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- WE WANT TO USE THE TABLE TO DETERMINE
IF THE FUNCTION IS INCREASING OR DECREASING,
AND WHETHER THE FUNCTION IS CONCAVE UP OR CONCAVE DOWN.
TO DETERMINE IF THE FUNCTION IS INCREASING OR DECREASING,
WE NEED TO SEE IF THE FUNCTION VALUES INCREASE OR DECREASE
AS X INCREASES.
SO LOOKING AT THIS FIRST TABLE, NOTICE AS X INCREASES
THE FUNCTION VALUES ARE DECREASING EACH TIME.
AND THEREFORE, THIS TABLE REPRESENTS
A DECREASING FUNCTION.
BEFORE WE TALK ABOUT CONCAVITY,
LETS TAKE A LOOK AT OUR SECOND TABLE.
NOTICE AS THE X VALUES INCREASE
THE FUNCTION VALUES IN THIS TABLE ARE DECREASING AS WELL,
SO BOTH OF THESE REPRESENT DECREASING FUNCTIONS.
NOW, TO DETERMINE IF THE FUNCTION IS CONCAVE UP
OR CONCAVE DOWN,
WE NEED TO DETERMINE IF THE RATE OF CHANGE OF THE FUNCTION
INCREASES OR DECREASES AS X INCREASES.
SINCE THIS IS AN ALGEBRA CLASS
AND WE DON'T HAVE THE EQUATION OF THE FUNCTION,
WE'LL HAVE TO FIND THE AVERAGE RATE OF CHANGE
TO DETERMINE IF THIS FUNCTION IS CONCAVE UP OR CONCAVE DOWN.
IF THE RATE OF CHANGE IS INCREASING AS X INCREASES,
THE FUNCTION IS CONCAVE UP.
IF THE RATE OF CHANGE IS DECREASING AS X INCREASES,
THE FUNCTION IS CONCAVE DOWN.
TO DETERMINE OUR AVERAGE RATES OF CHANGE,
WE'LL DETERMINE THE CHANGE IN THE FUNCTION VALUES
AND DIVIDE BY THE CHANGE IN THE X VALUES.
SO THERE'S GOING TO BE QUITE A FEW CALCULATIONS HERE,
SO I'VE ALREADY SET THIS UP ON THE NEXT SLIDE.
HERE WE HAVE THE AVERAGE RATE OF CHANGE FROM X = 0 TO 1,
FROM 1 TO 2, FROM 2 TO 3, 3 TO 4, AND 4 TO 5.
SO IF WE LOOK AT THE AVERAGE RATES OF CHANGE,
THERE ARE -17, -12, -7, -6, AND -4.
WHILE THESE ARE ALL NEGATIVE, THESE VALUES ARE INCREASING,
WHICH MEANS THE FUNCTION IS CONCAVE UP.
SO THIS IS A DECREASING FUNCTION THAT IS CONCAVE UP.
WE TAKE A LOOK AT THE SECOND TABLE,
AGAIN, HERE ARE THE AVERAGE RATES OF CHANGE,
THEY'RE -6, -11, -18, -23, AND -25.
WELL, THESE ARE GETTING SMALLER,
AND THEREFORE THE FUNCTION IS CONCAVE DOWN.
SO THIS FUNCTION IS DECREASING AND CONCAVE DOWN.
NOW LET'S VERIFY THIS BY GRAPHING THESE POINTS.
HERE'S THE FIRST TABLE.
NOTICE HOW IT'S GOING DOWN-HILL, SO THE FUNCTION IS DECREASING.
BUT NOTICE HOW IT ALSO FORMS AN UPWARD FACING CUP,
THEREFORE IT'S CONCAVE UP.
HERE'S THE GRAPH OF THE SECOND TABLE.
AGAIN, NOTICE HOW IT'S GOING DOWN-HILL,
THEREFORE IT'S DECREASING.
BUT NOTICE HOW THE POINTS FORM A DOWNWARD FACING CUP
AND THEREFORE IT'S CONCAVE DOWN.
LET'S TAKE A LOOK AT TWO MORE EXAMPLES,
SAME QUESTION, TWO DIFFERENT TABLES.
WE'LL FIRST DETERMINE IF THE FUNCTION IS INCREASING
OR DECREASING,
AND THEN DETERMINE THE CONCAVITY.
NOTICE IN THIS FIRST TABLE AS THE X VALUES INCREASE
THE FUNCTION VALUES ARE INCREASING EACH TIME AS WELL,
SO THIS IS AN INCREASING FUNCTION.
NOTICE IN THE SECOND TABLE THE SAME THING IS OCCURRING,
AS X INCREASES THE FUNCTION VALUES INCREASE AS WELL,
SO BOTH OF THESE TABLES ARE INCREASING FUNCTIONS.
AND, AGAIN, BECAUSE THIS IS AN ALGEBRA CLASS,
WE'LL NOW FIND THE AVERAGE RATES OF CHANGE FROM X = 0 TO 1,
1 TO 2, 2 TO 3, 3 TO 4, AND 4 TO 5.
AGAIN, I'VE ALREADY SET THIS UP.
HERE'S THE AVERAGE RATE OF CHANGE FROM X = 0 TO X = 1,
FROM X = 1 TO 2, FROM 2 TO 3, 3 TO 4, AND 4 TO 5.
NOTICE HOW THE RATES OF CHANGE ARE 7, 12, 18, 25, AND 32.
THESE VALUES ARE GETTING LARGER OR INCREASING,
WHICH MEANS THE FUNCTION IS CONCAVE UP.
SO THIS IS AN INCREASING AND CONCAVE UP FUNCTION.
AND FOR THE LAST TABLE,
THE RATES OF CHANGE ARE 43, 35, 25, 15, AND 6.
THESE VALUES ARE GETTING SMALLER OR DECREASING,
WHICH MEANS THE FUNCTION IS CONCAVE DOWN.
SO THIS FUNCTION IS INCREASING AND CONCAVE DOWN.
AGAIN, LET'S GO AHEAD AND VERIFY THIS GRAPHICALLY.
HERE'S THE FIRST INCREASING FUNCTION.
NOTICE HOW THE POINTS ARE GOING UP-HILL,
AND IT ALSO FORMS AN UPWARD FACING CUP,
THEREFORE IT'S INCREASING AND CONCAVE UP.
AND THE LAST TABLE, AGAIN IT'S GOING UP-HILL,
SO IT'S INCREASING.
BUT IT FORMS A DOWNWARD FACING CUP,
THEREFORE IT'S CONCAVE DOWN.
OKAY, I HOPE YOU FOUND THESE EXAMPLES HELPFUL.