1 MATH 16 A-LECTURE. OCTOBER 9, 2008. PROFESSOR: WELCOME BACK. HELLO, HELLO, TESTING, TESTING. SO WE'RE IN THE MIDDLE OF TALKING ABOUT HOW TO USE CALCULUS TO SOLVE OPTIMIZATION PROBLEMS. MINDING THE MAXIMA OR MINIMA OF THING THIS YOU CARE ABOUT. AND SO WE HAVE A A STANDARD APPROACH LAST TIME. LET ME PUT IT UP AGAIN, REVIEW IT BY DOING ONE EXAMPLE AND STRETCH IT A LITTLE BIT. SO HOW DO YOU SOLVE AN OPTIMIZATION PROBLEM? YOU DRAW A PICTURE. SO YOU KNOW EVERYTHING THAT'S GOING ON AND WHAT ALL THE THING ARE THAT CAN VARY. AND THEN YOU LABEL PARTS OF THE PICTURE WITH NAMES. MIGHT BE THE LENGTH OF SOMETHING, THE WIDTH OF SOMETHING, AREA OF SOMETHING. GIVES NAMES TO ALL THE IMPORTANT QUALITY. SO WRITE DOWN, USING THOSE VARIABLES A FORMULA TO OPTIMIZE. AND OPTIMIZE MEANS EITHER MINIMIZE, SOMETHING LIKE COST, HIS OR HER MAXIMIZE IF IT'S SOMETHING LIKE THE AREA OF A GARDEN. SO YOU WRITE DOWN A FORMULA. AND THEN YOU WRITE DOWN ANOTHER FORMULA WHICH REPRESENTS THE CONSTRAINTS ON YOUR VARIABLES. LIKE YOU CAN'T USE MORE THAN 40 FEET OF FENCE. WE DID AT THAT LAST TIME. YOU CAN'T SPEND MORE THAN SO MUCH MONEY. SO WRITE DOWN THE FORMULA FOR THE CONSTRAINTS. SOLVE THE CONSTRAINTS FOR ONE OF THE VARIABLES IN TERMS OF OTHER. SO SOLVE FOR ONE OF THE VARIABLES. AND THEN YOU, LET ME CALL THIS FORMULA ONE, FORMULA TWO, SOLVE THE CONSTRAINTS, THAT'S FORMULA TWO FOR ONE OF VARIABLES. AND THEN YOU SUBSTITUTE THAT INTO FORMULA ONE TO GET AN EXPRESSION IN ONE VARIABLE. SO NOW I HAVE A FUNCTION, AND EXPRESSION IN ONE 2 VARIABLE AND I KNOW HOW TO OPTIMIZE THAT. FINALLY WE USE CALCULUS TO OPTIMIZE. SO THAT WAS THE GENERAL APPROACH WE USED LAST TIME, IN WORDS. AND LET ME JUST ILLUSTRATE IT AGAIN, GIVING YOU A NEW EXAMPLE. DOES ANYBODY ABOUT ANY QUESTIONS ABOUT THAT GENERAL APPROACH BEFORE I GO USE IT AGAIN? OKAY. SO HERE'S AN EXAMPLE. THE RULE, IT'S WILL I I LOOKED IT UP ON THE WEB, THE LOOK HAD IT WRONG. THE U.S. POST OFFICE RULES SAYS IF YOU WANT TO MAKE A PACKAGE, YOU MUST HAVE A CERTAIN SIZE LIMIT. TO GET A CERTAIN GOOD RATE. AND WHAT IS THE RULE? IT'S THE LENGTH PLUS WHAT THE POST OFFICE CHOOSES TO CALL THE GIRTH WHICH I'LL DEFINE IN A MINUTE, THAT'S TO BE AT MOST 108 INCHES. THE BOOK SAID 84, OBVIOUSLY AN OLD BOOK. AND THE GIRTH IS SORT OF THE PERIMETER GOING AROUND THE NARROW WAY. OKAY. SO HERE'S A TYPICAL EXAMPLE. SO HERE'S A TUBE. AND IT MIGHT BE A BLANK L FROM ONE ENDS OF THE TUBE TO THE OTHER. THE THE TUBE HAS RADIUS R, THEN THE GIRTH IS JUST THE SISH CUNCHES OF THE CIRCLE. CIRCUMFERENCE OF THE CIRCLE. SO THAT'S THE, THE GIRTH IS JUST THE SISH COVERAGE OF THE CIRCLE. TWO PI R. ALL THE WAY AROUND. SO THAT'S THE KIND OF SHAPE THAT YOU'RE ALLOWED TO MAIL. SO THE QUESTION IS WHAT SHAPE CYLINDER, HAS THE LARGEST VOWEL. WHAT IS THE MOST AMOUNT OF STUFF THAT YOU CAN MAIL IN THE POST OFFICE WITH THESE RULES. HOW BIG CAN I MAKE THE VOLUME. I'M ALLOWED TO VARY L AND ALLOWED TO VARY R. SO LET'S WRITE DOWN THE CONSTRAINTS IN THE FORMULA TO OPTIMIZE. SO WHAT I WANT TO DO IS MAXIMIZE THE VOLUME. I'LL GIVE THAT A NAME, V SO. IF I KNOW THE 3 RADIUS OF A CYLINDERS AND THE LEARNLINGS OF A CYLINDERS WHAT'S THE VOLUME. THE LENGTH TIMES THE AREA OF THE CROSS SECTION. SOTHAT'S GOING TO BE THE LENGTH. AND WHAT IS THE AREA OF THE CROSS SECTION, SO IT'S A CIRCLE OF RADIUS R. SO PI R-SQUARED. OKAY. SO THERE'S THE VOLUME. AND THE CONSTRAINT SAYS THAT THE LENGTH PLUS THE GIRTH, IS TWO PI R, THE SIR CURCHES OF THE CIRCLE IS ONE OWE EIGHT. THAT'S THE FIRST THREE STEPS UP THERE. EVERYBODY BY THAT MODEL? SO I HAVE TWO VARIABLES L AND R. HERE'S THE CONSTRAINT AND I WANT IS IT PICK THEM TO MAXIMIZE THAT. STUDENT: CAN YOU IT BE LESS THAN ONE OWE EIGHT. PROFESSOR: I'M MAKING A LEAP HERE. I'M SAYING THAT DO YOU THINK THE VOLUME IS GOING TO BE THE BIGGEST WHEN I MAKE THIS AS BIG AS POSSIBLE? PROBABLY. SO EVEN THOUGH THIS IS WHAT THE POST OFFICE SAYS, I'M USING YOUR INTUITION TO SAY I MIGHT AS WELL MAKE IT AS BIG AS POSSIBLE, THAT WILL MAXIMIZE THE VOLUME. SO LET'S GO AHEAD AND DO THAT. SO I'M GOING TO SOLVE THE CONSTRAINT. STUDENT: IS THE CONSTRAINT ALWAYS GIVEN. PROFESSOR: IN THE PROBLEM WE'RE TALKING ABOUT YOU CON GET STARTED UNLESS YOU HAVE SOME CONSTRAINT LIKE THIS. HERE IT CAME FROM THE POST OFFICE IT. DEPENDS ON THE SITUATION YOU'RE IN. WE'LL DO OTHER EXAMPLES. SO LET'S FOLLOW THE CONSTRAINT FOR ONE OF THE VARIABLES. AND LET ME MAKE SURE I PICKED THE ONE I LIKED. SO PRETTY EASE TO TO SOLVE. HERE'S THE LENGTH. SUBSTITUTE THAT INTO C. OKAY. AND IF I MULTIPLY THAT OUT I GET 108 PI R-SQUARED 4 MINUS TWO PI SQUARED R-CUBED. THERE'S A FUNCTION OF EXACTLY ONE VARIABLE R-AND I WANT TO MAXIMIZE. SO NOW FINALLY EVERYTHING HAS BEEN ALGEBRA OR A TINY OF GEOMETRY SO FAR. NOW IT'S TIME TO DO CALCULUS. SO, WE CAN PLOT THIS. THAT'S A GOOD IDEA IT CHECK BUT LET ME DO IT THE WAY WE'VE LEARNED SO FAR AM I'M GOING TO SOLVE V PRIME EQUALS ZERO, LOOK FOR THE CRITICAL POINTS WHERE THE TANGENT IS HORIZONTAL. SO LET ME SOLVE IT. SO TWO TIMES 108, THERE'S A DERIVATIVE OF THE FIRST TERM. OKAY. EVERYBODY