1 MATH 16 A-LECTURE. SEPTEMBER 16, 2008. GSI: AN ANNOUNCEMENT. GSI: MY NAME IS CASPER AND THIS IS MELISSA. N-TAKE THIS OPPORTUNITIES TO INVITE ALL OF YOU TO THE FIRST EVENT THIS SEMESTER. THIS WILL BE A GOOD OPPORTUNITIES FOR YOU TO EXPLORE DIFFERENT INDUSTRIES OF BUSINESS. MANY OF OUR ALUMNI -- SO ONCE AGAIN EVENTS 730 IN WHEELER AND THANK YOU FOR YOUR TIME AND HOPE TO SEE YOU THERE. PROFESSOR: SO WELCOME BACK. I WAS IN THE MIDDLE OF TALKING ABOUT CONTINUITY IN DERIVATIVES. LET ME BACK UP A TINY LITTLE BIT AND REMINDS YOU OF A FEW OF THING I WAS TRYING TO SAY. THE QUESTION WE WERE ASKING OURSELVES WAS, HOW DO WE RECOGNIZE WHEN A FUNCTION F-OF X-HAS A DERIVATIVE. SO WE WROTE DOWN THE DEFINITION. WE HAD IT ASK OURSELVES, A DERIVATIVE AT A PARTICULAR POINTS A. SO WHAT WE DID WAS TOOK (INAUDIBLE). HERE'S THE DEFINITION. THE QUESTION WAS WHAT DOES IT TAKE FOR THIS TO EXIST. IF IT'S EXIST, THAT'S THE DEFINITION OF F-PRIME OF A. AND EASIEST THICK YOU CAN DO WITHOUT DOING ANY WORK AT ALL IS TO SAY AS H-GETS TINY AND DIVIDE BY ZERO SO THE NUMERATOR BETTER GET DINE I. OTHERWISE I'M GOING TO BE DIVIDING BY ZERO AND IT WON'T WORK. SO WE CLEARLY NEED A NUMERATOR BETTER GO TO ZERO. AND WE GAVE THAT NICE PROPERTY A NAME. WE CALLED IT CONTINUITY. SO F-OF X-IS CONTINUOUS AT A, THAT DENOMINATOR GOES TO ZERO. SO LET ME WRITE DOWN, LIMIT WHEN H-GOES TO ZERO (ON BOARD). AND THE INTUITION IS THAT LET'S SAY THAT'S A. WHAT'S, 2YOU CAN DRAW THE FUNCTION THROUGH THE POINT WITHOUT THE LIFTING YOUR CHALK OFF THE THE BOARD. THAT WAS VERY SIMPLE IDEA. YOU CAN PLOT F-OF X-NEAR X-EQUALS A-WITHOUT LIFTING YOUR PENCIL OR EXAMPLE, RUNNING OUT OF LEAD. BECAUSE THE FUNCTION WIGGLES SO MUCH. ON THE OTHER HAND, AND THIS IS JUST WHERE I ENDED LAST TIME, SO WHAT WE JUST SAID HERE IS THAT DIFFERENT SHABILITY IMPLIES CONTINUITY. SO WRITING DOWN IN ENGLISH THE SAME THING WE JUST SAID. SO IF IT'S DIFFERENTIABLE THE NUMERATOR THAT'S TO GO TO ZERO, IT HAS TO BE CONTINUOUS. ANOTHER WAY TO SAY THAT IS CONTINUITY IS NECESSARY FOR DIFFERENT SHAIBILITY. SO THE VERY LAST THING I SAID WAS IT'S NOT ENOUGH ALL BY ITSELF. NOT ENOUGH BY ITSELF. SO LET ME DO THE EXAMPLE THAT REMIND YOU OF THE EXAMPLE WE HAD LAST TIME. LET'S TAKE THE FUNCTION F-OF X-EQUALS ABSOLUTE VALUE OF X. IT ASKS, THE PLOT OF THIS IS THERE, THERE, THERE IT IS, OVER HERE, F-OF X-EQUALS X-AND OVER HERE F-OF X-EQUALS NEGATIVE X. SO THE QUESTION IS IS THIS FUNCTION DIFFERENTIABLE AT X EQUALS ZERO. AND SO LET ME APPEAL TO YOUR INTUITION. SO WHAT DOES THAT MEAN, I-E, DOES IT HAVE A TANGENT LINE. AND WHAT'S THE TANGENT LINE? IS THERE A LINE THROUGH THIS POINT, THROUGH X-COMMA F-OF X-WHICH IS JUST THE ORIGIN, IS THAT, SO WHAT THE TANGENT LINE HAVE TO BE THAT ONLY INTERSECTS THE CURVE, ONLY INTERSECTS THE GRAPH ONCE NEAR THAT POINT. CAN'T GO THROUGH IT TWICE. AND IT'S UNIQUE. THERE'S ONLY ONE WAY TO DRAW IT. ONLY ONE WAY TO DRAW A TANGENT LINE. SO LOOK AT THIS POINTS AND ASK CAN I DRAW A LINE THROUGH THERE THAT ONLY THE 3 INSECT AT ABSOLUTE VALUE PLOT ONCE. YEAH, SHIEWMPLET AND THERE'SONLY ONE WAY TO DO IT. NO, I CAN ALSO DRAW THAT ONE. THAT ONE WORKS, AND THAT ONE WORK. THERE'S NO SINGLE TANGENT LINE, THERE'S NO LIMIT. UNIQUE, THE DEFINITION OF A LIMIT SAYS THE SLOPE, THERE'S EXACTLY ONE SLOPE. WHEN YOU WRITE DOWN THAT LIMIT UP THERE, THAT THING, DIFFERENT VALUES. JUST ONE VALUE THAT IT APPROACHES. JUST ONE TANGENT LINE AM AND HERE YOU CAN SEE THERE'S A BUNCH OF THEM. SO THAT'S THE INTUITION THAT SAYS CORNERS ARE PROBLEMS. YOU DON'T GET TANGENTS IN CORNERS. SO THAT'S THE GEOMETRIC INTUITION. BUT LET JUST PLUG IT IN THE ALGEBRA. AND ASK DOES THIS LIMIT EXIST? SO HERE IT IS. THERE'S, THAT'S THE LIMIT YOU WANT TO TALK, EVALUATING IT AT X-EQUALS ZERO. SO LET ME JUST KEEP GOING. THIS EXIST. SO IF IT DOES, LET ME PLUG IN H-GOES TO ZERO, ABSOLUTE H-MINUS ABSOLUTE ZERO, AND, DIVIDED BY H. OKAY. (ON BOARD). SO THAT'S THE FUNCTION. WHAT DOES THIS FUNCTION LOOK LIKE? SO THERE'S THE FUNCTION. I WANT TO KNOW DOES IT APPROACH LIMIT AS P-GET SMALL. SO LET ME PLOT IT. GOING TO HAVE H-ON THAT AXIS. AND PLOT OF GRAPH OF THIS FUNCTION. WHAT DOES THIS FUNCTION EQUAL WHEN H-IS POSITIVE? AND WHAT HAPPENS WHEN H-IS NEGATIVE? THERE'S PLUS ONE. WHAT HAPPENS WHEN H-IS NEGATIVE? IT'S NEGATIVE ONE. AND SO THAT'S WHAT THE GRAPH OF THIS THING LOOKS LIKE. DOES IT HAVE A LIMIT? AS H-GOES TO ZERO? NO BECAUSE ON THIS SIDE IS GETS CLOSER TO ONE, IT'S ONE. ON THIS SIDE IT'S NEGATIVE ONE AM REMEMBER LIMIT HAVE TO APPROACH A SINGLE NUMBER AS YOU GET CLOSE 4 TO THE POINTS. SO YOU WANT TO GO BY YOUR INTUITION WHICH SAYS AT A CEARN THERE IS NO UNIQUE WAY TO DRAW A TANGENT. OR YOU DO THEALGEBRA. THIS FUNCTION DOESN'T HAVE A LIMIT. EITHER WAY YOU CONVINCED YOURSELF THAT IT WAS NOT ENOUGH THAT THIS FUNCTION WAS CONTINUOUS. THAT I CAN DRAW THE ABSOLUTE VALUE FUNCTION JUST BY LEAVING CHALK ON THE BOARD. SO ARE THERE ANY QUESTIONS ABOUT THAT? I SEE A FEW PUZZLED FACES SO IT'S OKAY TO ASK. FOR EXAMPLE, NO DERIVATIVES AT CORNERS. OKAY. SO YOU HAVE TO BE