- WE WANT TO USE THE GRAPH 
OF F OF X
TO ESTIMATE THE INSTANTANEOUS 
RATE OF CHANGE AT X = 4.
THE INSTANTANEOUS RATE OF CHANGE 
OF THE FUNCTION IN X = 4
IS EQUAL TO THE SLOPE 
OF THE TANGENT LINE AT X = 4.
IT'S ALSO EQUAL TO 
THE DERIVATIVE FUNCTION
EVALUATED AT X = 4 
OR F PRIME OF 4.
LET'S FIRST LOCATE THE POINT 
IN THE FUNCTION WHEN X = 4,
WHICH WOULD BE HERE.
NEXT WE'LL SKETCH 
THE TANGENT LINE
AND THEN FIND THE SLOPE 
OF THAT LINE.
SO HERE'S THE GRAPH 
OF THE TANGENT LINE
AND NOW TO FIND THE SLOPE 
OF THIS LINE,
WHICH IS THE INSTANTANEOUS 
RATE OF CHANGE,
WE'LL LOCATE TWO POINTS 
ON THIS LINE
AND WE'LL FIND TWO POINTS WHERE 
THE COORDINATES ARE INTEGERS.
SO THE POINT OF TANGENCY HERE 
LOOKS GOOD
WHERE THE COORDINATES ARE (4,5).
AND NOW LET'S USE THIS POINT 
HERE
WHERE THE COORDINATES LOOK LIKE 
THEY'RE APPROXIMATELY (5,7).
AND NOW TO FIND THE SLOPE 
OF THIS TANGENT LINE
WE WANT TO FIND THE CHANGE OF Y 
DIVIDED BY THE CHANGE OF X,
AND WE CAN DO THIS 
USING THE SLOPE FORMULA HERE
OR BY LOOKING AT THE GRAPH
NOTICE HOW TO GO FROM THE POINT 
ON THE LEFT
TO THE POINT ON THE RIGHT YOU 
WOULD HAVE TO GO UP TWO UNITS,
WHICH MEANS THE CHANGE OF Y 
IS +2.
AND THEN WE'D HAVE TO GO 
RIGHT ONE UNIT,
SO THE CHANGE OF X IS +1.
AND, THEREFORE, THE 
INSTANTANEOUS RATE OF CHANGE
AT X = 4 OR F PRIME OF 4 = 2 
DIVIDED BY 1 OR JUST 2.
THE OTHER OPTION IS TO USE 
THESE TWO ORDERED PAIRS
AND FIND THE SLOPE 
USING THIS FORMULA HERE.
AGAIN, WHERE WE HAVE THE CHANGE 
OF Y DIVIDED BY THE CHANGE OF X,
SO LET'S ALSO SHOW THAT.
LET'S CALL THESE THE 1s, 
MEANING (X SUB 1,Y SUB 1),
WE'LL CALL THESE THE 2s, 
MEANING (X SUB 2,Y SUB 2).
SO WE'LL GET THE SAME RESULT
BUT F PRIME OF 4 OR 
THE INSTANTANEOUS RATE OF CHANGE
AT X = 4 = Y SUB 2 - Y SUB 1,
THAT'S 7 - 5 DIVIDED BY X SUB 2 
- X SUB 1,
THAT'S 5 - 4, 
WHICH GIVES US 2/1.
WHEREAS WE HAD BEFORE +2.
TO UNDERSTAND WHAT THIS MEANS, 
THOUGH,
IT'S SIMPLE TO HAVE 
THE INSTANTANEOUS RATE OF CHANGE
IN FRACTION FORM,
IN THIS CASE, 2/1.
SO IF F PRIME OF 4 = 2/1,
WHERE THE 2 REPRESENTS 
THE CHANGE OF Y,
THE CHANGE OF FUNCTION VALUES,
AND THE 1 REPRESENTS THE CHANGE 
OF X,
THIS TELLS US THAT WHEN X = 4 
THE FUNCTION VALUE
OR Y VALUE IS CHANGING AT A RATE 
OF 2 UNITS PER 1 UNIT CHANGE
IN X.
I HOPE YOU HAVE FOUND 
THIS HELPFUL.
