Ex 2: The Difference Quotient (Quadratic Function)

Ex 2:  The Difference Quotient (Quadratic Function)


– WELCOME TO A SECOND EXAMPLE OF DETERMINING THE DIFFERENCE
QUOTIENT FOR A GIVEN FUNCTION. IN THIS EXAMPLE, THE GIVEN
FUNCTION IS A QUADRATIC FUNCTION AND AS MENTIONED IN EXAMPLE 1, THIS DIFFERENCE QUOTIENT
OR THIS QUOTIENT HERE IS A VERY IMPORTANT QUOTIENT
IN THE STUDY OF CALCULUS, BUT FOR THIS EXAMPLE WE’RE JUST
WORKING ON OUR ALGEBRA SKILLS TO SIMPLIFY THIS QUOTIENT. SO TO START WE WANT TO FIND
F OF THE QUANTITY X + H, WHICH MEANS THE QUANTITY X + H IS THE INPUT INTO
THE GIVEN FUNCTION. SO WHEREVER WE SEE X, WE’LL REPLACE X
WITH THE QUANTITY X + H. SO IF (F OF X)
=X SQUARED – (3X + 4), F OF THE QUANTITY X + H
WOULD BE, INSTEAD OF X SQUARED,
(X + H SQUARED), AND THEN INSTEAD OF – 3X WE’LL HAVE – 3
x (THE QUANTITY X + H) + 4. SO ALL OF THIS IS
F OF THE QUANTITY X + H. LET’S PUT THIS IN PARENTHESES
TO KEEP THINGS CLEAR AND NOW WE STILL HAVE
TO SUBTRACT (F OF X) SO WE’LL HAVE – (THE QUANTITY X
SQUARED) – (3X + 4). ALL THIS IS DIVIDED BY H. NOW WE’RE GOING TO MULTIPLY OUT
(F OF THE QUANTITY X + H) SO WE’LL START BY SQUARING
(THE QUANTITY X + H). THERE’S NO SHORTCUTS HERE. IF WE WANT TO SQUARE
(THE QUANTITY X + H) WE’LL HAVE TWO FACTORS
OF (X + H). WHEN MULTIPLYING THESE BINOMIALS
WE’LL HAVE 4 PRODUCTS, ONE, TWO, THREE AND FOUR. SO X x X=X SQUARED.
X x H=+HX. H x X IS ANOTHER +HX,
AND THEN FINALLY +H SQUARED. SO WE’LL HAVE X SQUARED PLUS,
WE HAVE 1HX + 1HX, THAT’S 2HX + H SQUARED. WHICH MEANS
(F OF THE QUANTITY X + H) WOULD BE (X SQUARED + 2HX)
+ H SQUARED, AND NOW WE’LL DISTRIBUTE -3, SO WE HAVE (-3X – 3H) + 4
AND WE STILL HAVE (-F OF X). SO – (THE QUANTITY X SQUARED
– 3X) + 4. THIS IS STILL DIVIDED BY H. NOW WE’LL CLEAR THE PARENTHESES
IN THE NUMERATOR AND COMBINE LIKE TERMS. SO IF IT’S HELPFUL WE CAN THINK
OF DISTRIBUTING A +1 HERE WHICH WON’T CHANGE ANYTHING,
BUT BECAUSE OF THE SUBTRACTION, WE CAN THINK OF DISTRIBUTING
A -1 IF THAT’S HELPFUL. BUT FOR THE FIRST PART
WE WOULD JUST HAVE (X SQUARED + 2HX) +
(H SQUARED – 3X) – (3H + 4). NOTICE NOTHING CHANGED
BUT FOR THE SECOND PART IF WE DISTRIBUTE -1 OR SUBTRACT
THIS ENTIRE FUNCTION, IT’S GOING TO CHANGE THE SIGN
OF EACH TERM, SO WE’LL HAVE – X SQUARED
AND THEN (-1 x -3X)=+3X AND THEN -1 x +4 WOULD BE -4. NOW LOOKING AT THE NUMERATOR NOTICE HOW WE HAVE
X SQUARED – X SQUARED, THAT’S 0. WE ALSO HAVE -3X + 3X,
THAT’S 0 AND THEN WE ALSO HAVE 4 – 4,
THAT’S 0. SO WE’RE LEFT WITH (2HX + H SQUARED)
– 3H DIVIDED BY H BUT WE’RE NOT DONE YET.
THIS DOES SIMPLIFY FURTHER. NOTICE EACH TERM IN
THE NUMERATOR CONTAINS A COMMON FACTOR OF H. SO WE’RE GOING TO FACTOR OUT THE COMMON FACTOR OF H
FROM THE NUMERATOR. THAT WILL LEAVE US WITH (H x THE QUANTITY 2X) + (H – 3)
STILL DIVIDED BY H, BUT NOW WE HAVE H/H,
THAT SIMPLIFIES TO 1, SO OUR SIMPLIFIED DIFFERENCE
QUOTIENT IS JUST (2X + H) – 3. OKAY. I HOPE YOU FOUND THIS
HELPFUL. THANK YOU FOR WATCHING.

2 Replies to “Ex 2: The Difference Quotient (Quadratic Function)”

  1. listening to your lips smack and the spit slosh around in your mouth during this video is making me want to drop out of college

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