE BETWEEN THIS CASE AND THAT CASE? BASED ON THE A-B-C? THIS IS A-BIGGER THAN ZERO. AND THIS IS A-LESS THAN ZERO, GOES DOWN. ALL RIGHT, WHAT ABOUT I'M ALMOST

26 DOWN HERE. B-SQUARED MINUS FOUR A-C-IS EQUAL TO ZERO? HOW MANY ROOTS DO I HAVE? THIS QUANTITY IS NOW ZERO. HOW MANY DIFFERENT ROOTS DO I GET? ONE. WHAT ARE THE PICTURES? WHAT'S, SO WHAT'S THE PICTURE OF PARABOLA LOOK LIKE NOW? BARELY TOUCHES THERE. IT LOOKS LIKE OR LIKE THIS. SO THAT'S CASE ONE. CASE TWO, AND THE LAST CASE WHICH I'LL CLOSE WITH, IT'S LESS THAN ZERO. SO IT'S A NEGATIVE NUMBER. ARE THERE ANY ROOTS? REAL ROOTS? OKAY. THERE ARE NO ROOTS. NO ZERO, SORRY. THE TWO WORDS ARE SYNONOMOUS. SO 30 WHAT DO THE PICTURES LOOK LIKE. THE PARABOLA IS ALL THE WAY UP THERE. NEVER CROSSES OR ALL THE WAY DOWN THERE. SO THOSE ARE THE PICTURES THAT GO WITH THE ALGEBRA. SEE YOU NEXT TIME.

27