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- WELCOME TO THE FIRST EXAMPLE
OF DETERMINING THE DIFFERENCE QUOTIENT FOR A GIVEN FUNCTION.
THIS QUOTIENT HERE IS CALLED THE DIFFERENCE QUOTIENT,
AND WHILE IT MIGHT SEEM RANDOM AT THIS POINT,
THIS IS ACTUALLY A VERY USEFUL QUOTIENT.
IF WE KNEW THE VALUE OF X AND THE VALUE OF H,
IT WOULD GIVE US A SLOPE OF A SECANT LINE TO A GIVEN FUNCTION.
AND IN CALCULUS, WHEN WE TAKE THE LIMIT OF THIS FUNCTION
AS H APPROACHES ZERO,
IT ACTUALLY GIVES US SOMETHING CALLED THE DERIVATIVE
OF THE GIVEN FUNCTION.
BUT FOR THIS EXAMPLE WE'RE JUST PRACTICING THE ALGEBRA.
SO TO START WE WANT TO FIND F OF THE QUANTITY X + H,
WHICH MEANS FOR OUR FUNCTION F
WE'RE GOING TO SUBSTITUTE THE QUANTITY X + H FOR X.
SO WE WOULD HAVE 2 x THE QUANTITY X + H - 5.
SO ALL OF THIS IS F OF THE QUANTITY X + H.
LETS PUT THIS IN PARENTHESIS.
NOW WE HAVE TO SUBTRACT F OF X.
SO WE'LL SUBTRACT THE QUANTITY 2X - 5,
AND NOW WE'LL DIVIDE ALL OF THIS BY H.
NOW LET'S BEGIN TO SIMPLIFY F OF THE QUANTITY X + H
BY DISTRIBUTING 2 HERE.
SO WE'D HAVE 2X + 2H - 5,
AND THEN - THE QUANTITY 2X - 5 DIVIDED BY H.
AND NOW WE'RE GOING TO CLEAR THE PARENTHESIS
AND SIMPLIFY THE NUMERATOR.
SO TO CLEAR THE FIRST SET OF PARENTHESIS
WE'LL THINK OF DISTRIBUTING A +1 WHICH WON'T CHANGE ANYTHING.
BUT THEN FOR THE SECOND PART,
BECAUSE OF THE SUBTRACTION WE CAN THINK OF DISTRIBUTING -1
WHICH WILL CHANGE THE SIGN OF EACH OF THESE TERMS.
SO WE'LL HAVE 2X + 2H - 5.
AND DISTRIBUTING -1 THAT WILL GIVE US -2X AND -1 x -5 = +5,
SO WE HAVE +5.
SO IT IS IMPORTANT THAT WE HAVE THESE PARENTHESIS HERE
TO MAKE SURE WE SUBTRACT THE ENTIRE FUNCTION F OF X.
NOW LET'S SIMPLIFY THE NUMERATOR.
WE HAVE 2X - 2X THAT'S ZERO.
WE ALSO HAVE -5 + 5 THAT'S 0.
SO THIS SIMPLIFIES NICELY.
WE HAVE 2H DIVIDED BY H, BUT H/H SIMPLIFIES TO 1,
SO OUR DIFFERENCE QUOTIENT = +2.
IN EXAMPLE 2 WE'LL FIND THE DIFFERENCE QUOTIENT
FOR A QUADRATIC FUNCTION.
I HOPE YOU FOUND THIS HELPFUL.