DIFFERENTIATE SOMETHING AT THE VERY BEGINNING THE COURSE I'LL ADD YOU QUESTIONS USING THEM. BUT PARTICULAR QUESTIONS AS YOU'LL SEE


 Emery Cooper
 7 months ago
 Views:
Transcription
1 1 MATH 16A LECTURE. OCTOBER 28, PROFESSOR: SO LET ME START WITH SOMETHING I'M SURE YOU ALL WANT TO HEAR ABOUT WHICH IS THE MIDTERM. THE NEXT MIDTERM. IT'S COMING UP, NOT THIS WEEK BUT THE NEXT WEEK. IT WILL COVER, SECTIONS, WELL ANYTHING IS FAIR GAME. IF I TOLD YOU HOW TO DIFFERENTIATE SOMETHING AT THE VERY BEGINNING THE COURSE I'LL ADD THAT. BUT IT'S BASICALLY GOING TO COVER STUFF THAT WE DID SINCE THE LAST MIDTERM. SECTIONS 1.7 TO 4.3. THERE ARE SAMPLE MIDTERM POSTED ALREADY. IN CLASS AS USUAL. SAME RULES. ALL LIKE YOU HAD LAST TIME. ANY QUESTIONS ABOUT THAT? STUDENT: DID YOU SAY WE MIGHT HAVE QUESTIONS FROM THE BEGINNING OF CLASS. PROFESSOR: IF I TAUGHT YOU SOMETHING, BASIC RULES OF ALGEBRA OR CALCULUS THIS SHOWED UP IN SECTIONS UP TO 1.6 I'M ALLOWED TO ASK YOU QUESTIONS USING THEM. BUT PARTICULAR QUESTIONS AS YOU'LL SEE FROM THE SAMPLE MIDTERM COME FROM THOSE SECTIONS. STUDENT: DID THE GRADE BREAKDOWN CHANGE FROM THE FIRST MIDTERM. PROFESSOR: THE TARGET IS STILL THE CURVE THAT I HAD ON WEB PAGES UNTIL WE FIND OUT HOW PEOPLE DO. I DON'T KNOW IF I NEED TO RECAP LATE. SO THE FIRST QUESTION, QUESTION FIVE I EXPERIMENTED WITH. MAYBE A LITTLE HARDER THAN LAST TIME.  SO LET ME REVIEW A TINY BIT. SO I CAN KEEP GOING. WE LOOKED AT EXPONENTIAL FUNCTIONS. AND I THINK I TOLD YOU THAT IF YOU PLOT THEM THEY ALWAYS LOOK THE SAME. AS LONG AS THEY DON'T BOTHER TO LABEL THE AXIS, THIS IS X, THIS IS FUNCTION BTO THE XWILL BE SOME FUNCTION BIGGER THAN 2
2 ONE. AND I'M PARTICULARLY INTERESTED IN THE SLOPE OF THE TANGENT LINE RIGHT THERE BECAUSE THEY ALWAYS GO THROUGH.0 COMMA ONE. THAT'S IF YOU TAKE  ALWAYS GO THROUGH THAT POINT. THIS WAS SORT OF GENERIC PICTURE OF WHAT ANY EXPONENTIAL FUNCTION LOOKED LIKE. LAST TIME I SHOWED THAT IF YOU TAKE THIS EXPONTENTIAL FUNCTION, AND YOU DIFFERENTIATE IT, THERE WAS ONE FORMULA, AND IT WAS VERY EASY, YOU GOT THE SAME FUNCTION BACK MULTIPLIED BY THE SLOPE OF THAT LINE. THAT WAS GEOMETRICALLY WHAT WE DID LAST TIME. WE DIFFERENTIATE. GET THE SAME FUNCTION BACK. POSSIBLY MULTIPLIED BY ONE NUMBER, WHICH IS THAT SLOPE. THEN WE CHOSE ONE MAGIC NUMBER, E, WHICH GOES ON FOREVER AT SOME IRRATIONAL NUMBER. TO MAKE THE SLOPE ONE. SO FOR A PARTICULAR VALUE OF E, WE GOT THAT DDXTO THE EXWAS ETO THE X. IT CAME BACK BECAUSE THE SLOPE AT THIS POINT WAS EQUAL TO ONE. OKAY. AND THAT SUMMARIZES A WHOLE LOT OF WHAT WE DID LAST TIME. THAT'S THE PICTURE FOR BTO THE XWHEN BIS GREATER THAN ONE. IF YOU KEEP MULTIPLIED BY B IT KEEPS GET BEING BIGGER. WE ALSO DREW THIS PICTURE. AND THIS IS BTO THE XWHEN BWAS LESS THAN ONE. BECAUSE IF YOU KEEP MULTIPLYING NUMBER LESS THAN ONE BY ITSELF IT KEEPS GETTING SMALLER. THIS IS THE SAME PICTURE BACKWARDS. IT STILL GOES THROUGH ZERO COMMA ONE. THAT WAS A SUMMARY. AND NOW I WANT TO SPEND MORE TIME IF WE BELIEVE ALL THIS TO HOW TO DIFFERENTIATE FUNCTIONS THAT LOOK LIKE ETO THE XTHAT ARE VARIATIONS ON ETO THE X. USE ONE MAGIC NUMBER SO LET'S GO ON TO SECTION 4.3. WHICH WE STARTED QUICKLY LAST TIME. SO RULES, ALL BASED ON 3 THE CHAIN RULE FOR DIFFERENTIATING FUNCTIONS THAT LOOK LIKE ETO THE X. SO LET'S START BY USING THE CHAIN RULE TO DIFFERENTIATE
3 ETO THE KTIMES XWHERE KIS GOING TO BE A CONSTANT. I THINK I DID THIS LAST TIME. BUT LET ME DO IT AGAIN. TO USE THE CHAIN RULE I'M GOING TO WRITE THIS AS FOF GOF XWHERE FOF X, THE FIRST THING I DO IS TAKE XAND MULTIPLY IT BY K. THAT'S THE FUNCTION G AND THEN I TAKE THAT AND I EXPONENTIATE IT. SO IF I DO FOF GOF XI GET ETO THE KX. IF I WANT TO USE THE CHAIN RULE, DDXETO THE KX I KNOW THE CHAIN RULE TELLS ME THIS. (ON BOARD). I BETTER FIGURE OUT WHAT GPRIME AND FPRIME ARE. FPRIME IS MEANT TO BE PARTICULARLY EASY. THAT FUNCTION DOESN'T CHANGE WHEN DIFFERENTIATE IT, THAT'S WHY WE CHOSE ETHE WAY WE DID. GPRIME IS JUST A CONSTANT TIMES X. SO IF I PLUG THAT ALL IN I GET THAT THIS IS JUST ETO THE KXTIMES K. (ON BOARD). THERE'S K. AND THERE'S ETO THE KX. SO DIFFERENTIATING A FUNCTION WHERE YOU TOOK A CONSTANT UP THERE, YOU BRING IT DOWN TO THE FRONT. TO SUMMARIZE, (ON BOARD). JUST BRING THE CONSTANTS DOWN TO THE FRONT. AWFULLY EASY FUNCTION TO DIFFERENTIATE. AND SO HERE, LET ME DO ONE EXAMPLE WE DID LAST TIME. DDXTO ETO THE MINUS XTHAT APPLIES HERE BECAUSE I HAVE TO THINK THAT MINUS XIS, OF COURSE, NEGATIVE ONE TIMES X. AND APPLY THE RULE AND YOU GET NEGATIVE ONE TIMES ETO THE MINUS X. SO THERE'S THE EXAMPLE. AND IT'S NOT MUCH FARTHER TO WRITE DOWN THE GENERAL CHAIN RULE FOR TAKING DDXTHE DERIVATIVE OF ETO THE ANY OLD FUNCTION GOF X. WE'RE GOING TO BE DOING THAT. 4 THAT TELLS US IT'S THE SAME RULE AS UP THERE. FPRIME OF GOF X. THERE'S THE FPRIME OF GOF XTIMES GPRIME. SO DIFFERENTIATE A FUNCTION ETO ANYTHING, YOU'LL LEAVE IT ALONE AND COPY IT, BRING DOWN THE EXPONENTIAL AND DIFFERENTIATE IT. AND THAT'S JUST THE
4 CHAIN RULE TOO. IT'S EXACTLY THE SAME CHAIN RULE THAT'S UP THERE. HERE'S FOF GOF X(ON BOARD). I LEFT GALL ALONE. STUDENT: SO WHAT EQUATION IS F OF X PROFESSOR: FOF XIS A FUNCTION. AND I CHOSEN IT TO BE THE SAME AS IT WAS UP THERE. FOF XIS JUST ETO THE X. IT'S FOR THIS PARTICULAR EXAMPLE. AND SO ITS DERIVATIVE IS UNCHANGED. WORLD'S SIMPLEST FUNCTION TO DIFFERENTIATE. ANY FUNCTION YOU CARE TO N. GOF X. USE THIS WHY GIS SOME OTHER FUNCTION. LET ME ILLUSTRATE. ETO THE POWER XSQUARED PLUS ONE. SO GOF XHERE'S GOF X. AND SO THIS IS GOING TO BE JUST COPY IT, THAT PART DOESN'T CHANGE. AND THEN I JUST HAVE TO MULTIPLY BY THE DERIVATIVE OF XSQUARED PLUS ONE. AND MULTIPLY BY TWO X. SO THAT'S WHAT THE RULE TELLS US. LET ME DO ANOTHER ONE LIKE THAT. (ON BOARD). THERE'S THE BIG EXPONENTS. AND TO DIFFERENTIATE IT WE COPY THAT WITHOUT ANY CHANGE. THAT PART OF THE FUNCTION STAYS THE SAME AND THEN I HAVE TO MULTIPLY BY THE DERIVATIVE OF WHATEVER'S UP IN THE EXPONENT AND NOW WE USE WHAT WE KNOW ABOUT THIS, THAT'S GOING TO BE 27 XCUBED MINUS ONE OVER XSQUARED, PLUS ONE OVER XSQUARED (ON BOARD). SO GOF XWAS THIS GUY UP HERE. SO THERE'S, THAT WAS G OF X. SO LET ME JUST WRITE DOWN THE CHAIN RULE AGAIN, JUST BECAUSE USE TWO DIFFERENT NOTATIONS 5 FOR IT IN THE PAST. STUDENT: WHY IS IT 27. PROFESSOR: DID I MULTIPLY IT BY  GOOD FOR CATCHING THAT. THANK YOU. (ON BOARD). MAKING SURE YOU WERE AWAKE. THANK YOU. (INAUDIBLE QUESTION FROM STUDENT.) CAN EVERYONE DIFFERENTIATE THREE TIME XCUBED NOW, GOOD, I HOPE, INCLUDING ME. LET ME
5 WRITE DOWN THE CHAIN RULE AGAIN USING SLIGHTLY DIFFERENT NOTATION. IT'S THE SAME THING. I WANT TO ILLUSTRATE WE WRITE IT DOWN BECAUSE WE'VE USEDS THIS NOTATION BEFORE, IF UIS A FUNCTION OF X, TAKE DDXOF ETO THE U, AND IT'S JUST GOING TO BE ETO THE UTIMES DUDX. SAME RULE JUST SLIGHTLY DIFFERENT NOTATION USING UTO REPRESENT THAT FUNCTION. THE BOOK'S USED IT BEFORE. I JUST THOUGHT I'D WRITE IT DOWN AGAIN, NO NEW IDEAS. EVERYBODY KNOWS ABOUT THE MIDTERM, SO I CAN ERASE THAT? SINCE WE'VE SHOWN THAT DDXETO THE XQ(ON BOARD) THERE'S ANOTHER WAY TO SAY THAT. ETO THE KX, THAT FUNCTION SATISFIES SOMETHING CALLED A DIFFERENTIAL EQUATION. (ON BOARD). NEW IDEA. AN EQUATION WHERE THE UNKNOWN YOU WANT TO SOLVE FOR IS A FUNCTION. SO THIS IS AN EQUATION. IT'S SATISFIED BY  STUDENT: SO IS THE DERIVATIVE OF ETO THE KXETO  STILL NEED (INAUDIBLE). PROFESSOR: WHERE DID I, THANK YOU FOR PAYING ATTENTION. THANK YOU FOR CATCHING THAT TYPO. I SHOULD SLOW DOWN. ANY YES. OTHER TYPOS. I LEFT THE KOUT OF THE EXPONENTS. WHENEVER YOU DIFFERENTIATE YOU ALWAYS COPY IT OVER. AND THEN YOU PULL OUT THE 6 DERIVATIVE. THANK YOU. SO HERE, I'VE WRITTEN DOWN AN EQUATION. IT'S SATISFIED BY SOME FUNCTION, YOF X. WE DON'T KNOW THE FUNCTION Y. SO HERE'S AN EQUATION TO SOLVE NOT FOR UNKNOWN AVAILABLE. WE DID THAT ALL THE TIME BUT FOR AN UNKNOWN FUNCTION. YOF X. SO I'M TELLING YOU THAT THIS AND WRITING DOWN THIS DIFFERENTIAL EQUATION BECAUSE IT COMES UP ALL THE TIME WHEN YOU WRITE DOWN EQUATIONS TO UNDERSTAND HOW THE WORLD WORKS AND THE ANSWER IS RIGHT THERE. THE ANSWER IS YOF XIS ETO THE
6 KTIMES XTIMES ANY CONSTANTS. WHEN I SAY WHAT IS ANY POSSIBLE FUNCTION THAT SATISFIES THAT EQUATION, THE ANSWER IS ETO THE KXBUT YOU GET TO MULTIPLY BY ANY CONSTANT YOU LIKE AND IT STILL WORKS. LET ME TRY THAT. LET'S VERIFY THAT. LET ME VERIFY. TAKE DDXTIMES SOME CONSTANT ETO THE KX. I CAN FACTOR CONSTANT IF I TAKE THE DERIVATIVE. SO I HAVE TO DIFFERENTIATE THAT PART. THAT'S HOW CONSTANTS WORK. NOW I GET CONSTANT TIMES KETO THE KX. IF I JUST REVERSE KAND C, I CAN MULTIPLY NUMBERS IN ANY ORDER I LIKE, IT'S THE SAME THING, I SEE I GET KTIMES YOF X. AND THAT IS WHAT I WANTED TO SHOW. I WANTED TO SHOW DDXOF YOF XIS THIS CONSTANT KTIMES YOF X. SO IT WORKS. SO THAT'S THE FUNCTION. SO FOR EXAMPLE, I CAN TAKE ANY OLD 37 ETO THE KXSATISFIES THE EQUATION IN THE BOX. SO HOW DO WE PICK C? YOU NEED SOME, YOUR PROBLEM YOU'RE TRYING TO SOLVE HAS TO HAVE SOME MORE INFORMATION. SO FOR EXAMPLE, FIND A SOLUTION TO THAT SAME EQUATION, I'LL WRITE IT DOWN AGAIN, TO DDXYOF XEQUALS KTIMES YOF XSATISFYING, I CAN GIVE YOU 7 MORE INFORMATION, YOF ZERO EQUALS FOUR. OKAY. THERE'S A LITTLE BIT OF EXTRA INFORMATION. IN ADDITION TO THAT AND I WANT TO SOLVE THIS BUT I'LL TELLING YOU THE FUNCTION, THE ANSWER HAS TO GO THROUGH THAT PARTICULAR POINT. SO HOW DO WE DO THAT? WE KNOW THAT YOF XHAS TO EQUAL SOME CONSTANTS. TIMES ETO THE KXFOR SOME CONSTANTS CWHICH WE DON'T KNOW YET. HOW DO I SOLVE FOR IT? PLUG IN XEQUALS ZERO. I GET THAT YOF ZERO WHICH HAD BETTER EQUAL FOUR, IS GOING TO BE THIS CONSTANT TIMES ETO THE KTIMES ZERO, AND SO, AND THAT'S CONSTANT ETO THE ZERO, AND ETO THE ZERO IS? ONE. SO IT'S C. THAT HAS TO EQUAL, THEREFORE, FOUR
7 BECAUSE THAT'S THE EXTRA INFORMATION I GAVE IN THE PROBLEM. SO THE ANSWER IS YOF XIS FOUR TIME ETO THE KX. IF YOU GIVE ME ONE POINT ON THE CURVE I CAN FIGURE OUT C STUDENT: WHAT ABOUT K PROFESSOR: SO KWAS GIVEN TO ME AS PART OF PROBLEM. IF THAT HAD BEEN SEVEN, I JUST DO IT FOR A GENERAL K. SO FOR THE PROBLEM TO WORK YOU HAVE TO HAVE POINT ON THE CURVE, K, WHICH IS CONSTANT. STUDENT: WHAT IF XWASN'T THERE. PROFESSOR: LET'S DO THAT. LET'S SEE, LET'S SEE HOW THAT WORKS. I MIGHT NEED SOMETHING FROM SECTION 4.4. LET'S SOLVE THE SAME THING. DDXYOF XEQUALS, I THINK CONSTANT MAYBE SPECIFIC SEVEN TIMES YOF X. AND YOF, SAY ONE EQUALS FOUR JUST TO BE SPECIFIC. SHOULD I TRY THAT ONE? THAT I KNOW THAT YOF XIS GOING TO BE SOME CONSTANTS TIMES ETO THE POWER SEVEN X. I KNOW THAT FOR SURE. WHAT I HAVE TO PICK IS A CONSTANT C. SO I PLUG IN AND GET 8 YOF ONE, THAT BETTER BE FOUR, AND THAT'S CTIMES ETO THE SEVEN TIMES ONE. I THINK YOU CAN SOLVE THAT FOR C. SO SOLVE THAT FOR C. AND WHAT DO I GET? I GET THAT CIS GOING TO BE FOUR DIVIDED BY ETO THE SEVEN TIMES ONE OR FOUR DIVIDED BY ETO THE SEVENTH POWER. SOME NUMBER WE CAN COMPUTE. SO THERE'S HOW I WOULD SOLVE THAT ONE. WE CAN SOLVE THOSE TOTALLY GENERALLY. THEY COME UP A LOT IN PRACTICE. STUDENT: I DON'T SEE WHERE YOU GET IN BOTH OF THOSE THE POWER OF FOUR. WHERE DID THE FOUR COME? PROFESSOR: THIS IS JUST EXTRA INFORMATION THAT WHEREVER THE PROBLEM COMES FROM YOU NEED TO KNOW THIS. THIS SAYS WHEN YOU DRAW THE CURVE BEING SOMEBODY HAS TO GIVE YOU A POINT ON IT, AND
8 I'M TELLING YOU IT HAS TO GO THROUGH THE POINT ONE WHICH IS FOUR. THE STATEMENT OF THE PROBLEM HAS TO TELL YOU THAT OTHERWISE YOU CAN'T DO IT. YOU HAVE TO KNOW ONE POINT ON THE CURVE AND THEN YOU CAN FIGURE OUT THIS CONSTANT. SO LET ME, OKAY, SO I'VE SAID ALL THAT. WHAT I WANT TO DO NOW, IT'S JUST A COUPLE OF LINES, IS TELL YOU WHY THAT'S THE ONLY ANSWER. SO THAT THE SOLUTION IS UNIQUE. THERE'S ONLY ONE ANSWER. AND IT'S GOING TO BE A CONSTANT TIMES ETO THE KX. LET ME JUST, SO WHY AM I DOING THIS? NOT EVERY EQUATION THAT YOU CAN WRITE DOWN HAS ONE SOLUTION. FOR EXAMPLE, HOW ABOUT XSQUARED EQUALS TWO. THERE'S AN EQUATION WHERE YOU DON'T KNOW THE X. THERE'S TWO SOLUTIONS, PLUS THE SQUARE ROOT OF TWO OR MINUS THE SQUARE ROOT OF TWO. BUT THAT PARTICULAR EQUATION I'M GOING TO PROVE THAT YOU NOW THERE'S 9 EXACTLY ONE SOLUTION. YOU DON'T HAVE TO WORRY ABOUT ANY OTHER ONE. BUT THIS ONE HAS, ONLY HAS A SOLUTION YOF XIS SOME CONSTANT WHICH WE STILL HAVE TO PICK, TIMES ETO THE KX. THAT'S THE ONLY SOLUTION. THAT'S WHAT I WANT TO SHOW YOU. THERE CAN NOT BE ANY OTHER FUNNY STUFF WHICH CAN HAPPEN. SO HERE, LET ME TRY TO PROVE THAT NOW. IT'S VERY EASY. SO LET'S YOF XBE SOME SOLUTION TO THIS DIFFERENTIAL EQUATION AND I WANT TO PROVE THERE'S ONLY THAT ONE WAY TO GET THE ANSWER. HERE'S THE IDEA. I'M GOING TO LET FOF XBE A NEW FUNCTION WHICH I'M GOING TO GET FROM THIS SOLUTION, MULTIPLIED BY ETO THE MINUS KX. I'M GOING TO PROVE TO YOU NOW THAT THIS FUNCTION HERE HAS TO BE A CONSTANT. IT'S GOING TO BE THE CONSTANT C. THAT'S GOING TO BE THE GAME PLAN. SO GOAL HERE IS SHOW THAT FOF XIS REALLY A CONSTANT. HOW DO YOU KNOW IF A FUNCTION IS CONSTANT? WHAT'S ITS DERIVATIVE
9 HAVE TO BE? ZERO. SO LET ME JUST GO AHEAD AND COMPUTE THE DERIVATIVE OF THAT FUNCTION AND CONFIRM THAT IT'S ZERO. AND IT HAS TO BE A CONSTANT. SO LET'S CONFIRM THAT THE DERIVATIVE OF THIS NEW FUNCTION IS ZERO. THAT WILL DO IT. THAT WILL SHOW IT'S A CONSTANT. THAT MEANS I HAVE TO DO DDXOF FOF XWHICH IS JUST THE PRODUCT RULE. GOING TO USE THE PRODUCT RULE HERE. (ON BOARD). SO HERE'S, WRITE DOWN THE PRODUCT RULE. (ON BOARD). AND NOW I JUST HAVE TO USE THE INFORMATION THAT I HAVE, DDXOF YOF X, WELL THAT, I'M TOLD WHAT THAT IS. I HAVE NO CHOICE. THIS THING HERE IS KTIME YOF X. THAT WAS GIVEN TO ME. AND WHAT'S DDXOF ETO THE MINUS KTIMES X? BY THE CHAIN RULE? SO 10 WE JUST TALKED ABOUT THAT. SO DDXOF THIS ETO THIS PARTICULAR CONSTANT TIMES XIS  I JUST PULLED CONSTANT OUT IN FRONT. SO IT'S GOING TO BE THIS CONSTANT WHICH IS MINUS KTIMES ETO THE MINUS KX. THAT'S JUST THE RULE I HAD BEFORE. SO I HAVE KTIMES YOF XTIMES ETO THE MINUS KXMINUS KTIMES YOF XTIMES ETO THE MINUS KX. THESE TWO TERMS ARE THE SAME. I SUBTRACT THEM AND GETS ZERO. SO THE FUNCTIONS'S DERIVATIVE IS ZERO. AND SO IT'S GOT TO BE A CONSTANT. SO FOF XIS SOME CONSTANT, LET ME GIVE IT A NAME. CALL IT C. BECAUSE THE ONLY FUNCTION WHO'S DERIVATIVE IS ZERO IS A CONSTANT. THAT'S THE WAY CONSTANTS ARE. AND SO FINALLY LET ME WRITE IT DOWN. I'LL FINISH THIS PROOF RIGHT UNDERNEATH THE STATEMENT. SO WHAT DO I HAVE? I HAVE THAT THIS CONSTANT IS WHAT I GET WHEN I MULTIPLY YOF XTIMES ETO THE MINUS KX. THAT'S WHAT'S SATISFIED BY YOF X. SO LET ME SOLVE FOR YOF X. (ON BOARD). LET ME USE THE LAW OF EXPONENTS HERE TO SIMPLIFY THIS. THAT'S THE CONSTANT TIMES ETO THE MINUS KXTO
10 THE MINUS ONE. THAT'S WHAT HAPPENS WHEN I TAKE IT TO THE DENOMINATOR. THE NEXT LAW OF EXPONENTS TELLS ME MULTIPLY THOSE TOGETHER. MINUS ONE TIMES THE OTHER EXPONENT. AND THAT'S ETO THE PLUS KX. WHICH IS WHAT I WANT. YEAH, THERE IT IS. SO THAT'S JUST REASSURING THAT WHENEVER YOU SEE THIS DIFFERENTIAL EQUATION UP THERE THERE'S ONLY ONE ANSWER TO WRITE DOWN WHICH IS NOT TRUE OF EVERY EQUATION. ALSO GOOD PRACTICE. ANY QUESTIONS ABOUT THAT? BEFORE I GO ON? SO LET'S TALK SOME MORE ABOUT WHAT THESE GRAPHS LOOK LIKE. 11 FIRST I'LL REMIND YOU THE PROPERTIES OF E TO THE XTHEN I WANT TO DO GENERAL ETO THE X. WHAT CAN THE GRAPH OF THE FUNCTION ETO THE KTIMES XLOOK LIKE? WELL, IF KIS GREATER THAN OR EQUAL TO ZERO, THAT'S GOING TO BE INCREASING FUNCTION OF X. AND IT WILL LOOK AS IT HAS OVER HERE, WE'VE DRAWN THIS PICTURE MANY TIMES, ETO THE KXWHERE KIS POSITIVE. SO ETO THE KXGOES TO INFINITY AS XGOES TO PLUS INFINITY. BUT IF I GO IN THE OTHER DIRECTION, AS XGOES TO MINUS INFINITY, AS XGOES THIS WAY, THE FUNCTION GETS SMALLER AND SMALLER AND APPROACHES ZERO, SO IT HAS AN ASYMPTOTE. THIS DIRECTION HERE, AN ASYMPTOTE, THE XAXIS. AND, OF COURSE, IT'S ALWAYS POSITIVE. NEVER CROSSES THE XAXIS. AND THE SLOPE HERE, THE SLOPE IS WHAT? WE BUILD IT HERE SO IT'S EASY TO FIGURE OUT. WHAT IS THE SLOPE IF I DO DDXTO ETO THE K XIS THAT KTIME E TO THE KX. IF I PLUG THIS IN AS I WANT THE SLOPE AT X EQUALS ZERO, KTIMES ETO THE KZERO IS K. SO THE SLOPE THERE IS JUST K. JUST READ IT OFF THE FUNCTION. WHEN KIS LESS THAN ZERO YOU GETS A MIRROR IMAGE LITERALLY OF EVERYTHING ABOUT THE XAXIS. LET ME WRITE DOWN AGAIN WHAT IT LOOKS LIKE.
11 KXGOING TO BE DECREASING. SO THIS PICTURE IS THE MIRROR IMAGE OF THAT PICTURE. ETO THE KXIS GOING TO GO TO ZERO AS XGOES TO PLUS INFINITY. (ON BOARD). IT'S STILL TRUE THAT ETO THE KXIS (ON BOARD). JUST THE IMAGE IS REVERSED. ONE WAY YOU CAN THINK ABOUT THIS FUNCTION ETO THE KXWHEN KIS NEGATIVE, WHAT IF I WRITE IT THIS WAY? (ON BOARD). SORRY, LET ME WRITE IT THIS WAY, ETO THE MINUS KTIMES MINUS X. THAT DOESN'T CHANGE THE 12 FUNCTION. SO IF KIS NEGATIVE, THEN MINUS KIS POSITIVE. SO THERE I HAVE A POSITIVE THING. BUT NOW I JUST HAVE A FUNCTION WHERE I'VE CHANGED XTO MINUS X. SO IT'S GOING TO LOOK LIKE THAT EXCEPT I CHANGED XTO MINUS X. AND WHAT HAPPENS WHEN I PLOT A FUNCTION BY CHANGE THE ARGUMENT FROM XTO MINUS X. HOW DO YOU CHANGE THE GRAPH? IF YOU HAVE THE GRAPH OF ANY FUNCTION YOU LIKE, FOF X, I WILL DRAW ANY OLD FUNCTION. THEN THE GRAPH OF FOF MINUS XIS GOTTEN BY, WELL IT'S SIMPLY JUST THE MIRROR IMAGE IN THE YAXIS. SIMPLY DO IT BACKWARDS. SO THAT'S WHAT HAPPENS WHEN YOU CHANGE XTO MINUS X, IT JUST REVERSES. SO THAT FUNCTION IS JUST THE REVERSE OF THAT. IT'S AN EASIER WAY TO REMEMBER, IF I LIKE. OKAY. THE REASON I'M REMINDING YOU OF ETO THE KXIS ALL YOU NEED TO KNOW TO GRAPH ANY OTHER EXPONENTIAL FUNCTION. SO THIS OVER THERE IS ALL WE NEED TO GRAPH YEQUALS BTO THE XFOR ANY BGREATER THAN ZERO. SO LET ME SHOW YOU HOW THAT WORKS. SO LET ME DRAW ETO THE XFOR THE MOMENT. AND LET ME TAKE THE NUMBER B. I DON'T KNOW WHERE IT IS. IT'S SOMEWHERE AND BRING THERE, PLOT IT ON THE YAXIS. THERE'S B. I'M GOING TO DRAW A HORIZONTAL LINE THROUGH B. THAT'S EASY ENOUGH TO DO. BECAUSE OF THE WAY THIS FUNCTION ETO THE X, IT GETS AS CLOSE TO
12 ZERO AS YOU LIKE HERE, IT GOES OFF TO INFINITY OVER THERE. SO THIS HORIZONTAL LINE HAS TO INSECT SOMETHING. IT DOESN'T MATTER WHAT BIS. IF I TAKE BDOWN HERE, IT MIGHT INTERSECT OVER THERE. BUT SOMEWHERE THAT HORIZONTAL LINE HAS TO INTERSECT. WHY DO I CARE ABOUT THAT? BECAUSE THERE'S AT SOME POINT, ONCE YOU FIGURE 13 OUT THIS POINT YOU GO STRAIGHT DOWN AND THAT GIVES A YOU A K. AND SO ETO THE KEQUALS WHAT? SO THIS IS THE GRAPH THE CURVE OF THE FUNCTION ETO THE X. THERE'S K. THERE'S B. SO B, ETO THE KEQUALS B? OKAY. SO FOR ANY BTHERE'S ALWAYS A K. IT MIGHT BE DOWN HERE. OR IT MIGHT BE OVER THERE. WHEREVER, THERE'S ALWAYS GOING TO BE A K. SO WHAT ABOUT THE FUNCTION BTO THE X? BI JUST EXPLAINED TO YOU WAS ETO THE KTIMES X. AND NOW WHAT DOES THE LAW OF EXPONENTS TELL ME? TELLS ME MULTIPLY THE TWO EXPONENT TOGETHER. SO IN OTHER WORDS THE GRAPH OF THE FUNCTION BTO THE XIS THE SAME AS THE GRAPH OF ETO THE KXFOR SOME K. THIS IS THE ONLY FUNCTION YOU NEED TO UNDERSTAND. ETO THE KX, IT TELLS YOU EVERYTHING. ANY QUESTIONS. SO THIS IS ONE OF THE THINGS THAT MATH IS GOOD FOR, TAKE THIS WHOLE FAMILY OF FUNCTIONS AND SAYS I ONLY HAVE TO UNDERSTAND IT FOR ONE CASE, FOR THE VALUE OF E. SO LET'S DO A COUPLE OF EXAMPLES TO ILLUSTRATE THAT. I'LL USE MONEY JUST TO GET YOUR ATTENTION AGAIN. SO LET ME DO AN EXAMPLE. YOU INVEST A THOUSAND DOLLARS AT 5 PERCENT INTEREST, IF YOU'RE LUCKY I GUESS THESE DAYS, POSITIVE INTEREST THAT IS, PER YEAR. AND HOW MUCH DO YOU HAVE AFTER TYEARS? LET'S WRITE DOWN BASED ON KNOWLEDGE OF WHAT IT MEANS TO EARN INTEREST. HERE'S I'LL USE M FOR MONEY, MONEY IS A FUNCTION OF TIME. START WITH A THOUSAND DOLLARS AND EVERY YEAR MULTIPLY
13 IT BY ONE PLUS 5 PERCENT. AND AFTER TYEARS THAT'S WHAT IT EQUALS. SO THERE'S THE FAMILIAR FORMULA. I IMAGINE. AND SO I'M GOING TO WRITE THIS AS BTO THE TWHERE B IS SO 14 THERE'S THE FUNCTION. AND THIS IS IF I ASK, IF I WRITE BAS ETO THE K, WHAT'S K, YOU CAN DO THAT PARTICULAR ARITHMETIC OR LOOK UP IN A TABLE AND KIS IT HAPPENS TO BE THAT. THAT TELLS YOU HOW MUCH MONEY YOU HAVE AS A FUNCTION OF TIME. IT'S AN EXPONENTIAL FUNCTION, 1,000 ETO THE POWER.0488 T. THERE'S YOUR BANK ACCOUNT. LET ME CHANGE THE PROBLEM SLIGHTLY. AND DO IT AGAIN BUT WHERE THEY COMPOUND INTEREST DAY. THIS IS ONE A YEAR YOU GET PAID 5 PERCENT. LET'S CHANGE IS VERY SLIGHTLY TO DO DAILY COMPOUNDING AND SEE HOW THAT CHANGES THE ANSWER. SO WHAT IS M OF TIF INTEREST IS COMPOUNDED DAILY? SO I'LL WRITE M OF TAGAIN. START WITH A THOUSAND DOLLARS. AND NOW, WHAT DOES COMPOUNDED DAILY MEAN? IT MEANS YOU GET ONE DAY'S WORTH OF YOUR INTEREST EVERYDAY. SO THAT MEANS INSTEAD OF GETTING FIVE PERCENT YOU GET, LET'S SAY THERE'S 365 DAYS IN THE YEAR YOU GET 5 PERCENT DIVIDED BY 365 BUT YOU GET IT 365 TIMES PER YEAR. IF IT WERE MONTHLY IT WOULD BE 12 AND 12. BUT THAT'S THE GENERAL IDEA. SO THAT'S GOING TO BE 1,000 TIMES BTO THE POWER 365 TWHERE BIS ONE PLUS.05. OVER 365. SO THAT'S I'M GOING TO WRITE ETO THE K. SO BEQUALS ETO THE K. BUT NOW 365 T. AND I'LL TELL YOU WHAT KIS FINALLY. KIS YOU CAN FIGURE OUT WHAT THAT IS AGAIN USING A CALCULATOR OR TABLE. NOW I'M GOING TO USE THE LAW OF EXPONENTS. WRITE THAT AS ETO THE KTIMES 365 TIMES T. MULTIPLY ALL THE THE EXPONENT TOGETHER. RULE OF EXPONENTS. AND SO WHAT I WANT TO DO IS FIGURE OUT WHAT
14 IS KTIMES 365. THAT'S GOING TO BE THE EXPONENT I HAVE UP IN 15 THERE. AND IT'S GOING TO BE.05002, FOUR DIGITS. (ON BOARD). THAT'S HOW FAST YOUR MONEY GROWS. NOW WE CAN COMPARE A THOUSAND TIME E TO THE.0500 TVERSUS ETO THE.0488 T. IT GROWS A LITTLE FASTER BUT NOT A HECK OF A LOT. HERE FILL IT IN, HERE, HOW MUCH MONEY DO I HAVE AFTER ONE YEAR? SO WE DON'T HAVE TO ACTUALLY WORK VERY HARD. YOU JUST PLUG IN TEQUALS ONE TO THE ORIGINAL FORMULA AND HOW MUCH MONEY DO YOU HAVE? 5 PERCENT MORE? AND THE ANSWER IS, WHAT'S 5 PERCENT OF A THOUSAND BUCKS? FIFTY BUCKS. THAT'S WHAT YOU GET IN THAT REGIME. LET'S COMPARE IT. AND SEE HOW MUCH FARTHER AHEAD WE COME OUT. NOW M OF ONE HERE, PLUG IN ONE AND ETO THE POWER.05 AND THE ANSWER IS YOU'RE NOT GOING TO BE MUCH RICHER. THAT'S WHAT DAILY COMPOUNDING GIVES YOU. STUDENT: WHAT'S THE POINT OF PUTTING IT INTO FORM OF ETO THE ONE WHEN YOU CAN KEEP IT. PROFESSOR: THERE ARE TWO REASONS. ONE IS I CAN COMPARE THIS TO THAT. THIS FUNCTION TO THAT FUNCTION MORE EASILY THAN THIS FUNCTION TO THAT FUNCTION. SO I CAN INDEED, WHEN THEY'RE ACTUALLY COMPUTED ON THE CALCULATOR OR THE SPREADSHEET THIS IS HOW IT'S REPRESENT INTERNALLY. WOULD YOU KNOW IF FROM STARTING OUT THIS HOW MUCH BIGGER FROM THAT? NOT SO OBVIOUS. OTHERWISE IT'S JUST TO PROVE TO YOU YET AGAIN THAT ALL YOU HAVE TO DO IS UNDERSTAND ONE EXPONENTIAL FUNCTION WHICH IS ETO THE KXAND YOU CAN DO ALL THE OTHER EXPONENTIAL FUNCTIONS. GOOD QUESTION. STUDENT: HOW DO YOU FIGURE OUT KAGAIN. 16
15 PROFESSOR: SO THAT'S SECTION 44. BUT IF YOU DO IT PICTORIALLY. GRAPH ETO THE X, DRAW A HORIZONTAL LINE THROUGH BAND DROP THE VERTICAL LINE IT K. THERE'S A NAME, ANOTHER NAME FOR K, IT'S CALLED THE LOGARITHM. THAT'S WHAT IT IS. THIS IS THE LOGARITHM. OKAY. INDEED THAT IS THE NEXT SECTION THAT I'M ABOUT TO DO. SECTION 4.4. THERE'S MORE THAN ONE LOGARITHM. I'M GOING TO START BY TALKING ABOUT K. IN PARTICULAR IT'S CALLED THE NATURAL LOGARITHM. NOW DRAW THE PICTURE AGAIN. RELABEL IT. (INAUDIBLE). CALL IT XCOMMA YON THE CURVE. HERE'S THE XCOORDINATE AND YCOORDINATE. AND NATURAL QUESTION IS GIVEN THE YCOORDINATE, WHAT IS X? HOW DO YOU FIND THAT? WHAT IS THAT FUNCTION? AND THE DEFINITION IS IF YEQUAL ETO THE XTHEN XIS CALLED THE NATURAL LOGARITHM OF YOR SOMETIMES THE LOGARITHM BASE EOF Y. YOU MAY HAVE ENCOUNTERED LOGARITHMS BASE TEN BEFORE. YOU CAN DO ANY BASE BUT I'M GOING TO START WITH BASE E. SO THE MAIN PROPERTY THAT WE JUST, AND IT'S WRITTEN, SO THE NOTATION IS XEQUALS L NY. LOG NATURAL OR SOMETHING LIKE THAT. L N. THAT'S THE NOTATION. SO THE MAIN PROPERTY OF LOGARITHM I CAN WRITE DOWN THIS WAY. JUST REPEATING WHEN I SAID THERE. WHICH IS YIS ETO THE XIS ETO THE NATURAL LOG OF Y. SO THAT'S TRUE FOR ANY OLD Y. AS LONG AS YGREATER THAN ZERO. THE LOGARITHM FUNCTION IS ONLY, MAKE SENSE FOR YGREATER THAN ZERO. YOU CAN ONLY GET POSITIVE NUMBERS FROM THE EXPONENTIAL FUNCTION. SO LET ME DO SOME EXAMPLE. ZERO COMMA ONE IS ON THE GRAPH. HERE IT IS. ZERO 17 COMMA ONE. WE KNOW THAT ONE EQUAL E TO THE ZERO, SO WHAT IS THE
16 LOG OF WHAT? TRANSLATING THAT, SOMETHING IS THE LOG OF SOMETHING. SO ZERO IS THE LOG OF ONE. ETO THE ZERO IS ONE. OKAY. LET ME DO ANOTHER EXAMPLE. HOW ABOUT TWO ESQUARED IS ON THE GRAPH. OF ETO THE XOVER THERE. IN OTHER WORDS, ESQUARED EQUAL ESQUARED. THAT'S NOT SO EXCITING. SO SOMETHING IS THE LOG OF SOMETHING. TWO IS THE LOG OF ESQUARED. AND FINALLY LET ME SUMMARIZE THIS PROPERTY. WHICH IS THAT SINCE XCOMMA ETO THE XIS ON THE GRAPH FOR ANY X, IF YOU TAKE THE LOG OF ETO THE X, YOU GET XBACK. SO THESE ARE, THAT PROPERTY AND THIS PROPERTY ARE THE TWO MAIN PROPERTIES OF LOGS AND EXPONENTS. LET ME WRITE THEM DOWN AGAIN. HERE WE HAVE THE TWO MAIN PROPERTIES ARE GOING TO BE IF I TAKE ETO THE XAND TAKE THE LOG TO GET XBACK. THE OTHER ONE SAYS TAKE THE LOG AND EXPONENTIATE IT, YOU GET THE NUMBER BACK AS WELL. ANOTHER WAY TO SAY THIS, IS THAT LOGARITHM AND EXPONENTIAL FUNCTIONS, THREES TWO FUNCTIONS ARE INVERSES, THEN UNDO ONE ANOTHER AM IF YOU DO THE LOG AND EXPONENTS AND GET BACK TO WHERE YOU STARTED. IF YOU DO THE EXPONENTS AND THE LOG YOU GET BACK TO WHERE YOU STARTED. THEY'RE INVERSES. THEY UNDO ONE ANOTHER. SO CAN YOU GIVE, WE'VE SEEN FUNCTIONS LIKE THIS BEFORE THAT UNDO ONE ANOTHER. CAN WE THINK OF SOME EXAMPLES THAT WE'VE SEEN BEFORE? TWO FUNCTIONS THAT UNDO ONE ANOTHER THAT GET YOU BACK TO WHERE YOU STARTED FROM? WHAT IF YOU DO THE FUNCTION YEQUALS XCUBED. WHAT IS THE INVERSE OF CUBING, CUBED ROOT. IF I CUBE 18 SOMETHING AND THEN TAKE THE CUBE ROOT I GET BACK TO WHERE I STARTED. OR I CAN TAKE THE CUBE ROOT FIRST, AND THEN CUBE IT I ALSO GET BACK TO WHERE I STARTED. THEY UNDO ONE ANOTHER, EITHER
17 OR. SO THIS IS OBVIOUS STUFF. IT'S, BUT FOR THE LOG AND THE EXPONENTS A LITTLE MORE INTERESTING. MORE COMPLICATED FUNCTIONS. THEY DO INVERSES OF ONE ANOTHER. LET US USE THAT FACT TO GRAPH THE LOGARITHM. LET'S USE THIS INVERSE PROPERTY TO GRAPH THE FUNCTION YEQUALS L NOF X. OKAY. SO THE PROPERTY IS GOING TO BE THAT IF XCOMMA YIS GOING TO BE ON THE GRAPH OF YEQUALS E TO THE XBECAUSE WE KNOW WHAT THAT GRAPH LOOKS LIKE. WE'VE DRAWN THAT A DOZEN TIMES. THEN THERE'S SOME OTHER POINT THAT'S GOING TO BE REALLY EASY FOR US TO WRITE DOWN, WHICH IS GOING TO BE ON THE GRAPH OF SOMETHING EQUALS LOGARITHM. BECAUSE THESE TWO ARE EQUIVALENT TO ONE ANOTHER. XIS THE LOGARITHM OF Y. SO WHAT POINTS IS GOING TO BE ON THAT GRAPH? SO IF I TELL YOU X COMMA YIS ON THE GRAPH OF ETO THE X, THEN WHAT, HOW DO I GET A POINT ON THE OTHER GRAPH? WHAT IS THE LOG OF WHAT? XIS THE LOG OF Y. SO XIS THE LOG OF X. THAT POINT'S GOING TO BE ON THE GRAPH OF XARE OF THE OTHER FUNCTION. IN OTHER WORDS LET ME DRAW IT HERE. SO HERE IS ETO THE X. AND SO THERE'S A TWO POINTS. IT'S GOING TO BE ON THE EXPONENTIAL ONE BUT THE OTHER ONE, ONE COMMA ZERO, THAT'S GOING TO BE ON THE LOG GRAPH BECAUSE THE LOG OF ONE IS ZERO. HOW ABOUT THE.1 COMMA ETO THE ONE? THAT'S ON THE EXPONENTIAL GRAPH SO I'M CLAIMING THAT THE POINT ETO THE ONE 19 COMMA ONE IS GOING TO BE ON THE LOG GRAPH BECAUSE LOG OF ETO THE ONE IS ONE. AND IF I TAKE SEVEN COMMA ETO THE SEVEN AND THERE'S GOING TO BE SOME POINT ETO THE SEVEN COMMA SEVEN THAT'S GOING TO BE ON THE LOGARITHM. IF I FILL THAT IN, IT'S GOING TO LOOK LIKE THIS. SO FOR EVERYONE POINTS UP HERE, I'M GOING TO GET, WHO'S
18 COORDINATES ARE XCOMMA WHY, FOR ANY POINT HERE WHO'S COORDINATES ARE XCOMMA YI'LL GET A COORDINATE DOWN HERE WHAT'S COORDINATES ARE YCOMMA X BECAUSE IF YIS THE EXPONENTIAL OF XTHEN  THEY UNDO ONE ANOTHER. SO THAT MEANS THAT I'M JUST TAKING EVERY POINT AND FLIPPING IT. REVERSING ITS COORDINATES. DID YOU AT SOME POINT LEARN IF SOMEBODY GIVES YOU A GRAPH HOW TO FLIP ITS COORDINATES BY DRAWING A PICTURE? JUST BY DOING SOME SORT OF FLIPPING OR FOLDING OF A PIECE OF PAPER. IF YOU DREW IT ON A PIECE OF PAPER, HOW DO YOU DO IT? I TAKE A MIRROR IMAGE OF THE GRAPH. REMEMBER THIS? THIS GRAPHY JUST FLIP IT IN THE DIAGONAL, THE LINE XEQUALS YAND I GET THIS GRAPH. THAT'S WHAT IT MEANS TO TAKE EVERY POINT XYAND FLIP IT TO YX. FOLD THE PIECE OF PAPER OVER, AND YOU GET THE PLOT. SO THAT'S JUST TAKE THE MIRROR IMAGE OF THE CURVE YEQUALS ETO THE XTO GET THE LOG PLOT. IS THAT SOMETHING, I THINK YOU'VE DONE THIS BEFORE. IF I PLOT YEQUALS XSQUARED, WE KNOW HOW TO PLOT THAT. WHAT IS THE GRAPH OF XEQUALS YSQUARED? ANOTHER PARABOLA, LOOKS LIKE THIS. AND THIS IMAGE IS JUST THE MIRROR IMAGE IN THAT DIAGONAL LINE. JUST FLIP THEM. IT'S REALLY EASY TO THINK ABOUT HOW TO GET FROM THIS GRAPH TO THAT GRAPH. SO THIS THING HERE IS THE LOGARITHM. 20 THERE'S THE GRAPH OF YEQUALS ETO THE X. (ON BOARD). SO LET ME WRITE DOWN, SUMMARIZE HERE THE PROPERTIES OF THE CURVE YEQUALS E TO THE X. AND THEN JUST BY REALIZING I'M TAKING A MIRROR THERE, READ OFF THE PROPERTIES OF Y, WELL, XEQUALS LOG OF Y. THIS IS THE SAME AS SAYING THIS. JUST SAME EQUATION WRITTEN DOWN SLIGHTLY DIFFERENTLY. ETO THE XIS ALWAYS GREATER THAN ZERO. ALWAYS ABOVE THE AXIS. THAT MEANS THAT THIS
19 FUNCTION IS ONLY DEFINED FOR YGREATER THAN ZERO. THAT CURVE IS ONLY DEFINED OVER THERE. YOU CAN ONLY TAKE THE LOG OF A POSITIVE NUMBER. SO BY, IF I TAKE XLESS THAN ZERO THAT MEANS THAT ETO THE XIS LESS THAN ONE. IF I LOOK AT, IF I TAKE XLESS THAN ZERO ETO THE XIS LESS THAN ONE. SO THAT'S RIGHT OFF. THE EQUIVALENT PROPERTY OVER THERE IS IF THE LOGARITHM OF YIS LESS THAN ZERO, IF AND ONLY IF, THESE TWO ARE EQUIVALENT TO ONE ANOTHER, IF I'M, IF THE ARGUMENT THERE, IF XIS LESS THAN ONE, SORRY, IF YIS LESS THAN ONE. SO I HAVE TO BE OVER HERE. AND IT'S GOING TO BE LESS THAN ZERO. XEQUALS ZERO, I GET ETO THE XIS ONE, ETO THE ZERO IS ONE. OVER HERE I GET THAT LOGARITHM OF THE NUMBER ZERO IS THE SAME THING AS SAYING THE NUMBER IS EQUAL TO ONE. THAT MEANS I'M, THERE'S THE POINT THERE. THE LOG OF, THE ARGUMENT IS ONE. AND XGREATER THAN ZERO MEANS THAT THE EXPONENTIAL IS BIGGER THAN ONE. AND THAT'S LIKE SAYING THE LOGARITHM IS POSITIVE IF THE ARGUMENT BIGGER THAN ONE. THAT SAYS I'M OVER HERE. ALL THOSE PROPERTIES GO TOGETHER BY TAKING THESE TWO MIRROR IMAGE OF ONE ANOTHER. THIS IS INCREASING 21 FUNCTION. IT GOES UP. AND THIS IS AN INCREASING FUNCTION, IT GOES UP. THAT'S SORT OF OBVIOUS FROM THE MIRROR PROPERTIES. BUT HERE'S SOMETHING THAT WORK OUT BACKWARDS, ETO THE XIS CONCAVE WHICH WAY? UP. AND HOW ABOUT THE LOGARITHM FUNCTION? CONCAVE DOWN. SO THAT PROPERTY WORKS KINDS OF THE OPPOSITE. IF THERE ARE NO QUESTIONS ABOUT THOSE BASIC PROPERTIES LET'S USE A FEW. AND DO SOME COMPUTATIONS AND CALCULATIONS OF LOGARITHMS. THIS IS JUST MAKE SURE WE KNOW THE INVERSE PROPERTIES. SO TAKE ETO THE NATURAL LOG OF FOUR, YOU GET FOUR. BECAUSE THESE CANCELING OUT.
20 HOW ABOUT ETO THE NATURAL LOG OF FOUR PLUS TWO TIME NATURAL LOG OF THREE? NOW, THERE ARE PROPERTIES OF LOGARITHMS. WE'LL GET TO THEM BUT I WANT TO USE THE LAW OF EXPONENTS. THE FIRST LAW OF EXPONENTS TELLS ME I CAN DO THIS. THAT'S JUST THE LAW OF EXPONENTS. BECAUSE IF I HAVE A SUM UP HERE I CAN BREAK INTO A PRODUCT. AND NOW, I CAN WRITE THAT AS, THIS IS FOUR, WE JUST FIGURED THAT OUT. THIS ONE, I'M GOING TO WRITE THIS WAY, DOESN'T MATTER WHAT ORDER I MULTIPLY THOSE TWO SCALERS IN. DOESN'T CHANGE THINGS. AND NOW I CAN USE THE LAW OF EXPONENTS AGAIN. WRITE IT THAT WAY. AND FINALLY I GET FOUR, NOW I CAN SEE THAT'S A THREE IN THERE AND NOW ALL THE LOGS AND EXPONENTS HAVE GONE AWAY AND I HAVE FOUR TIMES NINE OR 36. THE OTHER WAY YOU COULD HAVE DONE THIS ONE, WE RECALL WHICH PROPERTIES OF LOGARITHM I COULD HAVE RECOGNIZED THAT THIS EXPONENTS WAS ACTUALLY THE LOG OF 36 BUT WE'LL DO THAT EVENTUALLY. HOW ABOUT THIS ONE, SOLVE FIVE ETO THE POWER XMINUS THREE EQUALS FOUR. SO LET'S TRY TO SOLVE 22 THAT ONE. DIVIDE BOTH SIDE BY FIVE. MAKE THE EXPONENTIAL GO AWAY. SO TAKE THE LOG OF BOTH SIDES. LOG OF EXPONENTS IS WHAT? WHAT IS THIS? THIS TURNS INTO XMINUS THREE, EQUALS NATURAL LOG OF 28, AND FINALLY I GET XEQUALS THREE PLUS THE NATURAL LOG OF.8. AND WE CAN DO THAT, 2.78, SOMETHING LIKE THAT. DOT DOT DOT. STUDENT: HOW DO YOU KNOW (INAUDIBLE). PROFESSOR: SO I HAVEN'T TAUGHT US HOW TO COMPUTE THE DERIVATIVE YET. I WILL. AND RIGHT NOW I'M JUST GOING BY THE PICTURE WHICH SAYS IF I TAKE SOMETHING THAT'S CONCAVE UP, AND FLIP IT UPSIDE DOWN, IT'S CONCAVE DOWN. BUT I WILL COMPUTE THE SECOND
21 DERIVATIVE IN DUE COURSE. IT WILL BE EASY TO SEE WHETHER IT'S POSITIVE OR NEGATIVE. STUDENT: SO (INAUDIBLE). PROFESSOR: YOU'RE GOING TO BE DOING ALGEBRAIC MANIPULATIONS OF THIS. ONCE YOU GOT TO HERE, YOU'D BE DONE. I JUST FILLED IN THE REST FOR FUN. YOU WOULD BE DONE BY THE TIME YOU'VE DONE THE MANIPULATIONS TO GET THAT. ANY OTHER QUESTIONS? OKAY. SOLVE ANOTHER ONE HERE. TWO NATURAL LOG OF XPLUS SEVEN EQUALS ZERO. TWO NATURAL LOG OF XEQUALS MINUS SEVEN (ON BOARD). SO XIS ETO THE POWER MINUS 3.5. AT THAT POINT YOU'D BE DONE UNLESS YOU REALLY WANT TO KNOW THE ANSWER,.03. SO THIS IS AS I SAID THE LOGARITHM BASE E. AND FROM HIGH SCHOOL OR WHENEVER YOU LAST LOOKED AT LOGARITHM YOU PROBABLY LEARNED ABOUT LOGARITHMS BASE TEN. LET ME TELL YOU THEY'RE ALL RELATED TO 23 EACH OTHER. CONSTANT. TO GET TO THIS  ALL YOU HAVE TO DO IS MULTIPLY BY SO ALL THE LOG BASE, THEY ALL DIFFERENT BY MULTIPLYING BY (INAUDIBLE). SO THEY'RE EASY TO GET FROM ONE TO THE OTHER. OTHER LOGARITHM AND EXPONENTIAL FUNCTIONS. SO LET ME JUST REVIEW WHAT I SAID BEFORE. EARLIER, WE SAID THAT GIVEN ANY POSITIVE BTHERE WAS SOME NUMBER KSO THAT BEQUALED ETO THE K. I DREW THAT PICTURE A WHOLE BUNCH OF TIMES. SO WHAT DOES THAT MEAN? IF I TAKE BTO THE XTHAT'S GOING TO BE ETO THE KX. AND THAT'S BY THE LAW OF EXPONENTS THAT'S ETO THE KX. IF I COMPUTE ETO THE SOMETHING YOU CAN COMPUTE BTO THE SOMETHING BY USING IF I CAN FIGURE OUT THAT MAGIC NUMBER K. NOW WE WANT TO KNOW WHAT KIS. SO KIS WHAT? USING THE FUNCTION WE JUST DEFINED FOR THE
22 LAST FIVE MINUTES. IF BEQUALS ETO THE KKEQUALS NATURAL LOG OF B. THAT'S HOW YOU GET K. SO THIS IS EQUAL TO ENATURAL LOG OF BTIMES X. SO THAT'S HOW YOU GO FROM ONE EXPONENTIAL FUNCTION TO ANOTHER. THAT'S HOW EXPONENTIALS ARE RELATED. NOW LET'S DO LOGS. SO THERE'S THE PROPERTY I'VE WRITTEN DOWN MANY TIMES. WE CALL L NXLOG BASE E. AND LET ME TRY WRITING DOWN ANOTHER LOGARITHM THAT'S FAMILIAR. (ON BOARD). WHAT DOES LOG BASE TEN MEAN? IT MEANS THAT THAT'S I THINK THE RELATIONSHIP THAT WE'VE LEARNED BEFORE. LOG BASE TEN. (ON BOARD). SO THIS IS (ON BOARD) AND DEPENDING ON HOW YOU LEARNED IT THIS MIGHT HAVE BEEN CALLED THE COMMON LOG. AND THIS IS NATURAL LOG. TERMINOLOGY IS 24 EASIER TO SAY BASE EAND BASE TEN. SO YOU DON'T HAVE TO REMEMBER  MAYBE IN HIGH SCHOOL YOU LEARNED HOW TO YOU COMPUTATIONS WITH LOG BASE TEN. THE QUESTION IS HOW ARE THEE RELATED TO ONE ANOTHER. OR FOR THAT MATTER, BASE TWO, USE THAT A LOT, FOR FOR THAT MATTER BASE B, THE GENERAL. HOW ARE ALL THESE DIFFERENT LOGARITHMS RELATED TO ONE ANOTHER. AND THE ANSWER IS IT'S EASY. LET JUST DO BASE STEP, DO THE GENERAL CASE. THERE'S THE DEFINITION OF WHAT LOG BASE TEN OF XMEANS. WHATEVER NUMBERS WORKS THERE. NOW WHAT IS TEN? HOW DO I WRITE TEN ETO SOMETHING? WHAT GOES UNHERE? IF I JUST TEN TO ETO THE POWER, ANYBODY? NATURAL LOG OF TEN, THAT'S WHAT THE NATURAL LOG DOES. THAT'S X. AND THE OTHER WAY I CAN WRITE XIS ETO THE POWER NATURAL LOG OF X. THAT'S, SO THIS IS ETO THE POWER NATURAL LOG OF TEN, SOME NUMBER, TIMES LOG BASE TEN OF XWHICH IS THE THING I DON'T KNOW. AND I ALSO KNOW IT'S ETO THE NATURAL LOG OF
23 XBECAUSE THAT'S ANOTHER WAY TO WRITE X. SO IF THAT, IF ETO THIS NUMBER EQUALS ETO THAT NUMBER, WHAT ARE, CAN YOU TELL ME WHAT THESE TWO EXPONENTS? THEY'RE GOING TO BE THE SAME. IF, SO, IN FACT, SO IF, IF THIS IS TRUE, LET ME WRITE IT DOWN THIS WAY, WRITE DOWN EVERY STEP, I'LL BE CAREFUL, THAT'S WHAT I'VE GOTTEN SO FAR. AND SO IF THOSE TWO NUMBERS ARE EQUAL, THEN THEY'RE NATURAL LOGARITHMS ARE EQUAL. AND SO WHAT IS THE LOG OF THE ETO THE LOG? THESE TWO THINGS CANCEL. SO THIS IS JUST LOG X. MADE EGO AWAY. OVER HERE, LOG OF EXPONENTIAL, THESE TWO FUNCTIONS UNDO ONE ANOTHER. SO I GET (ON BOARD). WHAT DO I 25 WANT TO DO? SOLVE THIS EQUATION FOR LOG BASE TEN OF XAND DIVIDE BY THIS CONSTANT. IT'S GOING TO BE ONE DIVIDED BY THE NATURAL LOG OF TEN, SOME NUMBER, TIMES THE NATURAL LOG OF X. SO THE LOG BASE TEN IS JUST SOME CONSTANT TIMES NATURAL LOG OF X. THAT'S HOW THEY'RE WRITTEN. AND WHAT IS A CONSTANT? IT'S ABOUT.43 SOMETHING OR OTHER TIMES NATURAL LOG OF X. SO JUST, ONE THAT CHANGES ONE TOGETHER. IS THAT OKAY BIT OF ALGEBRA THERE TO RELATE THE TWO? YOU KNOW ONE LOGARITHM, YOU KNOW ALL OF THEM BECAUSE YOU HAVE TO JUST MULTIPLY THEM BY ONE OVER LOG OF TEN. NOW IN THAT ALGEBRA UP THERE WAS THERE ANYTHING SPECIAL ABOUT TEN THAT I USED? COULD IT HAVE BEEN ANY OTHER NUMBER? I DIDN'T REALLY USE ANY PROPERTIES WITH THE NUMBER TEN SO LET ME DO IT WITH ANY OLD BASE B. SO NOW DEFINE THE LOG BASE ANY POSITIVE NUMBER BOF X. SO THIS ONLY WORK IF BIS GREATER THAN ZERO. SO I HAVE THIS FUNCTION. I'M GOING TO WRITE ETO THE B. SO BIS GOING TO BE WRITTEN AS ETO THE LOG B. AND THIS IS GOING TO BE, WHICH IS XSO. THAT'S ETO THE LOG OF X. SO THAT EXPONENT AND
24 THIS EXPONENTS, THEY HAVE TO AGREE WITH ONE ANOTHER. SAME AS BEFORE. ETO THE THAT EQUALS ETO THE THAT. THESE TWO EXPONENT HAVE TO BE EQUAL. SO I LET LOG BTIMES LOG BOF X(ON BOARD). AGAIN REALLY SIMPLE RELATIONSHIP. I GET THAT THE LOG BASIC BOF XFOR ANY OLD BIS ONE OVER NATURAL LOG OF BTIMES NATURAL LOG OF X. (ON BOARD) AM SO YOU CAN COMPARE ANY BASE LOG BY MULTIPLYING BY THE SCALER OUT FRONT. ANY QUESTIONS ABOUT THAT? I THINK THAT'S A GOOD PLACE TO STOP. NEXT TIME WE'LL DIFFERENTIATE 26 LOGARITHMS.
25
MITOCW max_min_second_der_512kbmp4
MITOCW max_min_second_der_512kbmp4 PROFESSOR: Hi. Well, I hope you're ready for second derivatives. We don't go higher than that in many problems, but the second derivative is an important the derivative
More informationMITOCW big_picture_integrals_512kbmp4
MITOCW big_picture_integrals_512kbmp4 PROFESSOR: Hi. Well, if you're ready, this will be the other big side of calculus. We still have two functions, as before. Let me call them the height and the slope:
More informationMITOCW ocw f07lec02_300k
MITOCW ocw1801f07lec02_300k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.
More informationMITOCW ocw f08lec19_300k
MITOCW ocw18085f08lec19_300k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.
More informationOverview. Teacher s Manual and reproductions of student worksheets to support the following lesson objective:
Overview Lesson Plan #1 Title: Ace it! Lesson Nine Attached Supporting Documents for Plan #1: Teacher s Manual and reproductions of student worksheets to support the following lesson objective: Find products
More informationNote: Please use the actual date you accessed this material in your citation.
MIT OpenCourseWare http://ocw.mit.edu 18.06 Linear Algebra, Spring 2005 Please use the following citation format: Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology:
More informationTranscript: Reasoning about Exponent Patterns: Growing, Growing, Growing
Transcript: Reasoning about Exponent Patterns: Growing, Growing, Growing 5.12 1 This transcript is the property of the Connected Mathematics Project, Michigan State University. This publication is intended
More informationPROFESSOR: Well, last time we talked about compound data, and there were two main points to that business.
MITOCW Lecture 3A [MUSIC PLAYING] PROFESSOR: Well, last time we talked about compound data, and there were two main points to that business. First of all, there was a methodology of data abstraction, and
More informationAskDrCallahan Calculus 1 Teacher s Guide
AskDrCallahan Calculus 1 Teacher s Guide 3rd Edition rev 080108 Dale Callahan, Ph.D., P.E. Lea Callahan, MSEE, P.E. Copyright 2008, AskDrCallahan, LLC v3r080108 www.askdrcallahan.com 2 Welcome to AskDrCallahan
More informationSo just by way of a little warm up exercise, I'd like you to look at that integration problem over there. The one
MITOCW Lec02 What we're going to talk about today, is goals. So just by way of a little warm up exercise, I'd like you to look at that integration problem over there. The one that's disappeared. So the
More informationDescription: PUP Math Brandon interview Location: Conover Road School Colts Neck, NJ Researcher: Professor Carolyn Maher
Page: 1 of 8 Line Time Speaker Transcript 1. Narrator When the researchers gave them the pizzas with four toppings problem, most of the students made lists of toppings and counted their combinations. But
More information_The_Power_of_Exponentials,_Big and Small_
_The_Power_of_Exponentials,_Big and Small_ Nataly, I just hate doing this homework. I know. Exponentials are a huge drag. Yeah, well, now that you mentioned it, let me tell you a story my grandmother once
More informationPROFESSOR: I'd like to welcome you to this course on computer science. Actually, that's a terrible way to start.
MITOCW Lecture 1A [MUSIC PLAYING] PROFESSOR: I'd like to welcome you to this course on computer science. Actually, that's a terrible way to start. Computer science is a terrible name for this business.
More informationThe following content is provided under a Creative Commons license. Your support
MITOCW Lecture 17 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a
More informationNote: Please use the actual date you accessed this material in your citation.
MIT OpenCourseWare http://ocw.mit.edu 18.03 Differential Equations, Spring 2006 Please use the following citation format: Arthur Mattuck and Haynes Miller, 18.03 Differential Equations, Spring 2006. (Massachusetts
More informationMITOCW mit600f08lec17_300k
MITOCW mit600f08lec17_300k OPERATOR: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources
More informationMITOCW watch?v=vifkgfl1cn8
MITOCW watch?v=vifkgfl1cn8 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To
More information1/ 19 2/17 3/23 4/23 5/18 Total/100. Please do not write in the spaces above.
1/ 19 2/17 3/23 4/23 5/18 Total/100 Please do not write in the spaces above. Directions: You have 50 minutes in which to complete this exam. Please make sure that you read through this entire exam before
More informationDisplay Contest Submittals
Display Contest Submittals #1a  Original Message  From: Jim Horn To: rjnelsoncf@cox.net Sent: Tuesday, April 28, 2009 3:07 PM Subject: Interesting calculator display Hi, Richard Well, it takes
More information1 Lesson 11: Antiderivatives of Elementary Functions
1 Lesson 11: Antiderivatives of Elementary Functions Chapter 6 Material: pages 237252 in the textbook: The material in this lesson covers The definition of the antiderivative of a function of one variable.
More informationMITOCW Lec 3 MIT 6.042J Mathematics for Computer Science, Fall 2010
MITOCW Lec 3 MIT 6.042J Mathematics for Computer Science, Fall 2010 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer highquality
More informationECO LECTURE TWENTYTHREE 1 OKAY. WE'RE GETTING TO GO ON AND TALK ABOUT THE LONGRUN
ECO 155 750 LECTURE TWENTYTHREE 1 OKAY. WE'RE GETTING TO GO ON AND TALK ABOUT THE LONGRUN EQUILIBRIUM FOR THE ECONOMY. BUT BEFORE WE DO, I WANT TO FINISH UP ON SOMETHING I WAS TALKING ABOUT LAST TIME.
More informationMIT Alumni Books Podcast The Proof and the Pudding
MIT Alumni Books Podcast The Proof and the Pudding JOE This is the MIT Alumni Books Podcast. I'm Joe McGonegal, Director of Alumni Education. My guest, Jim Henle, Ph.D. '76, is the Myra M. Sampson Professor
More informationThe Definition of 'db' and 'dbm'
P a g e 1 Handout 1 EE442 Spring Semester The Definition of 'db' and 'dbm' A decibel (db) in electrical engineering is defined as 10 times the base10 logarithm of a ratio between two power levels; e.g.,
More informationAlgebra I Module 2 Lessons 1 19
Eureka Math 2015 2016 Algebra I Module 2 Lessons 1 19 Eureka Math, Published by the nonprofit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced, distributed, modified, sold,
More informationDominque Silva: I'm Dominique Silva, I am a senior here at Chico State, as well as a tutor in the SLC, I tutor math up to trig, I've been here, this
Dominque Silva: I'm Dominique Silva, I am a senior here at Chico State, as well as a tutor in the SLC, I tutor math up to trig, I've been here, this now my fourth semester, I'm graduating finally in May.
More informationBite Size Brownies. Designed by: Jonathan Thompson George Mason University, COMPLETE Math
Bite Size Brownies Designed by: Jonathan Thompson George Mason University, COMPLETE Math The Task Mr. Brown E. Pan recently opened a new business making brownies called The Brown E. Pan. On his first day
More informationMITOCW mit5_95js09lec07_300k_pano
MITOCW mit5_95js09lec07_300k_pano The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer highquality educational resources for
More informationSEVENTH GRADE. Revised June Billings Public Schools Correlation and Pacing Guide Math  McDougal Littell Middle School Math 2004
SEVENTH GRADE June 2010 Billings Public Schools Correlation and Guide Math  McDougal Littell Middle School Math 2004 (Chapter Order: 1, 6, 2, 4, 5, 13, 3, 7, 8, 9, 10, 11, 12 Chapter 1 Number Sense, Patterns,
More informationLesson 25: Solving Problems in Two Ways Rates and Algebra
: Solving Problems in Two Ways Rates and Algebra Student Outcomes Students investigate a problem that can be solved by reasoning quantitatively and by creating equations in one variable. They compare the
More informationThis past April, Math
The Mathematics Behind xkcd A Conversation with Randall Munroe Laura Taalman This past April, Math Horizons sat down with Randall Munroe, the author of the popular webcomic xkcd, to talk about some of
More informationd. Could you represent the profit for n copies in other different ways?
Special Topics: U3. L3. Inv 1 Name: Homework: Math XL Unit 3 HW 9/2810/2 (Due Friday, 10/2, by 11:59 pm) Lesson Target: Write multiple expressions to represent a variable quantity from a real world situation.
More informationThe following content is provided under a Creative Commons license. Your support
MITOCW Lecture 6 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer highquality educational resources for free. To make a donation
More informationRichard Hoadley Thanks Kevin. Now, I'd like each of you to use your keyboards to try and reconstruct some of the complexities of those sounds.
The sound of silence Recreating sounds Alan's told me that instruments sound different, because of the mixture of harmonics that go with the fundamental. I've got a recording of his saxophone here, a sound
More informationMITOCW MIT7_01SCF11_track01_300k.mp4
MITOCW MIT7_01SCF11_track01_300k.mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for
More informationMobile Math Teachers Circle The Return of the iclicker
Mobile Math Teachers Circle The Return of the iclicker June 20, 2016 1. Dr. Spock asked his class to solve a percent problem, Julia set up the proportion: 4/5 = x/100. She then crossmultiplied to solve
More informationUnit 7, Lesson 1: Exponent Review
Unit 7, Lesson 1: Exponent Review 1. Write each expression using an exponent: a. b. c. d. The number of coins Jada will have on the eighth day, if Jada starts with one coin and the number of coins doubles
More informationMAT Practice (solutions) 1. Find an algebraic formula for a linear function that passes through the points ( 3, 7) and (6, 1).
MAT 110  Practice (solutions) 1. Find an algebraic formula for a linear function that passes through the points ( 3, 7) and (6, 1). Answer: y = 2 3 + 5 2. Let f(x) = 8x 120 (a) What is the y intercept
More informationhow two exstudents turned on to pure mathematics and found total happiness a mathematical novelette by D. E. Knuth SURREAL NUMBERS A ADDISON WESLEY
how two exstudents turned on to pure mathematics and found total happiness a mathematical novelette by D. E. Knuth SURREAL NUMBERS A ADDISON WESLEY 1 THE ROCK /..,..... A. Bill, do you think you've found
More informationSection 2.1 How Do We Measure Speed?
Section.1 How Do We Measure Speed? 1. (a) Given to the right is the graph of the position of a runner as a function of time. Use the graph to complete each of the following. d (feet) 40 30 0 10 Time Interval
More informationCorrelation to the Common Core State Standards
Correlation to the Common Core State Standards Go Math! 2011 Grade 4 Common Core is a trademark of the National Governors Association Center for Best Practices and the Council of Chief State School Officers.
More informationHEAVEN PALLID TETHER 1 REPEAT RECESS DESERT 3 MEMORY CELERY ABCESS 1
Heard of "the scientific method"? There's a really great way to teach (or learn) what this is, by actually DOING it with a very fun game  (rather than reciting the standard sequence of the steps involved).
More informationMath Final Exam Practice Test December 2, 2013
Math 1050003 Final Exam Practice Test December 2, 2013 Note that this Practice Test is longer than the Final Exam will be. This way you have extra problems to help you practice, so don t let the length
More informationExample the number 21 has the following pairs of squares and numbers that produce this sum.
by Philip G Jackson info@simplicityinstinct.com P O Box 10240, Dominion Road, Mt Eden 1446, Auckland, New Zealand Abstract Four simple attributes of Prime Numbers are shown, including one that although
More informationOur Dad is in Atlantis
Our Dad is in Atlantis by Javier Malpica Translated by Jorge Ignacio Cortiñas 4 October 2006 Characters Big Brother : an eleven year old boy Little Brother : an eight year old boy Place Mexico Time The
More information6.034 Notes: Section 4.1
6.034 Notes: Section 4.1 Slide 4.1.1 What is a logic? A logic is a formal language. And what does that mean? It has a syntax and a semantics, and a way of manipulating expressions in the language. We'll
More informationMath 8 Assignment Log. Finish Discussion on Course Outline. Activity Section 2.1 Congruent Figures Due Date: InClass: Directions for Section 2.
082317 082417 Math 8 Log Discussion: Course Outline Assembly First Hour Finish Discussion on Course Outline Activity Section 2.1 Congruent Figures InClass: Directions for Section 2.1 082817 Activity
More informationA QUALITY IMPROVEMENT PROCESS IN, HEMLOCK DRYING
A QUALITY IMPROVEMENT PROCESS IN, HEMLOCK DRYING Neil Odegard Weyerhaeuser Corporation Snoqualmie, Washington The first thing I'd like to say is this; I'm not here to tell you what to do, or how and when
More informationpercents Common Core Standard 7.RP3 Use proportional relationships to solve multistep ratio and percent problems.
Intro All right, welcome to class everybody I need you guys to come in, take your seat Take out everything you need Put it on top of your desk I have something real special planned for you Now make sure
More informationEIGHTH GRADE RELIGION
EIGHTH GRADE RELIGION MORALITY ~ Your child knows that to be human we must be moral. knows there is a power of goodness in each of us. knows the purpose of moral life is happiness. knows a moral person
More informationSpeaker 2: Hi everybody welcome back to out of order my name is Alexa Febreze and with my co host. Speaker 1: Kylie's an hour. Speaker 2: I have you
Hi everybody welcome back to out of order my name is Alexa Febreze and with my co host. Kylie's an hour. I have you guys are having a great day today is a very special episode today we'll be talking about
More information2 nd Int. Conf. CiiT, Molika, Dec CHAITIN ARTICLES
2 nd Int. Conf. CiiT, Molika, 2023.Dec.2001 93 CHAITIN ARTICLES D. Gligoroski, A. Dimovski Institute of Informatics, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, Arhimedova
More informationMathematics Curriculum Document for Algebra 2
Unit Title: Square Root Functions Time Frame: 6 blocks Grading Period: 2 Unit Number: 4 Curriculum Enduring Understandings (Big Ideas): Representing relationships mathematically helps us to make predictions
More informationKey Maths Facts to Memorise Question and Answer
Key Maths Facts to Memorise Question and Answer Ways of using this booklet: 1) Write the questions on cards with the answers on the back and test yourself. 2) Work with a friend to take turns reading a
More information1 MR. ROBERT LOPER: I have nothing. 3 THE COURT: Thank you. You're. 5 MS. BARNETT: May we approach? 7 (At the bench, off the record.
167 April Palatino  March 7, 2010 Redirect Examination by Ms. Barnett 1 MR. ROBERT LOPER: I have nothing 2 further, Judge. 3 THE COURT: Thank you. You're 4 excused. 5 MS. BARNETT: May we approach? 6 THE
More informationThe Product of Two Negative Numbers 1
1. The Story 1.1 Plus and minus as locations The Product of Two Negative Numbers 1 K. P. Mohanan 2 nd March 2009 When my daughter Ammu was seven years old, I introduced her to the concept of negative numbers
More informationDigital Circuits 4: Sequential Circuits
Digital Circuits 4: Sequential Circuits Created by Dave Astels Last updated on 20180420 07:42:42 PM UTC Guide Contents Guide Contents Overview Sequential Circuits Onward FlipFlops RS Flip Flop Level
More informationOn the eve of the Neil Young and Crazy Horse Australian tour, he spoke with Undercover's Paul Cashmere.
Undercover Greendale (interview with poncho) Sometime in the 90's Neil Young was christened the Godfather of Grunge but the title really belonged to his band Crazy Horse. While Young has jumped through
More informationPrevious Lecture Sequential Circuits. Slide Summary of contents covered in this lecture. (Refer Slide Time: 01:55)
Previous Lecture Sequential Circuits Digital VLSI System Design Prof. S. Srinivasan Department of Electrical Engineering Indian Institute of Technology, Madras Lecture No 7 Sequential Circuit Design Slide
More informationHere s a question for you: What happens if we try to go the other way? For instance:
Prime Numbers It s pretty simple to multiply two numbers and get another number. Here s a question for you: What happens if we try to go the other way? For instance: With a little thinking remembering
More informationFun to Imagine. Richard P. Feynman. BBC 1983 transcript by A. Wojdyla
Fun to Imagine Richard P. Feynman BBC 1983 transcript by A. Wojdyla This is a transcript of the R.P. Feynman s Fun to imagine aired on BBC in 1983. The transcript was made by a nonnative english speaker
More informationFormalising arguments
Formalising arguments Marianne: Hi, I'm Marianne Talbot and this is the first of the videos that supplements the podcasts on formal logic. (Slide 1) This particular video supplements Session 2 of the formal
More information2 THE COURT: All right. You may. 4 MS. BARNETT: Thank you, Your Honor. 6 having been first duly sworn, testified as follows:
138 Jonathan French March 7, 2010 RecrossExamination by Mr. Robert Loper 1 (Witness sworn.) 2 THE COURT: All right. You may 3 proceed. 4 MS. BARNETT: Thank you, Your Honor. 5 APRIL PALATINO, 6 having
More information#029: UNDERSTAND PEOPLE WHO SPEAK ENGLISH WITH A STRONG ACCENT
#029: UNDERSTAND PEOPLE WHO SPEAK ENGLISH WITH A STRONG ACCENT "Excuse me; I don't quite understand." "Could you please say that again?" Hi, everyone! I'm Georgiana, founder of SpeakEnglishPodcast.com.
More informationElements of Style. Anders O.F. Hendrickson
Elements of Style Anders O.F. Hendrickson Years of elementary school math taught us incorrectly that the answer to a math problem is just a single number, the right answer. It is time to unlearn those
More informationFamous Quotations from Alice in Wonderland
Famous Quotations from in Wonderland 1. Quotes by What is the use of a book, without pictures or conversations? Curiouser and curiouser! I wonder if I've been changed in the night? Let me think. Was I
More information2003 ENG Edited by
2003 (This is NOT the actual test.) No.000001 0. ICU 1. PART,,, 4 2. PART 13 3. PART 12 4. PART 10 5. PART 2 6. PART 7. PART 8. 4 2003 Edited by www.buchonet.com Edited by www.buchonet.com Chose the
More informationDisplay Dilemma. Display Dilemma. 1 of 12. Copyright 2008, Exemplars, Inc. All rights reserved.
I visited friends in New York City during the summer. They took me to this HUGE WalMart store. There was a display of cookie boxes that I could not believe! The display was in a pyramid shape with at
More informationMITOCW watch?v=6wud_gp5wee
MITOCW watch?v=6wud_gp5wee The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To
More informationSPEED DRILL WARMUP ACTIVITY
SPEED DRILL WARMUP ACTIVITY Name the operation representative of each of the following: percent left reduction total more half less twice off lower each double Write the equivalents: 20% as a decimal
More informationMary Murphy: I want you to take out your diagrams that you drew yesterday.
Learning Vocabulary in Biology Video Transcript Mary I want you to take out your diagrams that you drew yesterday. We are in the middle of a unit talking about protein synthesis, so today's class focused
More informationVideo  low carb for doctors (part 8)
Video  low carb for doctors (part 8) Dr. David Unwin: I'm fascinated really by the idea that so many of the modern diseases we have now are about choices that we all make, lifestyle choices. And if we
More informationREJECTION: A HUMOROUS SHORT STORY COLLECTION BY JESSE JAMISON
REJECTION: A HUMOROUS SHORT STORY COLLECTION BY JESSE JAMISON DOWNLOAD EBOOK : REJECTION: A HUMOROUS SHORT STORY COLLECTION Click link bellow and free register to download ebook: REJECTION: A HUMOROUS
More information+ b ] and um we kept going like I think I got
Page: 1 of 7 1 Stephanie And that s how you can get (inaudible) Should I keep going with that? 2 R2 Did you do that last night? 3 Stephanie Last 4 R2 Last time 5 Stephanie Um 6 R2 Did you carry it further?
More informationWelcome Accelerated Algebra 2!
Welcome Accelerated Algebra 2! TearOut: Pg. 445452 (Class notes) Pg. 461 (homework) U6H2: Pg. 390 #2124 Pg. 448 #67, 911 Pg. 461 #68 Updates: U6Q1 will be February 15 th (Thursday) U6T will be March
More informationUnit 7, Lesson 1: Exponent Review
Unit 7, Lesson 1: Exponent Review Let s review exponents. 1.1: Which One Doesn t Belong: Twos Which expression does not belong? Be prepared to share your reasoning. 8 1.2: Return of the Genie m.openup.org/1/8712
More informationQ. But in reality, the bond had already been. revoked, hadn't it? It was already set at zero bond. before September 21st, specifically on September 
0 0 September st, correct? Q. But in reality, the bond had already been revoked, hadn't it? It was already set at zero bond before September st, specifically on September  A. The bond was revoked on
More informationNorth Carolina Standard Course of Study  Mathematics
A Correlation of To the North Carolina Standard Course of Study  Mathematics Grade 4 A Correlation of, Grade 4 Units Unit 1  Arrays, Factors, and Multiplicative Comparison Unit 2  Generating and Representing
More informationBBC Learning English Talk about English Webcast Thursday March 29 th, 2007
BBC Learning English Webcast Thursday March 29 th, 2007 About this script Please note that this is not a word for word transcript of the programme as broadcast. In the recording process changes may have
More informationI HAD TO STAY IN BED. PRINT PAGE 161. Chapter 11
PRINT PAGE 161. Chapter 11 I HAD TO STAY IN BED a whole week after that. That bugged me; I'm not the kind that can lie around looking at the ceiling all the time. I read most of the time, and drew pictures.
More informationThe unbelievable musical magic of the number 12
The unbelievable musical magic of the number 12 This is an extraordinary tale. It s worth some good exploratory time. The students will encounter many things they already half know, and they will be enchanted
More informationTHE BENCH PRODUCTION HISTORY
THE BENCH CONTACT INFORMATION Paula Fell (310) 4976684 paulafell@cox.net 3520 Fifth Avenue Corona del Mar, CA 92625 BIOGRAPHY My experience in the theatre includes playwriting, acting, and producing.
More informationOur Musical, Mathematical Brains Ideas to turn STEM to STEAM
Debbie Char, St Louis Community College at Forest Park dchar@stlcc.edu Our Musical, Mathematical Brains Ideas to turn STEM to STEAM What you heard... Seasons of Love (the musical Rent) Upside Down, Inside
More informationGetting Graphical PART II. Chapter 5. Chapter 6. Chapter 7. Chapter 8. Chapter 9. Beginning Graphics Page Flipping and Pixel Plotting...
05GPFTCh5 4/10/05 3:59 AM Page 105 PART II Getting Graphical Chapter 5 Beginning Graphics.......................................107 Chapter 6 Page Flipping and Pixel Plotting.............................133
More informationFILED: NEW YORK COUNTY CLERK 09/15/ :53 PM INDEX NO /2017 NYSCEF DOC. NO. 71 RECEIVED NYSCEF: 09/15/2017 EXHIBIT I
EXHIBIT I Page 9 2 Q. So I'll try to help you with that. 3 A. Okay. 4 Q. Okay. And do you recall when you 5 looked at the attachment to the consignment 6 agreement between your company and Ms. Lutz 7 that
More informationPUBLIC SERVICE COMMISSION OF WEST VIRGINIA CHARLESTON * * * * * * * * * v. * TC * * * * * * * * * HEARING TRANSCRIPT * * * * * * * * *
PUBLIC SERVICE COMMISSION OF WEST VIRGINIA CHARLESTON * * * * * * * * * AD OJI * v. * 0001TC VERIZON WEST VIRGINIA, INC.,* * * * * * * * * * HEARING TRANSCRIPT * * * * * * * * * BEFORE: RONNIE MCCANN,
More information,FR.. BURNE T SCAN FROM THE DIOCESE OF JOLIET N
,FR.. BURNE T SCAN FROM THE DOCESE OF JOLET N0.  Redacted April01. Released April01 1 1 1 1.! 1 1 Q. Alright. 'd like to have you tell us 1 Well, first of all, could you just hold up this 1 picture,
More information#hsmath wthashtag.com/hsmath
wthashtag.com/hsmath Transcript from March 31, 2011 to April 1, 2011 All times are Pacific Time March 31, 2011 11:45 am mathfour: You gonna be there??when should you teach what?? the #homeschool #math
More informationSCANNER TUNING TUTORIAL Author: Adam Burns
SCANNER TUNING TUTORIAL Author: Adam Burns Let me say first of all that nearly all the techniques mentioned in this tutorial were gleaned from watching (and listening) to Bill Benner (president of Pangolin
More informationA Children's Play. By Francis Giordano
A Children's Play By Francis Giordano Copyright Francis Giordano, 2013 The music for this piece is to be found just by moving at this very WebSite. Please enjoy the play with the sound of silentmelodies.com.
More informationContractions Contraction
Contraction 1. Positive : I'm I am I'm waiting for my friend. I've I have I've worked here for many years. I'll I will/i shall I'll see you tomorrow. I'd I would/i should/i had I'd better leave now. I'd
More informationHow to read a poem. Verse 1
How to read a poem How do you read a poem? It sounds like a silly question, but when you're faced with a poem and asked to write or talk about it, it can be good to have strategies on how to read. We asked
More informationTestimony of Kay Norris
Testimony of Kay Norris DIRECT EXAMINATION 2 3 BY MS. SHERRI WALLACE: 4 Q. Ms. Norris, are you sick? 5 A. I am very sick. I have got strep 6 throat. 7 Q. I'm sorry you have to be down here. I 8 will try
More informationThe Movies Written by Annie Lewis
The Movies Written by Annie Lewis Copyright (c) 2015 FADE IN: INT. MOVIE THEATER  NIGHT,, and, all of them 16, stand at the very end of a moderate line to the ticket booth. As they speak, they move forward,
More informationdb math Training materials for wireless trainers
db math Training materials for wireless trainers Goals To understand why we use db to make calculations on wireless links. To learn db math. To be able to solve some simple exercises. To understand what
More informationSTUCK. written by. Steve Meredith
STUCK written by Steve Meredith StevenEMeredith@gmail.com Scripped scripped.com January 22, 2011 Copyright (c) 2011 Steve Meredith All Rights Reserved INTOFFICE BUILDINGDAY A man and a woman wait for
More informationChapter 13: Conditionals
Chapter 13: Conditionals TRUE/FALSE The second sentence accurately describes information in the first sentence. Mark T or F. 1. If Jane hadn't stayed up late, she wouldn't be so tired. Jane stayed up late
More informationUnit Four: Psychological Development. Marshall High School Mr. Cline Psychology Unit Four AC
Unit Four: Psychological Development Marshall High School Mr. Cline Psychology Unit Four AC The Ego Now, what the ego does is pretty related to the id and the superego. The id and the superego as you can
More informationSonarWiz Layback  CableOut Tutorial
SonarWiz Layback  CableOut Tutorial Revision 6.0,4/30/2015 Chesapeake Technology, Inc. email: support@chesapeaketech.com Main Web site: http://www.chesapeaketech.com Support Web site: http://www.chestechsupport.com
More informationLecture 5: Tuning Systems
Lecture 5: Tuning Systems In Lecture 3, we learned about perfect intervals like the octave (frequency times 2), perfect fifth (times 3/2), perfect fourth (times 4/3) and perfect third (times 4/5). When
More information