"Annenberg Media" "¶" "DIFFERENTIALCALCULUS IS A POWERFUL MATHEMATICAL TOOL" "FORANALYZING HOWTHINGSCHANGE." "THEHANDLESON THISTOOL" "AREA FEWSIMPLERULES FOR CALCULATING DERIVATIVES." "SOMEWHERE AROUND 600 B.C.," "SOMEONE DISCOVERED THAT IF YOU WANT TO MAKE PLEASANT-SOUNDING CHORDS" "ON A STRINGED INSTRUMENT," "THEN THE LENGTHS OF THE STRINGS" "OUGHT TO BE IN THE RATIO" "OF SIMPLE NUMBERS LIKE 1:2." "THESE ARE CALLED THE PYTHAGOREAN HARMONIES." "THE DISCOVERY WAS IMPORTANT" "BECAUSE IT WAS THE FIRST DISCOVERY OF A CONNECTION" "BETWEEN MATHEMATICS AND THE PHYSICAL WORLD." "UNFORTUNATELY, THAT CONNECTION WAS FORGOTTEN" "AND HAD TO BE REDISCOVERED SLOWLY AND PAINFULLY" "THOUSANDS OF YEARS LATER." "ONE PERSON WHO CERTAINLY GRASPED IT, THOUGH," "WAS GALILEO GALILEI." "I WANT TO READ TO YOU SOMETHING THAT GALILEO WROTE." "THIS BOOK WAS PUBLISHED IN ROME IN 1623," "AND IT'S CALLED IL SAGGIATORE," "WHICH IS USUALLY TRANSLATED TO MEAN THE ASSAYER." "BUT I PREFER TO CALL IT THEEXPERIMENTALIST" "BECAUSE I THINK THAT'S MORE LIKE WHAT GALILEO HAD IN MIND." "GALILEO HAD THIS NASTY HABIT" "OF MAKING HIS FAMOUS REMARKS IN ITALIAN." "BUT I'LL TRANSLATE IT AS I GO ALONG." "HE SAID," "TRUEKNOWLEDGEIS WRITTEN INTHISENORMOUSBOOK," "WHICHISCONTINUOUSLYOPEN BEFOREOUREYES." "ISPEAKOF THEUNIVERSE." "BUT ONE CAN'T UNDERSTAND THAT" "UNLESSFIRSTONELEARNS" "TOUNDERSTANDTHELANGUAGE INWHICHIT 'SWRITTEN." "IT'SWRITTEN IN THE LANGUAGE OF MATHEMATICS." "SO TO PREPARE TO READ THE BOOK OF THE UNIVERSE," "WE MUST FIRST LEARN THE SYMBOLS AND THE VOCABULARY" "OF THE LANGUAGE OF MATHEMATICS." "IT'S A LANGUAGE OF PRECISION AND POETRY" "AND EVEN OF MUSIC." "FOR A LONG TIME," "PHYSICISTS HAVE USED THE LANGUAGE OF MATHEMATICS." "AND SINCE SOMEWHERE AROUND 600 B.C.," "SO HAVE MUSICIANS." "AS WITH JUST ABOUT ALL LANGUAGES" "INCLUDING MUSIC" "MATHEMATICS HAS ITS OWN VOCABULARY," "ITS OWN RULES AND SYMBOLS, ITS PRECISION AND ELEGANCE," "ITS POETRY AND ITS HISTORY." "ONE PART OF THAT HISTORY WAS GALILEO GALILEI," "WHO WAS SOMETHING OF A NONCONFORMIST." "IT WAS A TRAIT HE INHERITED FROM HIS FATHER VINCENZO," "WHO WAS AN ACCOMPLISHED MUSICIAN." "MUSICALLY," "VINCENZO REFUSED TO BE BOUND BY TRADITIONAL FORMS" "A POSITION THAT WOULD BECOME THE FAMILY TRADEMARK." "HE WROTE A BOOK" "OPPOSING THE USE OF PYTHAGOREAN HARMONIES" "AMONG HIS MUSICAL CONTEMPORARIES." "HE CONSIDERED THE ANCIENT GREEK CHORDS" "TOO SIMPLE FOR THE COMPLEX MUSICAL STRUCTURES" "OF THE ITALIAN RENAISSANCE." "LATER-- LIKE FATHER, LIKE SON" "GALILEO CONSIDERED THE GREEK STANDARDS OF MATHEMATICS" "TOO SIMPLE TO EXPRESS HIS IDEAS." "HE CREATED THE SCIENCE OF KINEMATICS," "A BRANCH OF MECHANICS" "DEALING WITH MOTION IN THE ABSTRACT." "AND IF ANY ABSTRACT IDEA IS TO BE EXPRESSED PROPERLY," "IT NEEDS APPROPRIATE LANGUAGE" "THE CONCEPTS AND SYMBOLS" "THAT GIVE AN IDEA ITS MEANING AND VALUE." "AS ADVANCED AS IT WAS, GALILEO'S NEW SCIENCE OF MOTION" "WAS STILL ROOTED IN THE SOIL OF THE ANCIENT GREEK INTELLECT." "AND WERE ANYTHING ENTIRELY NEW TO BLOOM" "IN THE FIELDS OF MATHEMATICS AND SCIENCE," "SCHOLARS NEEDED A LANGUAGE" "MORE SOPHISTICATED THAN THE ONE THAT HAD BEEN SPOKEN" "SINCE ARCHIMEDES AND EUCLID." "IN OTHER WORDS," "AFTER GALILEO, PHYSICS NEEDED AN ADVANCED LANGUAGE." "FEWER THAN 25 YEARS AFTER HIS DEATH," "THAT VERY LANGUAGE WOULD BE DISCOVERED" "AND IN ONE FORM OR ANOTHER, SPOKEN EVER AFTER." "IT WOULD COME TO BE CALLED DIFFERENTIAL CALCULUS." "DIFFERENTIAL CALCULUS IS VERY POWERFUL," "AND AS WITH ANY LANGUAGE," "IT DERIVES ITS POWER FROM THE IDEA BEHIND IT" "THE DERIVATIVE." "THE DERIVATIVE IS TO KINEMATICS WHAT THE WHEEL IS TO TRAVEL" "A SIMPLE YET VERY EFFECTIVE MEANS" "FOR GETTING FROM ONE PLACE TO ANOTHER." "AND TO GET JUST THE RIGHT PERSPECTIVE" "OF WHAT ELSE A DERIVATIVE IS," "NOTHING WORKS BETTER THAN A LITTLE EXERCISE." "A DERIVATIVE DOESN'T APPLY ONLY TO A BODY IN HORIZONTAL MOTION," "NOR FOR THAT MATTER," "ONLY TO A BODY IN VERTICAL MOTION." "UP, DOWN, ANY WAY" "A DERIVATIVE IS THE RATE OF CHANGE OF ANY FUNCTION" "AT ANY EXACT POINT OR INSTANT." "AS ILLUSTRATED BY GALILEO'S LAW OF FALLING BODIES," "SPEED IS THE DERIVATIVE OF DISTANCE." "BUT IT'S MORE THAN THAT." "A DERIVATIVE CAN REPRESENT THE RATE OF CHANGE OF ANYTHING" "FOR EXAMPLE," "THE POPULATION DENSITY OF DOLPHINS" "IN RELATION TO INCREASING OR DECREASING WATER TEMPERATURE." "OR THE RATE OF CHANGE IN THE VOLUME OF A BALLOON" "VERSUS ITS SURFACE AREA." "OR THE RATE OF CHANGE IN THE PRICE OF A PIZZA" "WITH RESPECT TO ITS SIZE." "CLEARLY THEN," "THE CONCEPT OF THE DERIVATIVE GOES FAR AND WIDE." "BUT THE MECHANICAL PROCESS OF THE DERIVATIVE" "DIFFERENTIAL CALCULUS" "NEEDS A PRACTICAL APPROACH" "OR THE CONCEPT GOES NOWHERE FAST." "EVENTUALLY, WITHOUT THE RULES OF DIFFERENTIATION," "THE SIMPLE CONCEPT OF THE DERIVATIVE" "BECOMES AN UPHILL STRUGGLE." "IN THE LONG RUN," "IT HELPS TO PICK UP A FEW MORE DEFINITIONS" "ALONG THE WAY." "SO BEFORE IT'S TOO LATE TO TURN BACK," "CONSIDER STEEPNESS." "ON THE INCLINE," "THE STEEPNESS IS THE RATIO OF THE CHANGE IN ELEVATION" "TO THE CHANGE IN HORIZONTAL DISTANCE." "THIS RATIO--A NUMBER-- IS CALLED THE SLOPE." "FOR EXAMPLE," "SUPPOSE THE ELEVATION OF AN INCLINE" "INCREASES 15 METERS EVERY 100 METERS." "THE RIDER MOVES UPWARD 15, HORIZONTALLY 100." "THE SLOPE IS .15." "A HILL WITH A SLOPE OF .30" "IS TWICE AS STEEP" "AS ONE WITH A SLOPE OF .15." "THE BIGGER THE SLOPE, THE STEEPER THE HILL." "WHEN THE SLOPE IS LARGE," "IT'S NO SMALL FEAT TO GET TO THE TOP." "WHEN THE SLOPE IS NEAR ZERO, IT'S EASY GOING." "AND WHEN THE SLOPE IS NEGATIVE, IT'S DOWNHILL ALL THE WAY." "ALTHOUGH MATHEMATICS CAN BE EASY GOING," "IT DOES HAVE ITS PEAKS AND VALLEYS," "AND IT ALWAYS HAS." "NOBODY KNOWS WHO FIRST ASKED" "THE BEST WAY TO GET FROM HERE TO THERE." "BUT THE ANSWER, IN ALGEBRAIC TERMS," "WAS FIRST OFFERED" "BY A FRENCH MATHEMATICIAN NAMED FERMAT." "IN 1629, HE CAME UP WITH THE IDEA" "OF FINDING THE TANGENT LINE TO AN ARBITRARY POINT ON A CURVE." "IN 1638, FERMAT SHARED HIS DISCOVERY" "WITH HIS COUNTRYMAN, RENE DESCARTES," "WHO HAD HIS OWN METHOD" "FOR FINDING TANGENTS TO ALGEBRAIC CURVES." "MANY OF THESE MATHEMATICAL IDEAS" "ESPECIALLY THOSE OF FERMAT" "WERE FURTHER DEVELOPED" "BY WILHELM LEIBNITZ AND ISAAC NEWTON" "INTO A GENERAL AND SYSTEMATIC METHOD OF MATHEMATICAL ANALYSIS" "DIFFERENTIAL CALCULUS." "SETTING HISTORY ASIDE-- AT LEAST FOR THE PRESENT" "SOME TIMELY QUESTIONS REMAIN." "FOR EXAMPLE," "ON A SMOOTHLY CHANGING CURVE," "THERE'S A CONSTANTLY CHANGING SLOPE." "HOW THEN, IN TODAY'S LANGUAGE," "IS A SLOPE CALCULATED AT ANY GIVEN POINT?" "TO DETERMINE THE SLOPE AT A PARTICULAR POINT" "HERE, FOR EXAMPLE" "SIMPLY TAKE ANOTHER POINT ON THE HILL." "IT DOESN'T MATTER WHERE." "NOW CONNECT THE TWO POINTS WITH A STRAIGHT LINE." "THAT LINE IS CALLED A CHORD," "AND ITS SLOPE DEPENDS ON" "THE LOCATION OF THE SECOND POINT." "IF THE FIRST AND SECOND POINTS ARE REASONABLY CLOSE," "THE CHORD IS A REASONABLY GOOD APPROXIMATION" "OF THE BIKE'S PATH." "NOW MOVE THE SECOND POINT" "CLOSER TO THE FIRST." "THE SLOPE IS A NUMBER." "AS THE POINTS GET CLOSER TOGETHER," "THE NUMBER GETS CLOSER TO A CERTAIN VALUE." "THAT NUMBER IS CALLED THE SLOPE OF THE HILL" "AT THAT POINT." "THE LINE WITH THAT SLOPE THROUGH THE POINT" "IS CALLED THE TANGENT LINE." "THE CHORD TURNS INTO THE TANGENT LINE" "AS THE POINTS GET CLOSER TOGETHER." "THE SLOPE OF THE TANGENT LINE AT THAT POINT" "IS THE SLOPE OF THE HILL." "INSTANTANEOUS SPEED CAN BE CALCULATED" "ALONG THE SAME LINES." "GALILEO'S LAW OF FALLING BODIES" "APPLIED HERE TO A RATHER RELUCTANT ONE" "IS MORE THAN AN EXPERIMENT IN TERROR." "IT'S DIFFERENTIAL CALCULUS TO THE RESCUE." "CHANGEINDISTANCE IS DIVIDED BY CHANGE IN TIME." "THERATIOIS THEAVERAGESPEED DURINGA GIVENTIMEINTERVAL." "WHEN THAT TIME SHRINKS TO ZERO," "THELIMITINGVALUE OFTHEAVERAGESPEED" "ISTHEINSTANTANEOUSSPEED." "CHANGEINELEVATION" "ISDIVIDEDBY  CHANGEINHORIZONTALDISTANCE." "THERESULT" "ISTHESLOPEOFA CHORD JOININGTWOPOINTS." "WHEN THE HORIZONTAL DISTANCE SHRINKSTOZERO," "THELIMITINGVALUE OFTHESLOPEOFTHECHORD" "ISTHESLOPEATTHAT POINT." "DIFFERENTIATION." "THE GOALS AND CALCULATIONS DIFFER," "BUT NOT THE ESSENTIAL CONCEPT NOR THE PROCESS." "SPEED IS THE DERIVATIVE OF DISTANCE" "WITH RESPECT TO TIME." "SLOPE IS THE DERIVATIVE OF ELEVATION" "WITH RESPECT TO" "HORIZONTAL DISTANCE." "IN ANY CASE," "A DERIVATIVE IS WHAT HAPPENS TO A QUOTIENT" "A RATIO OF TWO NUMBERS" "AS BOTH TOP AND BOTTOM SHRINK TO ZERO" "BEFORE THEY ACTUALLY REACH ZERO." "SMALL NUMBERS ARE MARKED BY THE GREEK LETTER DELTA." "DELTA y IS A SMALL CHANGE IN y." "DELTA x, A SMALL CHANGE IN x." "SO DELTA y/DELTA x" "IS MERELY A RATIO OF TWO SMALL NUMBERS." "WHEN THE SMALL NUMBERS SHRINK TO ZERO," "THAT RATIO BECOMES A DERIVATIVE." "AND THE DELTAS BECOME A NEW SYMBOL, dy/dx" "THE SYMBOL OF THE DERIVATIVE" "WHICH MEANS THE DERIVATIVE WITH RESPECT TO x" "OF THE QUANTITY y." "ONCE THE SIMPLE MECHANICS ARE MASTERED," "FINDING THE DERIVATIVE FOR JUST ABOUT ANYTHING" "IS NO HARDER THAN FLIPPING A SWITCH." "THE DERIVATIVE OF A FUNCTION" "IS THE SLOPE OF ITS TANGENT ATEACHPOINT." "THE DERIVATIVE OF A FUNCTION ISITSELFAFUNCTION." "IFTHEFUNCTIONISLINEAR," "THESLOPEIS CONSTANT" "ANDTHEDERIVATIVE" "ISJUSTTHATCONSTANT." "IFy = sinex," "THENdy/dx=cosinex." "IFy = cosinex," "THENdy/dx=- sinex." "TAKING DERIVATIVES TAKES A LITTLE PRACTICE," "BUT IT'S WELL WORTH THE EFFORT." "AND CONSIDERING" "ANY NUMBER OF CONTEMPORARY DERIVATIVE MACHINES," "IT'S BECOME A MODERN PRACTICE." "A SPEEDOMETER IS A DERIVATIVE MACHINE." "IT MEASURES" "THE DERIVATIVE OF THE DISTANCE TRAVELED AT EACH INSTANT." "THE RATE OF CHANGE OF POSITION" "IS THE INSTANTANEOUS SPEED EXPRESSED AS MILES PER HOUR." "OF COURSE," "WHEN THE VEHICLE ISN'T MOVING, NO DISTANCE IS BEING TRAVELED." "HERE, THE POSITION IS CONSTANT," "AND THE DERIVATIVE OF A CONSTANT IS 0." "MATHEMATICS IS A LANGUAGE BUILT ON GRAMMATICAL STRUCTURE," "A COLLECTION OF RULES THAT BOTH BUILDS AND BREAKS DOWN" "THE COMPOSITION OF THE TASK AT HAND." "WHATEVER THE MASTER WORK" "FROM BUILDING A HOUSE TO COMPOSING A SYMPHONY" "THE MOST COMPLICATED TASK" "CAN BE BROKEN DOWN IN MUCH THE SAME WAY." "NEWTON AND LEIBNITZ DEVELOPED THE TOOLS OF CALCULUS" "THAT PERMIT THE MOST COMPLEX FUNCTION TO BE DIFFERENTIATED" "BY BREAKING IT DOWN INTO SIMPLE PARTS." "ONE OF THE BASIC RULES OF DIFFERENTIATION" "IS THE SUM RULE." "SUPPOSE ONE PAINTER" "CAN PAINT 90 SQUARE METERS OF WALL PER HOUR" "AND THE OTHER 100." "THOSE ARE THE RATES" "AT WHICH AREAS OF THE WALL ARE CHANGING COLOR." "IN OTHER WORDS, THEY'RE DERIVATIVES." "THEREFORE, EVERY HOUR, 190 SQUARE METERS OF WALL ARE BEING PAINTED ALTOGETHER." "THAT'S HOW THE SUM RULE WORKS." "THE DERIVATIVE OF THE SUM IS THE SUM OF THE DERIVATIVES." "ANOTHER HANDY TOOL IS THE PRODUCT RULE," "WHICH IS USED TO FIND THE DERIVATIVE" "OF THE PRODUCT OF TWO FUNCTIONS." "FOR EXAMPLE, THE AREA OF ANY BOARD" "IS THE PRODUCT OF ITS LENGTH TIMES ITS WIDTH." "IF THE LENGTH IS SHORTENED..." "THE CHANGE IN AREA IS THE WIDTH TIMES THE CHANGE IN LENGTH." "IF THE WIDTH IS REDUCED," "THE CHANGE IN AREA IS THE LENGTH TIMES THE CHANGE IN WIDTH." "THE TOTAL CHANGE IN AREA IS THE SUM OF THESE." "THAT'S TRUE OF THE CARPENTER'S PRODUCT," "AND IT'S JUST AS TRUE" "IN THE LANGUAGE OF DIFFERENTIAL CALCULUS." "THEDERIVATIVE OFTHEPRODUCTyTIMESz" "ISy TIMESTHEDERIVATIVEOFz" "+z TIMESTHEDERIVATIVEOFy." "USINGTHISRULE," "IT'SPOSSIBLETO FIND THEDERIVATIVEOF x-SQUARED..." "OROFxTOTHE3rdPOWER." "OROFANYPOWEROFx." "THEDERIVATIVEOF x TOTHEPOWERn" "ISn TIMESxTOTHEPOWER n-1." "OFTEN, ONE OPERATION IS DEPENDENT ON ANOTHER." "FOR EXAMPLE," "SUPPOSE A VEHICLE" "HAS A FUEL EFFICIENCY OF 17 MILES PER GALLON." "THAT IS A DERIVATIVE." "IF y IS THE DISTANCE TRAVELED" "AND x THE AMOUNT OF FUEL CONSUMED," "THEN 17 MILES PER GALLON = dy/dx." "SAY IT USES 2 GALLONS EVERY HOUR." "2 GALLONS PER HOUR = dx/dt." "A VEHICLE'S SPEED IN MILES PER HOUR" "IS EQUAL TO THE MILES PER GALLON IT GETS" "TIMES THE GALLONS PER HOUR IT USES." "THIS IS THE CHAIN RULE." "IT'S USED WHEN y DEPENDS ON x AND x DEPENDS ON t." "THE SUM RULE..." "THE PRODUCT RULE..." "AND THE CHAIN RULE." "THESE THREE RULES" "REPRESENT THE GRAMMAR OF DIFFERENTIAL CALCULUS," "AND THE VALUE OF DIFFERENTIAL CALCULUS" "CAN BE SEEN IN THE VARIETY" "OF ITS APPLICATIONS." "FOR EXAMPLE, WHEN A ROCKET MOVES" "WITH DISPLACEMENT s AT TIME t," "THE DERIVATIVE" "OF THE DISPLACEMENT" "IS THE VELOCITY" "POSITIVE FOR UPWARD MOTION..." "AND NEGATIVE" "FOR DOWNWARD MOTION." "THE DERIVATIVE OF THE VELOCITY" "IS THE ACCELERATION" "WHICH IS THE SAME AS TAKING" "THE DERIVATIVE OF A DERIVATIVE" "THAT IS," "THE SECOND DERIVATIVE OF s." "THE ACCELERATION IS CAUSED" "BY THE FIRING OF THE ROCKET." "THE RULES OF DIFFERENTIAL CALCULUS" "AND THEIR APPLICATIONS TO PHYSICS." "[ORCHESTRA PLAYS]" "EACH STANDS AS A SOLO INSTRUMENT" "THAT PLAY IN THE ART AND SCIENCE OF MATHEMATICS." "WORKING TOGETHER, HARMONIZING," "THEY CAN BLEND INDIVIDUAL NOTES OR NUMBERS" "INTO AN EXQUISITE MELODY." "I'VE RECEIVED A LETTER" "FROM A MUSICIAN NAMED ALBERT EINSTEIN." "HE SENT IT IN 1912." "THE MAIL IS SLOW SOMETIMES." "ACTUALLY, HE DIDN'T SEND IT TO ME" "BUT TO A FRIEND OF HIS." "I JUST GOT IT AT THE LIBRARY." "ANYWAY, HE SAID," "I OCCUPY MYSELF EXCLUSIVELY WITH THE PROBLEM OF GRAVITATION" "ANDNOWBELIEVEIWILL  OVERCOMEALLDIFFICULTIES." "IHAVEBECOMEIMBUED" "WITHA GREATRESPECT FORMATHEMATICS" "THESUBTLEPARTSOFWHICH, INMYIGNORANCE," "IHADUNTILNOW  REGARDEDASPURELUXURY." "EINSTEIN WORKED ON GRAVITATION FOR FOUR MORE YEARS," "RESULTING IN THE GENERAL THEORY OFRELATIVITY." "IT IS THE MOST DIFFICULT MATHEMATICAL THEORY IN PHYSICS." "WHAT DID EINSTEIN MEAN" "BY SAYING THE SUBTLE PARTS OF MATHEMATICS" "HAD SEEMED A LUXURY?" "DID HE REALLY BELIEVE" "HE COULD GET ALONG WITHOUT DOING CALCULATIONS?" "WELL, OF COURSE NOT." "THE POINT IS," "PHYSICISTS HAVE A CERTAIN ARROGANCE ABOUT MATHEMATICS." "YOU MAY HAVE THE IMPRESSION THAT BY FOLLOWING SIMPLE RULES," "YOU CAN TAKE THE DERIVATIVE OF ANY FUNCTION." "THAT'S NOT QUITE TRUE." "SUPPOSE YOU HAD A FUNCTION" "WHICH LOOKED LIKE AN EGYPTIAN PYRAMID." "IT'S EASY TO SEE WHAT THE SLOPE IS HERE AND HERE." "BUT AT THE PEAK, YOU'D BE IN TROUBLE." "IT HAS NO SLOPE AT THAT POINT." "THE FUNCTION HAS NO DERIVATIVE AT THAT POINT." "I NEVER SAID ANYTHING" "THAT WOULD MAKE YOU BELIEVE THAT COULD EVER HAPPEN." "FOR PHYSICISTS, MATHEMATICS IS JUST A TOOL." "IT'S TO BE USED TO ACCOMPLISH SOMETHING ELSE." "BUT A REAL MATHEMATICIAN" "IS THE GUARDIAN OF PRECISION OF THOUGHT." "WHAT INTERESTS THE MATHEMATICIAN IS THE MATHEMATICS ITSELF." "WHEN A MATHEMATICIAN MAKES A STATEMENT ABOUT DERIVATIVES," "THE STATEMENT TAKES INTO ACCOUNT EVERY EXCEPTION," "NO MATTER HOW UNUSUAL" "LIKE THE PEAK OF THE PYRAMID." "THAT'S THE KIND OF SUBTLETY EINSTEIN WAS WORRIED ABOUT." "I'LL SEE YOU NEXT TIME." "CAPTIONING PERFORMED BY THE NATIONAL CAPTIONING INSTITUTE, INC." "CAPTIONS COPYRIGHT 1985 CALIFORNIA INSTITUTE OF TECHNOLOGY" "THE CORPORATION FOR COMMUNITY COLLEGE TELEVISION AND THE ANNENBERG/CPB PROJECT" "PUBLIC PERFORMANCE OF CAPTIONS PROHIBITED WITHOUT PERMISSION OF NATIONAL CAPTIONING INSTITUTE" "Annenberg Media" "¶" "For information about this and other Annenberg Media programs call 1-800-LEARNER and visit us at"