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THE DIFFERENCE QUOTIENT
f ( x h ) f ( ) x h
VOCABULARY
A quotient is the answer to a division problem. When one writes a fraction, one is really writing a division problem. Think about it.
Isn’t 124 the same thing as saying 12 divided by 4? Both produce the answer “3”.
So when you look at f^ (^ x^^ +^ h h )^^ −^ f^ ( ) x , think of it as a fraction. Hence it produces a quotient.
What does the word “ Difference ” mean?
For example, what’s the difference between 8 and 5? Three, eh? How did you arrive at the answer? Oh! You subtracted.
So the word “ difference ” is a synonym for “ subtraction ”.
In the top (numerator) of the quotient f^ (^ x^ h )^^ f^ ( ) x h
- − (^) , there is subtraction, a difference.
You’ll see later that there is subtraction in the bottom (denominator) too.
Since there is a “ difference ” in the “ quotient ”,
We call this special fraction, f^ (^ x^ h )^^ f^ ( ) x h
- − (^) , the DIFFERENCE QUOTIENT.
FUNCTION NOTATION.
The notation f ( ) x is function notation. It is used to define a rule that is used to determine values for the function. If you see an algebraic expression on the right side of an equation involving f ( ) x , you replace the “x” in the expression by whatever appears in the parentheses next to the f in the f ( ) x. The “(x)” does NOT mean that you’re multiplying “x”. It means that you write your answer using “x” as a variable. The “(3)” next to f would mean to replace “x” by “3”. It does NOT mean to multiply everything by “3”.
For example,
f ( x ) = 3 x^2 + 2 x + 5
f (3) means to replace the “x” in 3 x^2 + 2 x + 5 by “3” and do the arithmetic. 2 2
f x x x f
f tootsie ( ) means to replace the “ x ” in 3 x^2 + 2 x + 5 by “ tootsie ” and simplify. 2 2
( tootsie ) to ) 5
f x otsie tootsie
x f
x + + = + +
f ( x + h ) means to replace the “ x ” in 3 x^2 + 2 x + 5 by “ x+h ” and simplify.
2 2 2
2 2
2
3 ( ) 2 ( ) 5 3 6 3 2 2 5
x h x h x h x
x
xh h x h x xh h x h
f x x f + + +
THE DIFFERENCE QUOTIENT: f^ (^ x^^ +^ h h )^^ − f^ ( ) x
Think of this fraction as three different rooms. There are two rooms on the top floor separated by a minus sign, and one room on the bottom floor, the “h” room. We enter each room, do the work that needs to be done, and leave.
What follows are a linear example, a quadratic example, and a trigonometric example.
EXAMPLE 2: A Quadratic Expression
f ( ) x = x^2 + 2 x Find : f^ (^ x^ h )^^ f^ ( ) x h
Let’s begin with the two rooms in the top. f ( x + h ) − f ( ) x
With f ( ) x = x^2 + 2 x f ( x + h ) beco m es ( x + h ) 2 +2( x + h ) Simplifying,
2
2
2
x h x h x h x h x h x h xx xh xh hh x h x xh
f is
h x h
Still on the top floor, there is f ( ) x , and it has a minus sign in front of it. − f ( ) x becomes − ( x^2^ + 2 x ) which becomes − x^2 − 2 x
Now the top floor becomes the following:
f ( x +^ h ) −^ f ( ) x =^ x^2^^ +^2 xh^ +^ h^2^ +^2 x^ +^2 h −^ x^^2 − 2 x
Simplifying: x^2^ + 2 xh + h^2 + 2 x + 2 h − x^2^ − 2 x 2 2 2
x xh h 2 x h x 2 h h h
x x
Put this result back on the top floor over the “h” room on the bottom floor.
2
2
Factor the common factor of "h" in the top. 2 2 (2 2)
Cancel the common factor. (2 2) (^2 )
f x h f x xh h h h h
x h h h h x h
h
h h
x h (^) x h h
We’re done! In other words, if f ( ) x = x^2 + 2 x
Then
f ( x h ) f ( ) x (^2) x h 2 h
EXAMPLE 3: A Trigonometric Expression
f ( ) x = sin( ) x Find : f^ (^ x^ h )^^ f^ ( ) x h
Let’s begin with f ( x + h ) − f ( ) x
With f ( ) x =sin( ) x f ( x + h ) becomes sin( x + h )
We need the trig identity for the sine of a sum of two angles.
sin( A + B ) = sin A cos B +sin B cos A
This means that
f ( x + h ) = sin( x + h ) =sin( ) cos( ) x h +sin( ) cos( ) h x
Still on the top, there is f ( ) x , and it has a minus sign in front of it.
− f ( ) x becom es −sin( ) x
Now the top becomes the following:
f ( x + h ) − f ( ) x = sin( x + h ) −sin( ) x = sin( ) cos( ) x h + sin( ) cos( ) h x −sin( ) x
Collect together the terms with “sin(x)” in them. Use the commutative property of addition to rearrange terms.
f ( x + h ) − f ( ) x = s in( ) x cos( ) h + sin( ) cos( ) h x −sin( ) x = sin( ) x cos( ) h −sin( x ) +sin( ) h cos( ) x
Now factor “sin(x)” from the first two terms.
= sin( ) x [cos( ) h − 1 ] +sin( ) cos( ) h x
