1 MATH 16A LECTURE. OCTOBER 23, 2008. PROFESSOR: WELCOME BACK. IT SEEMS TO WORK. SO I ALMOST FINISHED CHAPTER THREE LAST TIME. PRODUCT RULE, INCLUSION RULE. AND AS SOON AS YOU'RE QUIET I HAVE ONE MORE EXAMPLE TO DO. TO FINISH CHAPTER THREE. ALL RIGHT. ALMOST. IS THAT WORKING BETTER? OKAY. I NEED STIFFER SHIRTS. LET'S TRY IT THIS WAY AND SEE WHAT HAPPENS. SO LET ME DO ONE MORE EXAMPLE RELATED RATES. AND THE EXAMPLE HAS TO DO WITH ECONOMICS. AND SO LET ME JUST REMIND YOU THAT PRICE AND DEMANDS ARE RELATED. SO THE PRICE P-AND THE DEMAND X-OF ANY PRODUCT ARE RELATED. AND HOW ARE THEY RELATED THE PRICE GOES UP THE DEMAND GOES DOWN. SO LET ME PICK A PARTICULAR EXAMPLE OF THAT. THERE'S A VERY SIMPLE FORMULA THAT RELATES FOR A PARTICULAR ECONOMIC EXAMPLE. HOW THE PRICE AND DEMAND ARE RELATED TO ONE ANOTHER. IF I PLOT IT I'LL GET SOMETHING THAT LOOK LIKE THIS WHICH SAYS THAT IF THE DEMANDS GOES UP, THE PRICE GOES DOWN AND VICE VERSA. SO WHAT I'D LIKE TO KNOW IS, AND I'M GOING TO PICK A PARTICULAR POINT HERE, SO LET ME SAY P-IS MEASURED IN DOLLARS PER, LET'S SAY YOU SELL THINGS IN THE THOUSAND THINGS AT A TIME, A THOUSAND UNIT. AND X-IS GOING TO BE MEASURED IN UNITS OF A THOUSAND PRODUCTS THAT YOU'RE TRYING TO SELL, THEY COME PACKAGED. WHAT I'D LOOK TO DO IS ASK THE FOLLOWING QUESTION. AND THINK A PARTICULAR POINT TO ASK IT AT. SO THERE'S A PARTICULAR POINT ON THE CURVE. SUPPOSE THE PRICE P-STARTS FALLING STARTING AT THAT POINT ON THE CURVE AND IT STARTS FALLING BY 40 CENTS A WEEK. SO 2 YOU LOOK AT THE PRICE OF OIL. AND IT'S GOING DOWN A LOT FASTERTHAN THAT. GOING DOWN BY 40 CENTS A WEEK. AND THE QUESTION IS HOW FAST DOES THE DEMAND INCREASE? OKAY. THEY'RE RELATED TO ONE ANOTHER. IF YOU START MOVING ON THAT CURVE BY LOWERING THE PRICE AT SOME RATE, 40 SENT A WEEK, THE DEMANDS IS GOING TO GO UP. HOW FAST DOES IT GO UP. IF YOU'RE RUNNING A BUSINESS YOU HAVE TO START CHANGING YOUR BUSINESS AT A CERTAIN RATE TO ACCOMMODATE THE DEMAND. SO THAT'S THE QUESTION WE WANT TO ASK. SO I'M GOING TO THINK OF BOTH THE PRICE AND THE DEMAND ARE FUNCTIONS. OF TIME AND SO THEY SATISFY THAT EQUATION. SO LET ME WRITE DOWN THAT EQUATION AGAIN AM PRICE PLUS TWO TIMES DEMAND PLUS DEMAND TIMES PRICE EQUALS 38. THAT'S GOING TO BE TRY AT EVERY TIME, YOU'RE GOING TO BE AT SOME POINT ON THAT CURVE. WHAT I WANT TO DO NOW IS I'M GIVEN THAT EQUATION. I'M ALSO GIVEN THAT THE PRICE IS FALLING. WHAT DOES THAT TELL ME? THERE'S SOMETHING ELSE IT TELLS ME. I KNOW SOMETHING ABOUT P-PRIME OF T-. THAT TELLS ME HOW FAST IT'S FALLING. SO WHAT DOES THE ENGLISH SENTENCE TELL YOU ABOUT P-PRIME OF T? IT'S EQUAL TO WHAT? IS IT POSITIVE OR NEGATIVE? IT IT'S NEGATIVE BECAUSE IT'S FALLING. SO MINUS .4. AND WHAT DO WE WANT TO COMPUTE? I WANT TO KNOW HOW FAST IS THE DEMAND INCREASE AND WHAT QUANTITY IS THAT? I WANT TO KNOW HOW FAST SOMETHING INCREASES. THAT IS THE DERIVATIVE. X-PRIME OF T-IS WHAT I WANT TO KNOW. LET ME COME OVER HERE AND ACTUALLY DO THAT NOW. IT'S GOING TO BE THE SAME IMPLICIT DIFFERENTIATION WE DID BEFORE, SO I'LL WRITE 3 DOWN THE EQUATION. P-OF T-TIMES TWO X-OF T-EQUALS 38. (ON BOARD). AND WHAT I'LL GOING TO DO IS TAKE BOTH SIDES OF EQUATION AND DIFFERENTIATE WITH RESPECT TO T. IF I TAKE AN EQUATION AND DIFFERENTIATE IT, I GET ANOTHER EQUATION. LET ME DO THAT., OFCOURSE, OVER HERE I GET ZERO, IT'S A DERIVATIVE OF A CONSTANT. AND IF I JUST DIFFERENTIATE TERM BY TERM, BE CAREFUL ABOUT IT ALL, (ON BOARD) THAT'S WHAT I'M GOING TO GET. SO LET'S JUST DO IT. THIS TURNS THIS P-PRIME OF T. WHAT DOES THIS BECOME? TWO X-PRIME OF T. AND WHAT ABOUT THIS GUY? PRODUCT RULE. ANYBODY? X-OF T-TIMES P-PRIME OF T-PLUS X-PRIME OF T-TIMES P-OF T-ALL OF THAT EQUAL ZERO. IS THAT OKAY? JUST USING THE PRODUCT RULE. AND NOW I KNOW X-OF T-BECAUSE I'M GOING TO BE AT THAT .4 COMMA SIX. I KNOW P-PRIME OF T-BECAUSE I'VE BEEN TOLD THE PRICE IS FALLING AT MINUS .4. SO I'M GOING TO DO THIS NOW, SOLVE THIS GUY FOR X-PRIME. THAT'S WHAT I WANT. I WANT TO KNOW THE DEMAND. JUST SOLVE THAT EQUATION FOR THE DEMAND. AND SO THERE ARE TWO TERMS THERE. SO COLLECT THEM OVER HERE. PUT EVERYTHING ELSE IN THE OTHER SIDE OF THE EQUAL SIGN. AND SO I CAN FACTOR X-PRIME OUT OF THIS. (ON BOARD). AND NOW ALL I HAVE TO DO IS DIVIDE THROUGH BY THIS MULTIPLYER TO GET X-PRIME OF T. AND I GET X-PRIME OF T. I CAN FACTOR OUT HERE TOO. LET ME JUST DO THAT. I HAVE A P-PRIME IN BOTH CASES. SO I HAVE MINUS ONE, MINUS X OF T. AND NOW I'M GOING TO DIVIDE BOTH SIDES BY THIS. AND I GET MINUS ONE, MINUS X-OF T-DIVIDED BY TWO PLUS P-T-ALL TIMES P-PRIME OF T. (ON BOARD). SO THIS WAS JUST LITTLE BIT OF ALGEBRA. BUT 4 I FINALLY GOT THE FORMULA FOR X-PRIME OF T-. AND THAT'S WHAT I WANTED, SO IF I PLUG IF I GET X-PRIME OF T-IS GOING TO BE MINUS ONE MINUS, WHAT IS X? IS EQUAL TO FOUR. SO AT X-OF T-COMMA P-OF T-EQUALS FOUR COMMA SIX, THEN WHAT HAPPENS I PLUG THAT IN AND I GET ONE PLUS SIX TIMES ONE, MULTIPLY THAT BY MINUS FOUR, AND SO ALL TOGETHER THAT TURNS INTO ONE QUARTER. IF I DID THAT RIGHT. I DID SOMETHING, TWO, SORRY, TWO PLUSES. SORRY. THAT TURNS INTOONE QUARTER. IS