1 MATH 16 A-LECTURE. OCTOBER 2, 2008. PROFESSOR: WELCOME. WELCOME BACK. I'D LIKE TO START WITH A BELIEF ANNOUNCEMENT. STUDENT: MY NAME IS BRIAN. I'M A FIFTH YEAR STUDENT AT CAL. I'M RECRUITER. I WANTED TO LET YOU KNOW BOTH THE WALT DISNEY WORLD RESORTS ARE LOOKING FOR INTERNS SUCH AS YOURSELF TO INTERN IN EITHER RESORTS. IN ADDITION TO THE INTERNSHIP DISNEY OFFERS A PROGRAM WHERE THEY DO PROFESSIONAL DEVELOPMENT. AND IN ADDITION YOU GET TO MEET SOME OF THE EXECUTIVES THE COMPANY. THE PROGRAM IS CALLED THE DISNEY RESORTS COLLEGE PROGRAM. I HAVE FLIERS I'LL PASS OUT. GO TO DISNEY COLLEGE PROGRAM.COM. AND PRESENTATION ON-LINE ABOUT WHAT EXACTLY YOU WOULD DO AS AN INTERN AND PAY RATE. IF YOU HAVE ANY QUESTIONS AT ALL YOU CAN GO AHEAD AND ACCEPTED AN E-MAIL, AT DREAM COME TRUE @BERKELEY.EDU. AS PART OF INTERNSHIP NEXT YEAR YOU'LL GET PARKS, CALLED WHATWILL YOU CELEBRATE. COME FOR FREE, ON THEIR BIRTHDAY. IF YOU HAVE ANY QUESTIONS GO AHEAD AND E-MAIL ME. YOU DO ALSO GET FREE ADMISSION TO BOTH PARKS. THANK YOU. PROFESSOR: OKAY. SO MAKE THAT HOMEWORK ASSIGNMENT I GAVE YOU ABOUT WHAT HAPPENS TO A ROCKET THAT YOU SHOOT UP WOULD BE APPROPRIATE PREPARATION FOR THAT INTERNSHIP. THEY DO A LOT OF FIRE WORK THERE. FIGURE OUT WHERE THEY LAND. OKAY. SO WHAT I'D LIED TO DO TODAY IS KEEP TALKING ABOUT PLOTTING GRAPHS. AND I'LL WAIT UNTIL I HAVE YOUR UNDIVIDED ATTENTION. SO I'M GOING TO TALK ABOUT USING CALCULUS TO PLOT GRAPHS A BIT MORE. AND I'D 2 LIKE TO REVIEW FROM LAST TIME SINCE I PUT UP AN ALL OF LOT OF DEFINITIONS, SO LET ME REMINDS YOU OF WHAT THEY ALL ARE. SO. I'M GOING TO COMBINE GOING UP AND GOING DOWN IN ONE DEFINITION IT MAKE IT EASIER. A FUNCTION, I'LL WRITE IT THIS WAY IS INCREASINGON AN INTERVAL, X-ONE COMMA X-TWO I-E- IS GOING UP IF ITS DERIVATIVE IS POSITIVE ON THE INTERVAL. SO FOR X-BETWEEN X-ONE AND X-TWO AND THE OTHER COULD YOU REPEAT THE PART TO THAT IS DECREASING ON THAT INTERVAL, GOING DOWN IF ITS DERIVATIVE IS NEGATIVE. GOING UP AND GOING DOWN ARE THE TWO MOST OBVIOUS QUESTIONS TO ASK ABOUT A GRAPH. SO WE'RE GOING IT FIGURE OUT WHAT THAT IS BY ASKING WHEN IS A DERIVATIVE POSITIVE OR NEGATIVE. SO THE NEXT DEFINITION BY WAY OF REVIEW IS LOOKING FOR WHEN THE FUNCTION REACHES A MAXIMUM OR WHEN IT REACHES A MINIMUM. SO A FUNCTION F-OF X-HAS A RELATIVE MAXIMUM AT A POINT X-IF IT'S FLAT THERE. SO FOR THE REST OF TODAY UNTIL I TELL YOU OTHERWISE, I'M GOING TO ASSUME MY FUNCTION DOESN'T HAVE ANY CORNERS. SO FUNCTION CAN LOOK LIKE THAT. SO I WANT TO HAVE DERIVATIVES EVERYWHERE. SO IT HAS TO BE FLAT THERE. AND, WELL THERE ARE TWO WAYS TO DO IT. F-PRIME OF X-IS SO IF IT'S A MAXIMUM, THAT MEANS IT'S INCREASING ON ONE SIDE, ALL RIGHT, WRITE IT THIS WAY. F-PRIME GOES FROM POSITIVE, INCREASING TO NEGATIVE, AT THAT POINT, SO IT GOES FROM POSITIVE TO NEGATIVE, THAT'S ONE WAY TOSAY IT. OR YOU CAN ASK WHAT IS THE SECOND DERIVATIVE THERE. AND THE SECOND DERIVATIVE HERE HAS TO BE NEGATIVE. THAT MEANS IT'S TURNING AROUND AND GOING DOWN. THAT'S ONE THING BY WAIVE REVIEW 3 FROM LAST TIME. THAT'S WHAT IT HAS A RELATIVE MAXIMUM. AND THE OTHER THING WE TALKED ABOUT IS WHEN DOES IT HAVE A RELATIVE MINIMUM ON POINT X. SO AGAIN HAS TO BE FLAT THERE. AND F-PRIME IS NOW GOING TO GO FROM NEGATIVE TO POSITIVE, THAT MEANS IT LOOKS LIKE THAT, IT GOES DOWN AND BACK UP AGAIN, AND THE OTHER WAY WE TALKED ABOUT RECOGNIZING THAT WAS IF THE SECOND DERIVATIVE WAS POSITIVE AT THIS POINT X. THERE'S X. SO THAT WAS JUST LAST TIME'S LECTURE IN ONE BOARD. TO A LARGE EXTENT. MAKE SURE I DIDN'T SKIP ANYTHING. SO AND THERE'S ONE OTHER DEFINITION FROM LAST TIME TO REVIEW. WE SAY THAT F-OF X-IS CONCAVE UPWARDS, IN OTHER WORDS, IT LOOKS LIKE A SMILE IF YOU LIKE, THAT'S CONCAVE UPWARDS, ON AN INTERVAL IF THE DERIVATIVE ISAS A FUNCTION IS INCREASING ON THAT INTERVAL. SO IT GOES FROM SLOPING DOWN, NEGATIVE DERIVATIVE TO SLOPING UP A POSITIVE DERIVATIVE. SO THE F-PRIME IS INCREASING ALONG THERE. OR ANOTHER WAY TO SAY IT, IS THAT IF F-PRIME OF X-IS POSITIVE ON THE INTERVAL, THAT'S ANOTHER WAY WE DESCRIBED IT, OR THE CURVE, WHEREVER YOU DRAW A TANGENT LINE IT LIES ABOVE, F-OF X-IS ABOVE ANY TANGENT LINE. SO HERE I'VE DRAWN TWO TANGENT LINES AND YOU SEE CURVED LINES COMPLETELY ABOVE IT, THAT'S CONCAVE UPWARDS. AND ALONG WITH THAT GOES JUST SORT OF FLIP IT UPSIDE DOWN, IS LOOKS LIKE A FROWN INSTEAD. IT'S CONCAVE DOWNWARDS ON THAT INTERVAL. THAT'S GOING TO BE THE CASE IF F-OF X-IS DECREASING SO, SLOPE HERE IS POSITIVE, THEN IT TURNS INTO ZERO AND THEN GOES NEGATIVE THAT MEANS SLOPE IS DECREASING. ANOTHER WAY TO 4 RECOGNIZE IS THAT IS IF THE SECOND DERIVATIVE IS NEGATIVE. AND THE OTHER WAY YOU CAN SEE FROM THIS SPICT THAT F-OF X-IS BELOWANY TAKEN LINE. ANY TANGENT LINE I DRAW, THE CURVE IS ALWAYS BLE. SO ALL OF THOSE ARE KIND OF EQUIVALENT TO EACH OTHER. AND WE ALSO ALONG THE SAME SIDE, ALONG THE SAME IDEA OF BEING CONCAVE UP AND CONCAVE DOWN WE ASK WHERE DOES THE FUNCTION CHANGE EXR BEING CONCAVE UP TO CONCAVE DOWN. THOSE POINTS ARE INTERESTING TOO, WHERE IT CHANGES FROM ONE TO THE OTHER. SO IF F, IF THE SECOND DERIVATIVE IS ZERO, THAT'S THIS POINT HERE, AND F-DOUBLE PRIME IS OF LET'S SAY, LET ME MAKE THIS POINT A-JUST TO BE, IF F-DOUBLE PRIME OF A-IS ZERO AND F-DOUBLE PRIME OF X-IS LESS THAN ZERO ON ONE SIDE OF A,