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- WELCOME TO A THIRD EXAMPLE
OF DETERMINING THE DIFFERENCE QUOTIENT
OR THIS QUOTIENT HERE, GIVEN A QUADRATIC FUNCTION.
SO FOR THE FIRST STEP,
WE WANT TO DETERMINE (F OF THE QUANTITY X) + H
WHICH MEANS X + H WILL BE THE INPUT INTO THE FUNCTION
SO WE'LL SUBSTITUTE X + H WHEREVER WE SEE AN X.
SO INSTEAD OF 3X SQUARED
WE'LL HAVE 3 x (THE QUANTITY X) + H SQUARED,
AND THEN INSTEAD OF - 5X, WE'LL HAVE - 5 x (THE QUANTITY X) + H
AND THEN WE HAVE + 2.
SO THIS IS (F OF THE QUANTITY X) + H.
LET'S GO AHEAD AND PUT THIS IN PARENTHESES
TO KEEP THINGS ORGANIZED.
AND NOW WE'LL SUBTRACT (F OF X), SO WE'LL SUBTRACT
(THE QUANTITY 3X SQUARED) - (5X + 2).
IT IS IMPORTANT THAT WE HAVE THESE PARENTHESES
AROUND (F OF X)
SO THAT WE MAKE SURE WE SUBTRACT THE ENTIRE FUNCTION
AND ALL THIS IS DIVIDED BY H.
NOW WE'RE GOING TO MULTIPLY OUT (F OF THE QUANTITY X) + H
SO LET'S FIRST FIND 3 x (THE QUANTITY X) + H SQUARED.
LET'S GO AHEAD AND DO THIS ON THE SIDE.
3 x (THE QUANTITY X) + H SQUARED
= 3 x (THE QUANTITY X) + H x (THE QUANTITY X) + H.
LET'S GO AHEAD AND FIND THE PRODUCT
OF THE TWO BINOMIALS FIRST.
SO WE'LL HAVE FOUR PRODUCTS. ONE, TWO, THREE AND FOUR.
SO WE HAVE 3 TIMES-- X x X = X SQUARED.
THE NEXT TWO PRODUCTS WILL BE LIKE TERMS.
WE HAVE X x H THAT'S HX + H x X, THAT'S ALSO HX.
SO WE HAVE + 2HX + H SQUARED AND NOW WE'LL DISTRIBUTE 3.
SO WE HAVE 3X SQUARED + 6HX + 3H SQUARED.
WHICH MEANS WE WOULD HAVE 3X SQUARED + 6HX + 3H SQUARED
AND WE'LL DISTRIBUTE -5, SO (-5X - 5H) + 2.
AGAIN, - F OF X.
ALL THIS IS STILL DIVIDED BY H.
NOW FOR THE NEXT STEP WE'LL CLEAR THE PARENTHESES
AND COMBINE THE LIKE TERMS IN THE NUMERATOR.
SO IF IT'S HELPFUL WE CAN THINK OF DISTRIBUTING A +1 HERE
AND BECAUSE OF THE SUBTRACTION
WE CAN THINK OF DISTRIBUTING -1 HERE.
SO NO SIGNS WILL CHANGE IN THIS FIRST SET OF PARENTHESES.
BUT WHEN YOU DISTRIBUTE -1
IT'S GOING TO CHANGE THE SIGN OF EACH OF THESE TERMS.
SO WE'LL HAVE - 3X SQUARED.
-1 x -5X = + 5X AND -1 x 2 = -2. ALL DIVIDED BY H.
NOW LOOKING AT THE NUMERATOR,
WE HAVE 3X SQUARED - 3X SQUARED, THAT'S 0.
WE ALSO HAVE -5X + 5X, THAT'S 0.
AND WE HAVE + 2 - 2, THAT'S 0.
SO WE'RE LEFT WITH 6HX + 3H SQUARED
AND THEN - 5H DIVIDED BY H BUT WE'RE NOT DONE.
NOTICE HOW EACH OF THE TERMS IN THE NUMERATOR
CONTAINS A FACTOR OF H.
SO NOW WE'LL GO AHEAD AND FACTOR THE NUMERATOR TO SIMPLIFY.
IF WE FACTOR OUT H WE'RE LEFT WITH 6X + 3H - (5 DIVIDED BY H).
IN THIS FORM WE HAVE H/H THAT SIMPLIFIES TO 1.
IT WILL BE SIMPLIFIED DIFFERENCE QUOTIENT
WOULD JUST BE 6X + 3H - 5.
I HOPE YOU FOUND THIS HELPFUL.