IN THIS VIDEO I'LL SHOW
HOW TO USE THE TI-84 GRAPHING CALCULATOR
TO FIND THE VALUE OF A DERIVATIVE
AT A GIVEN VALUE OF X.
AND WE'LL TAKE A LOOK AT TWO EXAMPLES.
WE WANT TO FIND THE VALUE OF THE DERIVATIVE OF F OF X WHEN X = 2,
AND HERE'S OUR GIVEN FUNCTION.
SO, AGAIN, OUR GOAL IS TO FIND THE VALUE
OF THE DERIVATIVE FUNCTION WHEN X = 2.
THE TI-84 CALCULATOR IS NOT GOING TO GIVE US
THE DERIVATIVE FUNCTION,
BUT IT WILL GIVE US THE VALUE OF THE DERIVATIVE
AT A GIVEN VALUE OF X.
AND THERE ARE TWO WAYS OF DOING THIS.
ONE WAY IS FROM THE HOME SCREEN.
SO IF WE PRESS THE MATH BUTTON,
WE WANT TO SCROLL DOWN TO OPTION 8 OR PRESS 8.
WE'RE LOOKING FOR THIS COMMAND HERE, N DERIVE OR N DERIVE.
AND PRESS ENTER, THEN WE TYPE IN THE RIGHT SIDE OF OUR FUNCTION.
SO WE HAVE X TO THE THIRD - 5X SQUARED + 2.
NOW WE HAVE TO PRESS COMMA X, AND THEN COMMA,
THE VALUE OF X THAT WE WANT TO EVALUATE THE DERIVATIVE AT,
WHICH IS 2.
CLOSE PARENTHESIS, PRESS ENTER.
A COMMON ERROR IS TO FORGET TO PUT THIS COMMA X
IN THE COMMAND.
NOW, THE NEXT THING IS NOTICE HOW IT'S GIVING US -7.999
AND SO-ON.
THE CALCULATOR'S NOT PERFECT.
WE NEED TO RECOGNIZE THIS AS -8.
SO LET'S GO AHEAD AND WRITE THIS DOWN
AND THEN I'LL SHOW YOU A SECOND WAY TO FIND THE SAME VALUE.
REMEMBER THIS WOULD BE SLOPE WITH A TANGENT LINE
WHEN X IS EQUAL TO +2,
WHICH WE'LL SHOW IN JUST A MOMENT.
THE SECOND WAY TO DO THIS IS FROM THE GRAPH SCREEN.
SO FIRST WE'LL PRESS Y EQUALS.
NOTICE HOW I'VE ALREADY ENTERED THE FUNCTION IN HERE,
SO NOW I'M GOING TO PRESS GRAPH.
HERE'S THE GRAPH OF OUR FUNCTION,
SO FROM THIS SCREEN WE CAN ALSO JUST PRESS SECOND TRACE
FOR THE CALCULATION MENU.
AND THEN SELECT OPTION SIX FOR DY, DX.
ONCE WE SELECT OPTION 6 WE NEED TO TYPE IN THE X VALUE OF 2.
SO WE PRESS 2, ENTER.
AND NOTICE HOW THE DERIVATIVE IS EQUAL TO THE SAME VALUE
THAT WAS ON THE HOME SCREEN,
WHICH WE NEED TO RECOGNIZE AS -8.
OF COURSE WE COULD VERIFY THIS BY HAND.
F PRIME OF X USING THE POWER RULE
WOULD JUST BE 3X SQUARED - 10X AND THEN + 0.
AND THEN WE COULD JUST SUB IN 2.
SO WE'D HAVE 3 x 2 SQUARED - 10 x 2.
WELL, THIS WOULD BE 4 x 3 THAT'S 12 - 20, WHICH IS EQUAL TO -8.
AND, AGAIN, WHAT THIS VALUE TELLS US IS,
GOING BACK TO THE GRAPH OF OUR FUNCTION, AT X = +2
OR AT THIS POINT HERE, THE SLOPE WITH A TANGENT LINE WOULD BE -8.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
WE WANT TO FIND THE VALUE OF THE DERIVATIVE OF OUR FUNCTION
AT THE POINT (-1,1/2).
AGAIN, HERE THEY'RE GIVING US THE POINT,
BUT TO DO THIS WE ACTUALLY ONLY NEED THE VALUE OF X = -1.
LET'S GO BACK TO OUR CALCULATOR
AND IT WILL SHOW IT BOTH WAYS AGAIN.
SO FROM THE HOME SCREEN WE'RE GOING TO PRESS MATH,
OPTION 8, ENTER.
TYPE IN OUR FUNCTION.
WE NEED TO BE CAREFUL HERE, WE WANT PARENTHESIS
AROUND THE NUMERATOR AND DENOMINATOR.
SO WE HAVE AN OPEN PARENTHESIS, (X - 1), CLOSE PARENTHESIS,
DIVIDED BY OPEN PARENTHESIS (X - 3, X, THE X VALUE'S -1),
CLOSE PARENTHESIS, PRESS ENTER.
AGAIN, THE CALCULATOR IS NOT PERFECT,
WE NEED TO RECOGNIZE THIS AS X = -0.125.
SO F PRIME OF -1 IS EQUAL TO -0.125 OR 0.125.
WE MAY WANT THIS IN FRACTION FORM,
SO IF WE GO BACK TO THE CALCULATOR
IT WON'T CONVERT THIS TO A FRACTION FOR US,
BUT IF WE TYPE IN -0.125 AND THEN PRESS MATH, ENTER, ENTER,
IT'S EQUAL TO -1/8, WHICH HOPEFULLY WE DO RECOGNIZE.
AND LETS ALSO SHOW THIS AGAIN USING THE GRAPH SCREEN.
SO WE'LL PRESS Y EQUALS, CLEAR OUT THE OLD FUNCTION,
TYPE IN THE NEW FUNCTION.
AGAIN, THESE PARENTHESIS ARE REQUIRED.
PRESS GRAPH, AND THEN FROM HERE WE PRESS SECOND TRACE
FOR CALCULATION,
AND THEN OPTION SIX.
AND THEN WE TYPE IN THE X VALUE, WHICH IS -1.
AND, AGAIN, NOTICE HOW IT'S GIVEN US -0.125,
WHICH IS EQUAL TO -1/8.
IF WE DID WANT TO VERIFY THIS ONE BY HAND
WE'D HAVE TO USE THE QUOTIENT RULE GIVEN HERE BELOW.
LET'S GO AND JUST TAKE A MOMENT AND DO THAT.
NOTICE OUR DENOMINATOR IS A DENOMINATOR SQUARED.
THE NUMERATOR IS GOING TO BE THE DENOMINATOR X - 3
X THE DERIVATIVE OF THE NUMERATOR,
WHICH IS JUST 1 - THE NUMERATOR
x THE DERIVATIVE OF THE DENOMINATOR, WHICH IS JUST 1.
LET'S GO AHEAD AND SIMPLIFY THIS.
SO HERE WE HAVE X - 3.
THIS WOULD BE - X + 1, SO THAT SIMPLIFIES TO -2.
SO THEN WE SUB IN -1 FOR X, WE'D HAVE -2/-1 - 3 SQUARED.
AND THIS ENDS UP BEING -2/16, WHICH IS -1/8.
AND, AGAIN, GRAPHICALLY WHAT THIS IS TELLING US
IS AT THIS POINT HERE ON OUR FUNCTION
THE TANGENT LINE WOULD HAVE A SLOPE OF -1/8.
LET'S GO AHEAD AND VERIFY THAT.
HERE'S THE POINT WE'RE REFERRING TO.
THIS RED TANGENT LINE DOES HAVE A SLOPE OF -1/8.
OKAY. I HOPE YOU FOUND THIS HELPFUL.