- 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.