1 MATH 16A LECTURE. NOVEMBER 13, 2008. PROFESSOR: WELCOME BACK TO CHAPTER FIVE IN MATH 16. SO DID YOU ENJOY YOUR BREAK OR YOUR MIDTERM MORE. STUDENT: BREAK. PROFESSOR: OH. MY QUESTIONS ABOUT THE MIDTERM? BEFORE WE GET -- ALL THE ANSWERS ARE POSTED. I'D SAY THE ONLY BAD NEWS ABOUT THE MIDTERM IS MAYBE PEOPLE DID SO MUCH BETTER THAN THE FIRST ONE I MIGHT HAVE TO MAKE THE FINAL HARDER. I HAVEN'T DECIDED. BUT YOU'LL GET THEM ALL BACK. IN DUE COURSE. THEY'VE ALL BEEN GRADED OBVIOUSLY BY NOW. YELL AT ME IF IT ISN'T. SO START CHAPTER FIVE WHICH I BEGAN ACTUALLY IN THE LAST THREE MINUTES OF LAST LECTURE. I'LL REMIND YOU OF THAT EXAMPLE. APPLICATIONS OF EXPONENTIAL AND LOG FUNCTIONS THEM COME UP IN ALL SORTS OF DIFFERENT PLACES AM WE'LL SPEND MOST OF OUR TIME TALKING ABOUT ALL EXAMPLES THAT COME UP IN ALL WALKS OF LIFE. ECONOMICS, BIOLOGY. TALKS ABOUT A MODEL, PREDICT THINGS LIKE POPULATION. AND I BEGAN THAT WITH A LITTLE HISTORICAL EXAMPLE LAST TIME. A FELLOW NAMED THOMAS -- PREDICT PESTILENCE. RADIOACTIVE DECAY. IT'S GOING TO BE THE SAME FORMULA. AND WE'RE GOING TO LEARN HOW TO USE A TECHNIQUE CALLED RADIO CARBON DATING. DIG UP SOMETHING VERY OLD, HOW DO YOU TELL HOW OLD IT IS, SAME IDEA. AND IF YOU'RE A ECONOMIC MAJOR IT'S THE SAME FORMULA FOR COMPOUND INTEREST. AND I'LL BE TALKING ABOUT THAT IN A LITTLE BIT MORE DETAIL. AND MANY MANY MORE AM SO THESE FUNCTIONS DESCRIBE MANY THING IN THE WORLD. AND THIS CHAPTER JUST TO MAKE IT MORE, CLEAR 2 THAT WE'RE GOING TO BE CONNECTED TO THE WORLD, WE'RE GOING TOCHANGE NOTATION AND THE VARIABLE WE'VE BEEN USING ALL SEMESTER X-TO INDICATE THE INDEPENDENT VARIABLE, WE'RE GOING TO CHANGE IT TO T-BECAUSE IT'S ALWAYS GOING TO BE TIME. BECAUSE WE'RE GOING TO BE ASKING HOW THING CHANGE WITH RESPECT TO TIME SO I MIGHT AS WELL USE T-TO REMINDS US OF THAT. SO THE STARTING POINT IS SOMETHING WE LEARNED LAST CHAPTER SO LET ME REMINDS YOU OF IT. THAT IF WE WRITE DOWN THIS DIFFERENTIAL EQUATION, THAT'S AN EQUATION WHERE THE UNKNOWN IS A FUNCTION AND IT'S Y-PRIME OF T-EQUAL THE CONSTANT K-TIMES Y-OF T-. THERE'S ONLY ONE SIMPLE SOLUTION. LOOKS LIKE THERE'S ANOTHER CONSTANT TIMES -- WHERE C-IS ANY CONSTANT YOU WANT TO PICK (ON BOARD). SO THIS FUNCTION SATISFIES THAT DIFFERENTIAL EQUATION. SO WHERE DOES C-COME FROM? WELL, THE SIMPLEST THING IS THAT IF YOU KNOW THE VALUE OF THE STARTING POINT TIMES T-EQUAL ZERO, THEN WHAT WE HAVE, WE HAVE Y-OF ZERO EQUALS C-TIMES E-TO THE K-TIMES ZERO. JUST LIKE IN T-TO THE ZERO. AND WHAT'S E-TO THE ZERO? ONE. SO THIS IS JUST EQUAL TO C. SO C-EQUALS Y-ZERO. AND WE HAVE AN EXACTLY THE SOLUTION. SO THAT'S THE SIMPLEST EXAMPLE. WE WILL DO MORE. SO WE'RE GOING TO USE THAT TO SOLVE DIFFERENTIAL EQUATIONS. LET'S START WITH EXAMPLE OF WHERE THAT DESCRIBES THINGS IN THE WORLD, IN THE WAY THEY GROW. AND WE'RE GOING TO CALL THAT EXPONENTIAL GROWTH, IF THE VALUE OF Y-IS GROWING IN TIME OR DECAY. SO POPULATIONS YOU'LL SEE THAT'S A PARTICULAR MODEL THAT'S GOING TO GROW. BUT WHEN SOMETHING DECAYS, RADIO DECAY, IT'S GOING TO BE 3 THE SAME EQUATION BUT WE'RE GOING TO CALL IT DECAY INSTEAD. SO LET ME JUST SORT OF SUMMARIZE ALL THE EXAMPLES. ALL THE EXAMPLES WE HAVE, Y-OF T-REPRESENTS THE AMOUNT OF SOME QUANTITY. THAT WE'RE TRYING TO COUNT. SO IT COULD BE THE AMOUNT OF SOME CHEMICAL. IT COULD BE THE NUMBER OF, IT COULD BE THE POPULATIONOF SOME SET OF ORGANISMS. SO AGAIN, THAT WOULD BE SOME NUMBER. IF YOU'RE THOMAS MAL FUSS IT COULD BE PEOPLE OR BACTERIA. ANY SORT OF ORGANISM THAT YOU CAN GET YOUR HAND ON. IT COULD BE MONEY. ANYTHING WHOSE QUANTITY IS INTERESTING THING TO COUNT. AND IN ALL THESE CASES, THE RATE OF CHANGE OF Y-OF T-WHICH IS, OF COURSE, RATE OF CHANGE IS JUST A SYNONYM FOR Y-PRIME IS GOING TO BE PROPORTIONAL TO THE POPULATION. SO IN OTHER WORDS, Y-PRIME OF T-IS GOING TO BE A CONSTANT TIMES Y-OF T-. THAT IS DIFFERENTIAL EQUATION. THERE'S GOING BE A DIFFERENT PHYSICAL REASON FOR EVERY ONE OF THESE CASES BUT IN ALL THOSE CASES Y-PRIME OF T-IS GOING TO BE PROPORTIONAL TO THAT. AND I'LL GIVE YOU SOME EXAMPLES WHY. AND NOW LET ME CHANGE NOTATION AGAIN. JUST TO BE CONSISTENT WITH THE BOOK. SINCE WE'RE COUNTING POPULATIONS MOST OF THE TIME, I'M GOING TO USE CAPITAL P-OF T-INSTEAD OF Y-OF T-TO MEAN POPULATION. SINCE P-JUST TO REMIND YOU THAT IT'S POPULATION. SO WHAT I'M SAKE IN ALL THESE CASES, P-PRIME OF T-IS GOING TO BE PROPORTIONAL TO THE CONSTANT P-OF T-AND THIS CONSTANT HAS A NAME, IT'S CALLED THE GROWTH CONSTANTS. JUST A WORD FOR THAT PARTICULAR QUANTITY, GOING TO TELL YOU HOW FAST THE POPULATION GROWS. IF K-IS BIGGER THIS FUNCTION GROWS OBVIOUSLY. 4 SO WHY IS THIS EQUATION TRUE? LET ME PUT DOWN TWO EXAMPLE AND APPEAL TO YOUR INTUITION THAT IT'S COMMON. SO LET'S SUPPOSE WE'RE TALKING ABOUT POPULATION OF BACTERIA. SUPPOSE THERE'S SOME FRACTION OF THAT POPULATION, SAY 1 PERCENT. POPULATION OF, WRITE DOWN POPULATION T-T. THAT'S HOW MANY BACTERIA THERE ARE, TIMES T, SO IF 1 PERCENT OF THAT POPULATION REPRODUCES EVERY HOUR, SO NOW T-IS GOING TO BE MEASURED IN HOURS, THAT'S GOING TO BE THE UNIT, THEN LET'S TRY TO WRITE THAT