1 MATH 16A LECTURE. NOVEMBER 4, 2008. PROFESSOR: SO SHALL WE GET STARTED? SOMEONE IS GIVING THE DEPARTMENT A TALK THIS AFTERNOON IN MY DEPARTMENT AND I SUGGESTED ON HIS SLIDE HE HAVE A LITTLE ICON ON THE BOTTOM THAT SAYS ELECTRICAL VOTES SO FAR SO PEOPLE WOULD ACTUALLY PAY ATTENTION. TODAY IS GOING TO BE A REVIEW. FOR MIDTERM TWO. AND IT'S THE SAME RULES AS MIDTERM ONE. IF YOU DON'T REMEMBER THEY'RE ALL ON THE WEB PAGE BUT IN PARTICULAR BRING PHOTO ID. AND ALL THE OTHER RULES ARE THERE TOO. SO IT'S GOING TO COVER SECTIONS 1.7 TO 4.3. WE'VE LECTURED PAST THAT BUT THAT'S WHAT WILL BE COVERED. BUT EARLIER MATERIAL IS FAIR GAME. I WILL ASSUME YOU SHOULD KNOW WHAT THE DEFINITION OF A DERIVATIVE IS, SO THAT WAS EARLIER. SHOULD KNOW WHAT A LIMIT IS, LAWS OF EXPONENTS. ALL THAT GOOD STUFF. SO I WILL JUST USE THAT. QUADRATIC FORMULA. OKAY. SO HERE'S WHAT I WANT TO COVER. AND LAST TIME I SUGGESTED THAT MAKING A TABLE OF YOUR OWN PERSONAL LIST OF THE MOST IMPORTANT THINGS WAS A GREAT WAY OF REVIEW. HERE'S MY PERSONAL MOST FAVORITE LIST. I'LL WRITE DOWN THE LIST HERE AND THEN GO THROUGH IN DETAIL. SO WE LEARNED IN A A LOT OF DIFFERENT SECTIONS, A LOT OF DIFFERENT RULES FOR DIFFERENTIATION. SINCE THEY'RE SPREAD OUT OVER A BUNCH OF CHAPTERS, MAKE ONE LIST, PUT THEM ALL IN ONE PLACE. MAKES IT EASIER TO REMEMBER. SO I WILL DO THAT. THEN HOW TO GRAPH A FUNCTION. AND THERE ARE LOTS OF DIFFERENT THINGS THERE BUT IT INCLUDES FINDING MINIMA AND MAXIMA. SO THAT'S IF YOU LIKE OPTIMIZATION, BUT IT'S PART OF GRAPHING TOO. 2 BUT THEN A VARIATION OF THAT IS OPTIMIZATION. MAKING SOMETHING, MAXIMIZING OR MINIMIZING BUT WITH CONSTRAINTS. A LOT OF DIFFERENT EXAMPLES OF THAT SORT. I'LL REVIEW IT. THEN WE HADIMPLICIT DIFFERENTIATION. AND THAT HAD TO DO WITH DIFFERENTIATING FUNCTIONS THAT YOU CAN'T WRITE DOWN. GIVEN BY AN EQUATION INSTEAD MUCH A EXPLICIT FORMULA. THEN A WHOLE BUNCH OF STUFF ON EXPONENTIALS. SO I WILL ALSO BECAUSE IT'S EASY TO REVIEW AT THE SAME TIME I WILL REVIEW THE RULES OF LOGARITHM EVEN THOUGH THEY'RE NOT ON THE EXAM. IT'S THROUGH THE END OF 4.3. JUST THROUGH THE EXPONENTIAL. AND THEN INTERSPERSED THROUGHOUT ALL THIS WERE APPLICATIONS. AND WORD PROBLEMS. THAT'S WHAT ALL WE COVERED. SINCE THE FIRST MIDTERM. NOW IF YOU WANT TO KNOW, IS THE EXAM GOING TO HAVE HOW MUCH OF THIS AND HOW MUCH OF THAT LOOK AT THE STUFF ON THE WEB PAGE. THAT'S THE BEST I CAN OFFER. LOOK AT THREE SAMPLE MIDTERM ON THE WEB PAGE AND YOU'LL SEE THIS KINDS OF STUFF. SO THAT'S WHAT I WANT TO COVER. ANYTHING THAT I MISSED SOMETHING THAT YOU REALLY LIKED. STUDENT: WHAT ABOUT 1.7 AND 1.8, IS THAT COVERED. PROFESSOR: THAT WAS APPLICATIONS OF DIFFERENT KINDS. STUDENT: NO IT WAS LIKE RATE OF CHANGE. PROFESSOR: I'M THINKING OF THAT AS APPLICATION OR WORD PROBLEM. INTERPRETS THE DERIVATIVE IS HOW FAST SOMETHING CHANGE BUT REALLY JUST USING DERIVATIVES. AND KNOWING THE LANGUAGE AND RATE OF CHANGE MEANS THE DERIVATIVES. STUDENT: OKAY. 3 PROFESSOR: SO LET'S GO ON TO RULES FOR DIFFERENTIATION., IN FACT, WHY DON'T I WRITE DOWN ALL THE RULES SINCE THE BEGINNING OF THE BOOK SINCE YOU HAVE TO KNOW THEM ALL SINCE THEY'RE ALL NECESSARY. FACTOR OUT CONSTANTS. WE DID THAT ONE BEFORE. IF YOU ADD THINGS OR SUBTRACT THEM THAT MEANS ADD OR SUBTRACT YOU JUST ADD OR SUBTRACT THE DERIVATIVES. SINCE I'M GOING TO BE LAZY, WRITE F-PRIME IS D-D-X-F-OF X. SO WE LEARNED THAT RULE. WE ALSO HAD THE POWER RULE. STILL GOING TO USE THAT ONE. BUT,SO THAT'S, YOU SHOULD KNOW THAT. HERE ARE THE NEW ONES MOVING FORWARD. WE HAVE THE PRODUCT RULE. WE SAID THAT D-D-X-OF F-TIMES G, USE SHORTHANDS. SO WE HAD THAT ONE AND WE HAD THE QUOTIENT RULE. (ON BOARD). IF YOU ONLY HAVE FOR SOME REASON THE ABILITY IT REMEMBER ONE REMEMBER THIS ONE AND THAT ONE BECAUSE THIS ONE WE FIGURED OUT BY USING THIS, F-TIMES G-TO THE MINUS ONE. SO IF PUT THESE TWO RULES TOGETHER YOU CAN FIGURE OUT THIS ONE. JUST IN CASE FOR SOME REASON IT SLIPS YOUR MEMORY. AND WE ALSO HAD HOW TO DIFFERENTIATE THE EXPONENTIAL FUNCTION WHICH IS ABOUT THE SIMPLEST RULE WE HAD SO FAR. DOESN'T CHANGE AT ALL. AND EVEN THOUGH THIS USES LOGARITHMS I'M GOING TO WRITE IT DOWN ANYWAY SINCE IT'S ALL PART THE SAME REVIEW. SO THE WAY WE DIFFERENTIATING B-TO ANY OLD POSITIVE CONSTANTS WAS WE WROTE IT AS B-AS EQUAL TO LOG B-AND WE GOT TIMES THIS, LOG B-TIMES B-(ON BOARD). SO WE HAD THAT RULE TOO. AND FINALLY, THEN WE ALSO HAD THIS ONE. I'M JUST PUTTING IT UP THERE FOR REVIEW. EVEN THOUGH IT'S NOT ON THE MIDTERM AND AND MAYBE THE MOST IMPORTANT ONE, ONE 4 OF THE MOST IMPORTANT ONES WAS THE CHAIN RULE. WHICH SAID THAT IF YOU COMBINE TWO FUNCTIONS, F OF G-OF X, THIS IS HOW YOU GET THE DERIVATIVE. SO THAT'S EVERYTHING. CAN FIT ON ONE SIDE. ONE BOARD. SO LET ME DO AN EXAMPLE. AND THE PURPOSE OF THIS EXAMPLE IS TO SAY IF SOMEBODY WRITE DOWN I'M GOING TO WRITE DOWN A FUNCTION THEY'RE GOING TO BE A LOT OF DIFFERENT WAYS IT APPLY THESE RULES TO GET THE SAME ANSWER BUT IF YOU'RE CLEVER YOU CAN SAVE YOURSELF SOME ALGEBRA BY REORGANIZING THE FUNCTION AND THEN IT'S EASIER TO APPLY THE