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Slopes and the Difference Quotient

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

Δym =

y2 – y1= x2 – x1Δx

Slopes and the Difference Quotient

(x1, y1)

(x2, y2)

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

Δym =

y2 – y1= x2 – x1Δx

Slopes and the Difference Quotient

(x1, y1)

(x2, y2)

Δy=y2–y1=rise

Δx=x2–x1=run

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

Δym =

y2 – y1= x2 – x1Δx

Let (x1,y1) = P = (x, f(x))

x

P=(x, f(x))

Slopes and the Difference Quotienty= f(x)

(x1, y1)

(x2, y2)

Δy=y2–y1=rise

Δx=x2–x1=run

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

x

P=(x, f(x))

Δym =

y2 – y1= x2 – x1Δx

Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))

Slopes and the Difference Quotienty= f(x)

(x1, y1)

(x2, y2)

Δy=y2–y1=rise

Δx=x2–x1=run

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

Δym =

y2 – y1= x2 – x1Δx

Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))

Slopes and the Difference Quotienty= f(x)

(x1, y1)

(x2, y2)

Δy=y2–y1=rise

Δx=x2–x1=run

h

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

Δym =

y2 – y1= x2 – x1Δx

Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))

be as shown for some y = f(x),

Slopes and the Difference Quotienty= f(x)

h

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

Δym =

y2 – y1= x2 – x1Δx

Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))

be as shown for some y = f(x), then the slope of the

cord connecting P and Q (in function notation) is

Slopes and the Difference Quotienty= f(x)

h

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

Δym =

y2 – y1= x2 – x1Δx

Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))

be as shown for some y = f(x), then the slope of the

cord connecting P and Q (in function notation) is

Δym =

f(x+h) – f(x) = (x+h) – x Δx

Slopes and the Difference Quotienty= f(x)

h

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

Δym =

y2 – y1= x2 – x1Δx

Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))

be as shown for some y = f(x), then the slope of the

cord connecting P and Q (in function notation) is

Δym =

f(x+h) – f(x) = (x+h) – x Δx

or m = f(x+h) – f(x)

h

Slopes and the Difference Quotienty= f(x)

h

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

Δym =

y2 – y1= x2 – x1Δx

Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))

be as shown for some y = f(x), then the slope of the

cord connecting P and Q (in function notation) is

Δym =

f(x+h) – f(x) = (x+h) – x Δx

or m = f(x+h) – f(x)

h

This is the "difference quotient" formula for slopes

Slopes and the Difference Quotienty= f(x)

h

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

Δym =

y2 – y1= x2 – x1Δx

Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))

be as shown for some y = f(x), then the slope of the

cord connecting P and Q (in function notation) is

Δym =

f(x+h) – f(x) = (x+h) – x Δx

or m = f(x+h) – f(x)

h

f(x+h)–f(x) = Δy

because f(x+h) – f(x) = difference in height

This is the "difference quotient" formula for slopes

Slopes and the Difference Quotienty= f(x)

Recall that if (x1, y1) and (x2, y2)

are two points then the slope m

of the line connecting them is

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

Δym =

y2 – y1= x2 – x1Δx

Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))

be as shown for some y = f(x), then the slope of the

cord connecting P and Q (in function notation) is

Δym =

f(x+h) – f(x) = (x+h) – x Δx

or m = f(x+h) – f(x)

h

h=Δx

f(x+h)–f(x) = Δy

because f(x+h) – f(x) = difference in height and

h = (x+h) – x = difference in the x's, as shown.

This is the "difference quotient" formula for slopes.

Slopes and the Difference Quotienty= f(x)

h

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

m = f(x+h) – f(x)

hh

f(x+h)–f(x)

The Algebra of Difference QuotientThe Difference Quotient Formula

y= f(x)

h

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

The goal of “simplifying”

the difference–quotient formula

is to eliminate the h in the denominator.

m = f(x+h) – f(x)

hh

f(x+h)–f(x)

The Algebra of Difference QuotientThe Difference Quotient Formula

y= f(x)

h

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

The goal of “simplifying”

the difference–quotient formula

is to eliminate the h in the denominator.

Examples of the algebra for manipulating this formula

are given below.

m = f(x+h) – f(x)

hh

f(x+h)–f(x)

The Algebra of Difference QuotientThe Difference Quotient Formula

y= f(x)

h

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

The goal of “simplifying”

the difference–quotient formula

is to eliminate the h in the denominator.

Examples of the algebra for manipulating this formula

are given below.

m = f(x+h) – f(x)

hh

f(x+h)–f(x)

The Algebra of Difference QuotientThe Difference Quotient Formula

Example A. (Quadratics) Given f(x) = x2 – 2x + 2,

f(x+h) – f(x) h . simplify its difference–quotient

y= f(x)

h

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

The goal of “simplifying”

the difference–quotient formula

is to eliminate the h in the denominator.

Examples of the algebra for manipulating this formula

are given below.

m = f(x+h) – f(x)

hh

f(x+h)–f(x)

The Algebra of Difference QuotientThe Difference Quotient Formula

Example A. (Quadratics) Given f(x) = x2 – 2x + 2,

f(x+h) – f(x) h =

f(x+h) – f(x) h . simplify its difference–quotient

y= f(x)

h

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

The goal of “simplifying”

the difference–quotient formula

is to eliminate the h in the denominator.

Examples of the algebra for manipulating this formula

are given below.

m = f(x+h) – f(x)

hh

f(x+h)–f(x)

The Algebra of Difference QuotientThe Difference Quotient Formula

Example A. (Quadratics) Given f(x) = x2 – 2x + 2,

f(x+h) – f(x) h =

(x+h)2 – 2(x+h) + 2 – [ x2 – 2x + 2]h

f(x+h) – f(x) h . simplify its difference–quotient

y= f(x)

h

x

P=(x, f(x))

x+h

Q=(x+h, f(x+h))

The goal of “simplifying”

the difference–quotient formula

is to eliminate the h in the denominator.

Examples of the algebra for manipulating this formula

are given below.

m = f(x+h) – f(x)

hh

f(x+h)–f(x)

The Algebra of Difference QuotientThe Difference Quotient Formula

Example A. (Quadratics) Given f(x) = x2 – 2x + 2,

f(x+h) – f(x) h =

(x+h)2 – 2(x+h) + 2 – [ x2 – 2x + 2]h

2xh – 2h + h2

h= 2x – 2 + h. =

f(x+h) – f(x) h . simplify its difference–quotient

y= f(x)

http://www.slideshare.net/math123

a/4-7polynomial-operationsvertical

h

The algebra for simplifying the difference quotient of

rational functions is the algebra for simplifying

complex fractions. To simplify a complex fraction,

use the LCD to clear all denominators.

The Algebra of Difference Quotient

Example B. (Rational Functions I)

Simplify the difference quotient of f(x) =

The algebra for simplifying the difference quotient of

rational functions is the algebra for simplifying

complex fractions. To simplify a complex fraction,

use the LCD to clear all denominators.

3 – x2

The Algebra of Difference Quotient

Example B. (Rational Functions I)

Simplify the difference quotient of f(x) =

The algebra for simplifying the difference quotient of

rational functions is the algebra for simplifying

complex fractions. To simplify a complex fraction,

use the LCD to clear all denominators.

3 – x2

The Algebra of Difference Quotient

f(x+h) – f(x) h

=

Example B. (Rational Functions I)

Simplify the difference quotient of f(x) =

The algebra for simplifying the difference quotient of

rational functions is the algebra for simplifying

complex fractions. To simplify a complex fraction,

use the LCD to clear all denominators.

3 – x2

–3 – (x + h)

23 – x

2

h

The Algebra of Difference Quotient

f(x+h) – f(x) h

=

Example B. (Rational Functions I)

Simplify the difference quotient of f(x) =

The algebra for simplifying the difference quotient of

rational functions is the algebra for simplifying

complex fractions. To simplify a complex fraction,

use the LCD to clear all denominators.

3 – x2

–3 – (x + h)

23 – x

2

h

(3 – x – h) (3 – x)

The Algebra of Difference Quotient

f(x+h) – f(x) h

=(3 – x – h) (3 – x)

Example B. (Rational Functions I)

Simplify the difference quotient of f(x) =

The algebra for simplifying the difference quotient of

rational functions is the algebra for simplifying

complex fractions. To simplify a complex fraction,

use the LCD to clear all denominators.

3 – x2

–3 – (x + h)

23 – x

2

h

(3 – x – h) (3 – x)

The Algebra of Difference Quotient

f(x+h) – f(x) h

=(3 – x – h) (3 – x)

(3 – x – h) (3 – x)

Example B. (Rational Functions I)

Simplify the difference quotient of f(x) =

The algebra for simplifying the difference quotient of

rational functions is the algebra for simplifying

complex fractions. To simplify a complex fraction,

use the LCD to clear all denominators.

3 – x2

–3 – (x + h)

23 – x

2

h

(3 – x – h) (3 – x)

The Algebra of Difference Quotient

f(x+h) – f(x) h

=(3 – x – h) (3 – x)

(3 – x – h) (3 – x)

2(3 – x) – 2(3 – x – h)

h(3 – x – h) (3 – x)=

Warning: It’s illegal to cancel the ( )’s,

we have to simplify the numerator,

simplify

The algebra for simplifying the difference quotient of

rational functions is the algebra for simplifying

complex fractions. To simplify a complex fraction,

use the LCD to clear all denominators.

–3 – (x + h)

23 – x

2

h

(3 – x – h) (3 – x)

The Algebra of Difference Quotient

f(x+h) – f(x) h

=(3 – x – h) (3 – x)

(3 – x – h)

2(3 – x) – 2(3 – x – h)

h(3 – x – h) (3 – x)= simplify

2hh(3 – x – h) (3 – x)=

2(3 – x – h) (3 – x)=

Example B. (Rational Functions I)

Simplify the difference quotient of f(x) = 3 – x2

(3 – x)

Example C. (Rational Functions II)

Simplify the difference quotient of f(x) =

The Algebra of Difference Quotient

3x + 12x – 3

Example C. (Rational Functions II)

Simplify the difference quotient of f(x) =

The Algebra of Difference Quotient

f(x+h) – f(x) h

=

3x + 12x – 3

Example C. (Rational Functions II)

Simplify the difference quotient of f(x) =

The Algebra of Difference Quotient

f(x+h) – f(x) h

=

3x + 12x – 3

–3(x + h) + 1 3x + 1

2x – 3

h

2(x + h) – 3

Example C. (Rational Functions II)

Simplify the difference quotient of f(x) =

(3x + 3h + 1)(3x + 1)

The Algebra of Difference Quotient

f(x+h) – f(x) h

=

3x + 12x – 3

–3(x + h) + 1 3x + 1

2x – 3

h

2(x + h) – 3

(3x + 3h + 1)(3x + 1)

Example C. (Rational Functions II)

Simplify the difference quotient of f(x) =

(3x + 3h + 1)(3x + 1)

The Algebra of Difference Quotient

f(x+h) – f(x) h

=

(3x + 3h + 1) (3x + 1)

3x + 12x – 3

–3(x + h) + 1 3x + 1

2x – 3

h

2(x + h) – 3

(3x + 3h + 1)(3x + 1)

Example C. (Rational Functions II)

Simplify the difference quotient of f(x) =

(3x + 3h + 1)(3x + 1)

The Algebra of Difference Quotient

f(x+h) – f(x) h

=

(3x + 3h + 1) (3x + 1)

h(3x + 3h + 1)(3x + 1) =

3x + 12x – 3

–3(x + h) + 1 3x + 1

2x – 3

h

2(x + h) – 3

(3x + 3h + 1)(3x + 1)

(2x + 2h – 3)(3x + 1) – (2x – 3)(3x + 3h + 1)

Example C. (Rational Functions II)

Simplify the difference quotient of f(x) =

(3x + 3h + 1)(3x + 1)

The Algebra of Difference Quotient

f(x+h) – f(x) h

=

(3x + 3h + 1) (3x + 1)

h(3x + 3h + 1)(3x + 1) =

11h=

3x + 12x – 3

–3(x + h) + 1 3x + 1

2x – 3

h

2(x + h) – 3

(3x + 3h + 1)(3x + 1)

(2x + 2h – 3)(3x + 1) – (2x – 3)(3x + 3h + 1)

h(3x + 3h + 1)(3x + 1)

Example C. (Rational Functions II)

Simplify the difference quotient of f(x) =

(3x + 3h + 1)(3x + 1)

The Algebra of Difference Quotient

f(x+h) – f(x) h

=

(3x + 3h + 1) (3x + 1)

h(3x + 3h + 1)(3x + 1) =

11h=

http://www.slideshare.net/math123b/2-5-complex-fractions

3x + 12x – 3

–3(x + h) + 1 3x + 1

2x – 3

h

2(x + h) – 3

(3x + 3h + 1)(3x + 1)

(2x + 2h – 3)(3x + 1) – (2x – 3)(3x + 3h + 1)

h(3x + 3h + 1)(3x + 1)

11=

(3x + 3h + 1)(3x + 1)

To rationalize square–root radicals in expressions

we use the formula (x – y)(x + y) = x2 – y2 and

(x + y) and (x – y) are called conjugates.

The Algebra of Difference Quotient

To rationalize square–root radicals in expressions

we use the formula (x – y)(x + y) = x2 – y2 and

(x + y) and (x – y) are called conjugates.

2x – 1

The Algebra of Difference Quotient

Example D. (Square–root Functions I) Simplify the difference quotient of f(x) =

To rationalize square–root radicals in expressions

we use the formula (x – y)(x + y) = x2 – y2 and

(x + y) and (x – y) are called conjugates.

h2(x + h) – 1 – 2x – 1

2x – 1

The Algebra of Difference Quotient

Example D. (Square–root Functions I) Simplify the difference quotient of f(x) =

f(x + h) – f(x) h

=

To rationalize square–root radicals in expressions

we use the formula (x – y)(x + y) = x2 – y2 and

(x + y) and (x – y) are called conjugates.

h2(x + h) – 1 – 2x – 1

2x – 1

The Algebra of Difference Quotient

Example D. (Square–root Functions I) Simplify the difference quotient of f(x) =

f(x + h) – f(x) h

=(2x + 2h –1 + 2x – 1)

(2x + 2h –1 + 2x – 1)

To rationalize square–root radicals in expressions

we use the formula (x – y)(x + y) = x2 – y2 and

(x + y) and (x – y) are called conjugates.

h2(x + h) – 1 – 2x – 1

=

2x – 1

The Algebra of Difference Quotient

Example D. (Square–root Functions I) Simplify the difference quotient of f(x) =

f(x + h) – f(x) h

=(2x + 2h –1 + 2x – 1)

(2x + 2h –1 + 2x – 1)

(2x + 2h –1)2 – (2x – 1)2

h(2x + 2h –1 + 2x – 1)

To rationalize square–root radicals in expressions

we use the formula (x – y)(x + y) = x2 – y2 and

(x + y) and (x – y) are called conjugates.

h2(x + h) – 1 – 2x – 1

=

2x – 1

The Algebra of Difference Quotient

Example D. (Square–root Functions I) Simplify the difference quotient of f(x) =

f(x + h) – f(x) h

=(2x + 2h –1 + 2x – 1)

(2x + 2h –1 + 2x – 1)

(2x + 2h –1)2 – (2x – 1)2

h(2x + 2h –1 + 2x – 1) =

(2x + 2h –1) – (2x – 1)

h(2x + 2h –1 + 2x – 1)

To rationalize square–root radicals in expressions

we use the formula (x – y)(x + y) = x2 – y2 and

(x + y) and (x – y) are called conjugates.

h2(x + h) – 1 – 2x – 1

=

2x – 1

The Algebra of Difference Quotient

Example D. (Square–root Functions I) Simplify the difference quotient of f(x) =

f(x + h) – f(x) h

=(2x + 2h –1 + 2x – 1)

(2x + 2h –1 + 2x – 1)

(2x + 2h –1)2 – (2x – 1)2

h(2x + 2h –1 + 2x – 1) =

(2x + 2h –1) – (2x – 1)

h(2x + 2h –1 + 2x – 1)

=2h

h(2x + 2h –1 + 2x – 1) =

2

(2x + 2h –1 + 2x – 1) http://www.slideshare.net/math123b/2-5-complex-fractions

We will see in calculus that the above algebra extend

the concept of “the slope of a line” to

“varying slopes of a curve”. The ones we did in turn

give the bases for general approaches for computing

“slopes for curves”.

The Algebra of Difference Quotient

Simplify the difference quotient of the following

functions by removing the h in the denominator.

Quadratic Functions:

1. f(x) = x2 – 2 2. f(x) = x2 – 2x + 5

3. f(x) = –x2 + 2x + 3 4. f(x) = –3x2 – 2x – 3

5 f(x) = ax2 + bx + c

Rational Functions:

1. f(x) =x – 2

3 2. f(x) =2 – 3x

4 3. f(x) =x – 23 – x

4. f(x) =2x – 31 – 5x 5. f(x) =

ax + b1

Square–root Functions:

1. f(x) = (x – 3)1/2 2. f(x) = (3x – 2)1/2

4. f(x) =(3x – 2)–1/2 5. f(x) = √ax + b

The Algebra of Difference Quotient

3. f(x) = (x – 3)–1/2