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Understanding the Difference Quotient: A Key Concept in Calculus, Study notes of Calculus

The concept of the difference quotient, a fundamental concept in calculus. It discusses the meaning of quotient and difference, function notation, and provides examples of linear, quadratic, and trigonometric expressions. It also explains how to simplify the difference quotient and cancel common factors.

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Uploaded on 09/27/2022

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THE DIFFERENCE QUOTIENT
()()fxhfx
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.
Isnt
12
4
the same thing as saying
12 divided by 4
? Both produce the answer 3.
So when you look at
()()fxhfx
h
+
, think of it as a fraction. Hence it produces a quotient.
What does the word Differencemean?
For example, whats the difference between
8 and 5
? Three, eh? How did you arrive at the
answer? Oh! You subtracted.
So the word differenceis a synonym for subtraction.
In the top (numerator) of the quotient
()()fxhfx
h
+
, there is subtraction, a difference.
Youll see later that there is subtraction in the bottom (denominator) too.
Since there is a “differencein the “quotient,
We call this special fraction,
()()fxhfx
h
+
, the DIFFERENCE QUOTIENT.
FUNCTION NOTATION.
The notation
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
, you replace the “xin the expression by whatever appears in the parentheses
next to the
f
in the
. The “(x)does NOT mean that youre multiplying x. It means that
you write your answer using x” as a variable. The (3)next to f would mean to replace “xby
3. It does NOT mean to multiply everything by 3.
pf3
pf4
pf5

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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 becomesx^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 hx^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