- WELCOME TO A LESSON ON NEWTON'S METHOD.
THE GOAL OF THIS VIDEO IS TO USE NEWTON'S METHOD
TO APPROXIMATE REAL ZEROS OF A FUNCTION.
NEWTON'S METHOD IS A TECHNIQUE
FOR APPROXIMATING THE REAL ZEROS OF A FUNCTION
THAT USES TANGENT LINES
TO APPROXIMATE THE GRAPH NEAR ITS X INTERCEPTS.
WHILE NEWTON'S METHOD IS A FAIRLY STRAIGHTFORWARD PROCEDURE
TO FOLLOW,
IT IS IMPORTANT TO REALIZE WHAT WAS HAPPENING GRAPHICALLY
WHEN WE USE NEWTON'S METHODS.
SO LET'S TAKE A LOOK.
LET'S SAY WE WERE TRYING TO APPROXIMATE
THE REAL ZERO OF THIS GRAPH
WHICH WOULD BE THE X VALUE OF THIS X INTERCEPT.
BUT INSTEAD OF DOING IT ALGEBRAICALLY,
WE WANTED TO APPROXIMATE IT USING TANGENT LINES.
WHAT WE WOULD DO IS START WITH AN INITIAL GUESS,
LET'S SAY WE THOUGHT IT WAS AROUND 2.5.
AT THIS LOCATION WE WOULD SKETCH THE TANGENT LINE TO THE FUNCTION
AS WE SEE HERE IN RED
AND WE WOULD DETERMINE THE ZERO
OR THE X INTERCEPT OF THE TANGENT LINE
TO APPROXIMATE THE REAL ZERO OF THE ORIGINAL FUNCTION.
AND YOU CAN SEE THAT THE REAL ZERO OF THIS TANGENT LINE
IS VERY CLOSE TO THE REAL ZERO OF THE BLUE FUNCTION.
NOW, IF OUR INITIAL GUESS FOR THE ZERO OF THE FUNCTION
WAS NOT AS GOOD
YOU CAN SEE THAT THE ZERO WITH THE TANGENT LINE
WOULD NOT BE AS CLOSE TO THE TRUE ZERO OF THE FUNCTION
AS WE MIGHT WANT.
SO WHAT WE COULD DO IS REPEAT THIS PROCESS
AND USE THIS AS OUR NEXT APPROXIMATION
AND IF WE USE THIS X VALUE
TO SKETCH AN ADDITIONAL TANGENT LINE
YOU CAN SEE THAT THE ZERO OF THE NEXT TANGENT LINE
BECOMES A BETTER APPROXIMATION,
AND WE COULD CONTINUE THIS PROCESS IF WE WANTED TO.
LET'S CHANGE THE FUNCTION NOW TO DETERMINE WHAT ELSE CAN HAPPEN.
NOTICE WITH THIS INITIAL GUESS
IT WOULD TAKE SEVERAL TRIES FOR THE ZEROS OF THE TANGENT LINES
TO COME VERY CLOSE TO THE ZEROES OF ORIGINAL FUNCTION.
HERE'S OUR FIRST TANGENT LINE, HERE'S OUR SECOND TANGENT LINE,
THIRD TANGENT LINE
AND THEN COMING ON OUR FOURTH TANGENT LINE
IS A VERY GOOD APPROXIMATION.
SO THIS INITIAL GUESS IS IMPORTANT
THAT IT'S FAIRLY CLOSE TO A REAL ZERO OF THE FUNCTION
IF WE WANT THIS WORK OUT WELL.
HERE'S AN EXAMPLE OF WHERE WE HAVE A BAD INITIAL GUESS.
IF WE'RE LOOKING FOR THIS REAL ZERO
YOU CAN SEE HOW THE SECOND TANGENT LINE WOULD HAVE A ZERO
THAT'S WAY OFF TO THE RIGHT.
BUT THIS IS THE MAIN IDEA OF NEWTON'S METHOD.
WE'RE GOING TO FIND THE ZERO OF A TANGENT LINE
TO APPROXIMATE THE REAL ZERO OF THE ORIGINAL FUNCTION.
SO IF WE LET X1 F OF X1 BE A POINT ON A FUNCTION
NEAR THE X INTERCEPT
THEN THE EQUATION OF THE TANGENT LINE WOULD LOOK LIKE THIS
IN POINT SLOPE FORM
WHERE F PRIME OF X1 IS THE SLOPE OF THE TANGENT LINE.
NOW, REMEMBER IF YOU WANT TO FIND THE ZERO X INTERCEPT
OF A LINE
YOU SET Y = TO 0 AND SOLVE FOR X.
LET'S GO AHEAD AND TRY TO DO THIS WITH THIS LINE.
WE'LL SET Y = TO 0 AND NOW WE'LL SOLVE FOR X.
WHAT WE'LL DO FIRST IS DIVIDE BY F PRIME OF X1
SO THIS SIMPLIFIES TO -F OF X SUB 1
DIVIDED BY F PRIME OF X SUB 1 = X - X SUB 1,
SO NOW WE CAN JUST ADD X SUB 1 TO BOTH SIDES
SO WE WOULD HAVE X = X SUB 1 - F OF X SUB 1
DIVIDED BY F PRIME OF X SUB 1.
AND THIS IS ACTUALLY THE FORMULA THAT WE WILL USE
TO APPLY NEWTON'S METHOD
WHERE X OF ONE WAS OUR INITIAL GUESS
AND THIS X HERE WOULD BE THE NEXT APPROXIMATION
TO THE REAL ZERO.
LET'S GO AHEAD AND FORMALIZE THIS.
IF F OF C = 0
THEN TO APPROXIMATE C WE CAN USE THE FOLLOWING STEPS,
WE MAKE AN INITIAL ESTIMATE X SUB 1 THAT IS CLOSE TO C,
A GRAPH IS HELPFUL TO DETERMINE THIS GUESS IF NOT GIVEN.
AND THEN TO DETERMINE A NEW APPROXIMATION
WE'RE GOING TO USE THIS EQUATION THAT WE JUST DERIVED
WHICH WOULD BE THE ZERO OF THE TANGENT LINE
WHEN X EQUALS X SUB 1.
SO NOTICE IF THIS IS X SUB 1 THIS WOULD BE X SUB 2
OR THE NEW APPROXIMATION.
THEN IF THE DIFFERENCE OF THE APPROXIMATIONS
IS WITHIN THE DESIRED ACCURACY
WE CAN LET X SUB AND SUB 1 SERVE AS OUR FINAL APPROXIMATION.
SO LET'S GO AHEAD AND GIVE THIS A TRY.
WHAT WE'RE GOING TO DO HERE
IS ORGANIZE ALL THE INFORMATION IN THE TABLE
AND WE'LL ALSO BE USING THE GRAPHING CALCULATOR.
SO THIS FIRST COLUMN N
REPRESENTS THE NUMBER OF ITERATIONS
SO THIS'LL BE ONE, TWO, THREE AND FOUR.
AND WHAT WE'RE GOING TO DO
IS USE NEWTON'S METHOD TO APPROXIMATE THE REAL ZERO
CLOSE TO X = 1
SO THAT'LL BE OUR INITIAL APPROXIMATION
UNTIL TWO SUCCESSIVE APPROXIMATIONS
DIFFER BY LESS THAN 0.005.
SO X SUB 1 IS GOING TO BE EQUAL TO 1.
NEXT, WE'RE GOING TO EVALUATE THE FUNCTION AT 1,
DIVIDE WITH THE DERIVATIVE AT 1, EVALUATE THIS QUOTIENT
AND THEN THIS IS THE FORMULA
THAT WILL GIVE US OUR NEW APPROXIMATION.
INSTEAD OF DOING THIS ALL BY HAND
WE'RE GOING TO USE THE GRAPHING CALCULATOR.
SO LET'S GO AHEAD AND GET THAT OUT.
WE'RE GOING TO TYPE IN THE FUNCTION INTO Y1
IN Y2 WE'RE GOING TO TYPE IN THE DERIVATIVE,
SO THE DERIVATIVE WOULD BE 4X.
FOR Y3 WE'LL USE THE CALCULATOR TO EVALUATE THIS QUOTIENT.
SO WE'LL HAVE THE FUNCTION DIVIDED BY THE DERIVATIVE
WHICH WOULD BE Y1 DIVIDED BY Y2.
SO WE'LL PRESS VARS, Y VARS, ENTER.
AGAIN OUR NUMERATOR WILL BE Y1 THE FUNCTION DIVIDED BY Y2
WHICH IS OUR DERIVATIVE.
SO PRESS VARS, RIGHT ARROW, ENTER.
DOWN ONCE AND THEN ENTER.
AND THE LAST COLUMN IS GOING TO BE X MINUS THIS QUOTIENT
SO WE'LL HAVE X - Y3 BECAUSE Y3 IS THE QUOTIENT.
SO WE HAVE X MINUS VARS, RIGHT ARROW, ENTER,
DOWN TO Y3, PRESS ENTER.
NOW, THE LAST THING WE HAVE TO DO
IS MAKE SURE THAT OUR TABLE FEATURE IS ON ASK.
SO PRESS SECOND WINDOW,
MAKE SURE THE INDEPENDENT VARIABLE IS ON ASK.
SO NOW WE'LL GO TO OUR TABLE FEATURE
AND JUST COMPLETE THIS TABLE.
SO SECOND GRAPH WE'LL TYPE IN X = 1
SO Y1 IS OUR FUNCTION VALUE THAT'S -1,
THE VALUE OF THE DERIVATIVE IS Y2, THAT'S 4.
NOW, WE'LL SCROLL OVER TO Y3
AND WE'LL GO AHEAD AND SHOW Y4 AS WELL.
THESE LAST TWO COLUMNS ARE -.25 AND 1.25.
SO AGAIN, WHAT HAPPENED 1 WAS OUR INITIAL APPROXIMATION
AND THEN WE SUBTRACTED THIS QUOTIENT
AND THIS QUOTIENT WAS -.25
SO WE SUBTRACTED A -1.25, 1.25 IS OUR NEXT APPROXIMATION.
SO WE'LL TAKE THIS VALUE OF 1.25
AND NOW WE'LL DO THE EXACT SAME THING AGAIN.
SO WE'LL GO DOWN TO THE SECOND ROW AND NOW WE'LL TYPE IN 1.25
AND FILL OUT THE TABLE AGAIN.
SO THE FUNCTION VALUE IS .125 AND THE DERIVATIVE VALUE IS 5.
LET'S RECORD THAT.
SO NOTICE HOW THE FUNCTION VALUE IS GETTING CLOSER TO ZERO.
NEXT, WE NEED TO FIND THE VALUE OF THIS QUOTIENT,
AND THEN OUR NEXT APPROXIMATION.
AND WE'LL FIND THOSE TWO VALUES IN Y3 AND Y4.
SO .025 AND 1.225.
SO NOTICE HOW OUR SECOND APPROXIMATION
IS NOT THAT MUCH DIFFERENT THAN 1.25
WHICH WE FOUND FROM FIRST ITERATION.
THE DIFFERENCE HERE IS STILL MORE THAN 0.005
SO WE'LL GO AHEAD AND DO THIS AGAIN.
THIS WILL BE OUR NEXT APPROXIMATION FOR THE ZERO
AND WE'LL GO AHEAD AND GO BACK TO OUR TABLE.
LET'S GO DOWN TO THE THIRD ROW AND NOW WE'LL TYPE IN 1.225.
AND NOTICE HOW AGAIN,
THE FUNCTION VALUE IS GETTING VERY CLOSE TO ZERO NOW
SO WE HAVE .00125 FOR THE FUNCTION VALUE
AND 4.9 FOR THE DERIVATIVE.
NOW WE'LL FIND THE QUOTIENT AND OUR NEXT APPROXIMATION.
SO WE'LL SCROLL OVER TO Y3 AND Y4.
AND NOTICE Y3 IS NOW IN SCIENTIFIC NOTATION.
SO WE'D HAVE TO MOVE THE DECIMAL FOUR PLACES TO THE LEFT
SO .00026
AND OUR NEXT APPROXIMATION IS 1.2247.
IF WE WANTED TO DO ANOTHER ITERATION
THIS WOULD BE OUR NEXT APPROXIMATION,
HOWEVER, THE DIFFERENCE BETWEEN THE PREVIOUS APPROXIMATION
AND THE NEW APPROXIMATION IS LESS THAN 0.005,
THEREFORE WE CAN STOP AND SAY THAT USING NEWTON'S METHOD,
OUR APPROXIMATION WILL BE 1.2247.
AND THAT SHOULD MAKE SENSE
BECAUSE IF YOU LOOK AT THE FUNCTION VALUE
IT'S VERY CLOSE TO ZERO.
SO THIS IS A VERY GOOD APPROXIMATION FOR THE ZERO
OF THE ORIGINAL FUNCTION NEAR X = 1.
OF COURSE, WE COULD HAVE CRUNCHED ALL THESE NUMBERS
BY HAND
BUT I HOPE YOU'LL AGREE
THAT USING THE TABLE FEATURE OF THE GRAPHING CALCULATOR
IS A GOOD USE OF TECHNOLOGY IN THIS CASE.
AND IF WE WANT TO USE NEWTON'S METHOD
TO APPROXIMATE A REAL ZERO OF ANY OTHER FUNCTION,
THE PROCESS HERE WOULD BE EXACTLY THE SAME.
I HOPE YOU FOUND THIS VIDEO HELPFUL.