1 MATH 16 A-LECTURE. OCTOBER 7, 2008. PROFESSOR: SO WELCOME BACK. IS THE MICROPHONE WORKING? BARELY. I MAY HAVE TO STOP IN THE MIDDLE AND MAKE A QUICK CALL FOR SOMEONE TO SHOW UP WITH BATTERIES. SO THERE'S AN ANNOUNCEMENT FOR A JOB FOR THIS COURSE. SO THERE'S A REQUEST FOR A DIFFERENT, ANOTHER NOTETAKER. AND IF YOU WANT TO GO GET PAID I BELIEVE IT'S $40 PER LECTURE FOR TAKING NOTES. YOU CAN SEE THE CLASS WEB PAGE FOR THE JOB ANNOUNCEMENT. SO I WILL LEAVE IT AT THAT. SO, LET'S GET STARTED. SO LAST TIME, LAST TIME, LAST TIME I WAS IN THE MIDDLE OF AN EXAMPLE AND I THINK I WAS GOING A LITTLE TOO FAST AT THE END. SO LET ME BACK UP AND DO THAT EXAMPLE AGAIN AND THEN KEEP GOING. SO HERE IT WAS. TO PLOT, HELLO, THE FOLLOWING CURVE. ONE OVER X-SQUARED PLUS X. AND SO THIS IS GOING TO USE SEVERAL OF THE IDEAS WE HAD BEFORE. THE FIRST THING TO NOTICE IS THISPROBLEM HAS A VERTICAL ASYMPTOTE. AT A PARTICULAR PLACE WHERE YOU DIVIDE BY ZERO. THAT'S HOW YOU FIND VERTICAL ASYMPTOTES. SO THIS FUNCTION IS GOING TO GO TO INFINITY AT X-EQUALS ZERO. SO WHEN X-IS SMALL, NEAR ZERO, THAT IMPLIES THAT ONE OVER X-SQUARED IS LARGE, AND THAT IMPLIES THAT THE WHOLE FUNCTION IF YOU ADD THESE TWO THINGS TOGETHER. ADDING SOMETHING SMALL TO SOMETHING LARGE. SO WHEN X-IS SMALL IT REALLY JUST BEHAVES LIKE THIS FUNCTION WHICH WE KNOW. WE PLOTTED THAT ONE BEFORE. IT'S LIKE A LITTLE BIT LIKE A HYPERBOLA. BUT IF X-IS LARGE, THEN WHEN WE DIVIDE BY IT THAT MEANS ONE OVER X-SQUARED HAS TO BE SMALL. SO NOW I'M 2 GOING TO BE ADDING SOMETHING SMALL TO SOMETHING LARGE AND IT'S THE LARGE ONE THAT MATTERS. I THINK EXACTLY WHAT THIS PLOT LOOKS LIKE WHEN SOMETHING'S LARGE? IT MEANS BIG AND LARGE, PLUS A THOUSAND OR MINUS A THOUSAND. LOOK LIKE THE FUNCTION OF THEPLOT -- SO FOR X-LARGE, Y-IS GOING TO LOOK LIKE ONE OVER X. COME OVER HERE AND DO THAT. SO LET ME START PLOTTING IT. TO TWO FUNCTIONS I'M GOING TO ADD. HERE IS Y-EQUALS X- STUDENT: WHAT DO YOU MEAN BY X-SMALL AND X-LARGE. PROFESSOR: SO X-LARGE MEANS X-IS VERY FAIR AWAY FROM ZERO IT'S WAY OVER THERE OR WAY OVER THERE. LARGE IN ABSOLUTE VALUE. AND THEN OUT THERE, THE FUNCTION Y-EQUALS X-WHICH IS PART OF WHAT WE'RE AFTER IS GOING TO BE WAY DOWN HERE OR WAY UP THERE. THAT'S WHAT THIS ONE LOOKS LIKE. SO NOW WHAT DO I WANT TO ADD TO THAT? I'M BREAKING UP THIS, TRYING TO PLOT ONE OVER X-SQUARED PLUS X-. SO TRYING TO UNDERSTAND THE TWO PARTS OF IT. X, THAT'S A STRAIGHT LINE, THAT'S EASY. I WANT TO UNDERSTAND ONE OVER X-SQUARED. SO WHAT DOES ONE OVER X-SQUARED LOOK LIKE? I THINK WE'VE DONE THAT ONE BEFORE. IT LOOK LIKE THIS. SO THERE, THIS IS THE PLOT OF ONE OVER X SQUARED. WHY IS THAT? WHEN X-GOES TO ZERO, YOU HAVE A VERTICAL ASYMPTOTE AM YOU'RE DIVIDING BY ZERO. IT BLOWS UP. WHEN IT GETS LARGE ONE OVER X SQUARED GETS SMALL.BUT WHAT I WANT TO DO IS ADD THESE TWO CURVES. THAT'S MY GOAL. WAY OUT HERE WHEN X-IS LARGE, TAKING THIS ONE AND ADDING IT TO THE SMALL ONE. THE ONLY THING HAD A MATTER IS THE LARGE NUMBER, AND SO MY CURVE X-PLUS ONE OVER X-SQUARED IS GOING TO BE VERY 3 VERY CLOSE TO Y-EQUALS X. ONLY ADDING THIS TINY BIT TO IT. OVER HERE, IT'S THE SAME STORY. HERE I'M, WHAT I'M GOING TO DO IS ADD X-PLUS ONE OVER X-SQUARED, X-IS WAY DOWN HERE, ONE OVER X-SQUARED IS TINY SO THE FUNCTIONS GOT TO LOOK LIKE THIS. IT'S GOT TO BE ALMOST ON TOP OF THE Y-EQUAL X-. BECAUSE I'M ADDING THIS NUMBER TO THAT NUMBER AND IT'S GOING TO BE RIGHT THERE. OVER HERE I'M ADDING THIS NUMBER TO THAT NUMBER. AND I GET RIGHT THERE. SO I'M ADDING BOTH, BY PICTURE. I HOPE THAT'S OKAY. WHEN X-IS SMALL, WHAT I'M SAYING IS WHEN YOU'RE RIGHT AROUND ZERO, THAT'S WHEN I MEAN BY X-SMALL. I WANT TO ADD THIS CURVE. THIS, THESETWO ASYMPTOTES TO THIS THING. I WANT TO ADD THAT TO THIS. THIS IS TINY, RIGHT? SO WHEN I ADD IT, I'M ALSO GOING TO GET SOMETHING BLOWING UP LIKE THIS. SO WHEN X-IS REALLY SMALL I KNOW THE CURVE HAS TO STILL GO UP. SO THOSE TWO PARTS I UNDERSTAND. I JUST DON'T KNOW WHAT IT LOOKS LIKE IN BETWEEN. I DON'T KNOW WHAT IT LOOKS LIKE IN THIS RANGE OR IN THIS RANGE. SO I'M, SO NOW WE'RE GOING TO USE ALL OUR CALCULUS TO FILL IN THE ENTIRE PLOT. BUT WHEN I'M SAYING IS YOU CAN LOOK AT THIS AND SAY I KNOW WHEN X-IS LARGE ENOUGH IT'S GOING TO BE UP HERE OR DOWN HERE. WHEN X-IS TINY IT'S GOING TO LOOK LIKE THE STUFF THAT'S BLOWING UP. SO I KNOW WHAT HAPPENS THERE. I HAVE TO FILL IN THE REST. STUDENT: (INAUDIBLE QUESTION). PROFESSOR: TO GET THIS NUMBER, I TAKE THAT DISTANCE PLUS THAT DISTANCE AND THAT MOVES ME UP A LITTLE BIT. THE VECTORS ARE ALL 4POINTING STRAIGHT UP. YOU CAN THINK OF IT AS VECTORS. IS THAT OKAY? SO, LET'S DO OUR USUAL THING. START WITH A FUNCTION. AND I WANT IT ASK WHERE IS IT INCREASING OR DECREASING. WHERE IS IT FLAT. WHERE'S THE MINIMA AND MAXIMA. SO KEEP ALL THE DERIVATIVES. SO WHAT IS THE FIRST DERIVATIVE? JUST THE POWER LAW. THIS IS X-TO THE MINUS TWO, SO THE DERIVATIVE IS (ON BOARD). AND THEN DERIVATIVE OF ONE. HOW ABOUT Y DOUBLE PRIME. IT'S -- DERIVATIVE IS SIX OVER X-TO THE FOURTH. SO IF I LOOK AT THIS, IF I PLUG IN ANY NUMBER, CAN YOU TELL ME WHETHER THIS IS GOING TO BE POSITIVE OR NEGATIVE? IT'S ALL GOING TO GO POSITIVE UNLESS, AS LONG AS I DON'T DIVIDE BY ZERO. IF X-ANY OTHER NUMBER, X-NOT FOURTH POWER IS POSITIVE, SO THAT