2. In order to discuss mathematics precisely, basic
geometric information and formulas concerning
graphs are given in function notation.
Slopes and the Difference Quotient
3. Given x, the output of a function is denoted as y or
as f(x).
In order to discuss mathematics precisely, basic
geometric information and formulas concerning
graphs are given in function notation.
Slopes and the Difference Quotient
4. Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a general point P
on the graph is often denoted as (x, f(x)).
In order to discuss mathematics precisely, basic
geometric information and formulas concerning
graphs are given in function notation.
Slopes and the Difference Quotient
5. Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a general point P
on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
In order to discuss mathematics precisely, basic
geometric information and formulas concerning
graphs are given in function notation.
y= f(x)
Slopes and the Difference Quotient
6. Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a general point P
on the graph is often denoted as (x, f(x)).
In order to discuss mathematics precisely, basic
geometric information and formulas concerning
graphs are given in function notation.
x
P=(x, f(x))
y= f(x)
f(x)
Note that the f(x) = the height of the point P.
Slopes and the Difference Quotient
7. Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a general point P
on the graph is often denoted as (x, f(x)).
Let h be a small positive value,
so x+h is a point close to x,
In order to discuss mathematics precisely, basic
geometric information and formulas concerning
graphs are given in function notation.
x
P=(x, f(x))
y= f(x)
f(x)
Note that the f(x) = the height of the point P.
Slopes and the Difference Quotient
8. Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a general point P
on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
Note that the f(x) = the height of the point P.
Let h be a small positive value,
so x+h is a point close to x,
x+h
In order to discuss mathematics precisely, basic
geometric information and formulas concerning
graphs are given in function notation.
f(x)
y= f(x)
Slopes and the Difference Quotient
9. Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a general point P
on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
Note that the f(x) = the height of the point P.
Let h be a small positive value,
so x+h is a point close to x,
x+h
then f(x+h) is the output for x+h,
and (x+h, f(x+h)) represents the
corresponding point, say Q,
on the graph.
In order to discuss mathematics precisely, basic
geometric information and formulas concerning
graphs are given in function notation.
f(x)
y= f(x)
Slopes and the Difference Quotient
10. Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a general point P
on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
Note that the f(x) = the height of the point P.
Let h be a small positive value,
so x+h is a point close to x,
x+h
then f(x+h) is the output for x+h,
and (x+h, f(x+h)) represents the
corresponding point, say Q,
on the graph.
In order to discuss mathematics precisely, basic
geometric information and formulas concerning
graphs are given in function notation.
Q=(x+h, f(x+h))
f(x)
y= f(x)
Slopes and the Difference Quotient
11. Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a general point P
on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
Note that the f(x) = the height of the point P.
Let h be a small positive value,
so x+h is a point close to x,
x+h
then f(x+h) is the output for x+h,
and (x+h, f(x+h)) represents the
corresponding point, say Q,
on the graph.
In order to discuss mathematics precisely, basic
geometric information and formulas concerning
graphs are given in function notation.
Note that the f(x+h) = the height of the point Q.
Q=(x+h, f(x+h))
f(x)
f(x+h)
y= f(x)
Slopes and the Difference Quotient
12. Recall that if (x1, y1) and (x2, y2)
are two points then the slope m
of the line connecting them is
Ξy
m =
y2 β y1
= x2 β x1Ξx
Slopes and the Difference Quotient
13. Recall that if (x1, y1) and (x2, y2)
are two points then the slope m
of the line connecting them is
Ξy
m =
y2 β y1
= x2 β x1Ξx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
as shown in the picture,
Slopes and the Difference Quotient
14. 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))
Ξy
m =
y2 β y1
= x2 β x1Ξx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
as shown in the picture,
Slopes and the Difference Quotient
15. 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))
Ξy
m =
y2 β y1
= x2 β x1Ξx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
as shown in the picture, then the slope of the cord
connecting P and Q (in function notation) is
Slopes and the Difference Quotient
16. 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))
Ξy
m =
y2 β y1
= x2 β x1Ξx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
as shown in the picture, then the slope of the cord
connecting P and Q (in function notation) is
Ξy
m =
f(x+h) β f(x)
= (x+h) β xΞx
Slopes and the Difference Quotient
17. 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))
Ξy
m =
y2 β y1
= x2 β x1Ξx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
as shown in the picture, then the slope of the cord
connecting P and Q (in function notation) is
Ξy
m =
f(x+h) β f(x)
= (x+h) β xΞx or
Slopes and the Difference Quotient
m = f(x+h) β f(x)
h (= Ξx)
18. 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))
Ξy
m =
y2 β y1
= x2 β x1Ξx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
as shown in the picture, then the slope of the cord
connecting P and Q (in function notation) is
Ξy
m =
f(x+h) β f(x)
= (x+h) β xΞx or
This is the "difference quotient" formula for slopes
Slopes and the Difference Quotient
m = f(x+h) β f(x)
h (= Ξx)
19. 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))
Ξy
m =
y2 β y1
= x2 β x1Ξx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
as shown in the picture, then the slope of the cord
connecting P and Q (in function notation) is
Ξy
m =
f(x+h) β f(x)
= (x+h) β xΞx or m = f(x+h) β f(x)
h (= Ξx)
f(x+h)βf(x) = Ξy
because f(x+h) β f(x) = difference in height
This is the "difference quotient" formula for slopes
Slopes and the Difference Quotient
20. 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))
Ξy
m =
y2 β y1
= x2 β x1Ξx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
as shown in the picture, then the slope of the cord
connecting P and Q (in function notation) is
Ξy
m =
f(x+h) β f(x)
= (x+h) β xΞx or
h=Ξx
f(x+h)βf(x) = Ξy
because f(x+h) β f(x) = difference in height and
h = (x+h) β x = Ξx = difference in the x's, as shown.
This is the "difference quotient" formula for slopes.
Slopes and the Difference Quotient
m = f(x+h) β f(x)
h (= Ξx)
21. Example A.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h, f(x+h))
with x = 2 and h = 0.2.
Slopes and the Difference Quotient
22. Example A.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h, f(x+h))
with x = 2 and h = 0.2.
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2.
Slopes and the Difference Quotient
23. Example A.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h, f(x+h))
with x = 2 and h = 0.2.
f(x+h) β f(x)
h
Using the difference
quotient, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
Slopes and the Difference Quotient
24. Example A.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h, f(x+h))
with x = 2 and h = 0.2.
f(x+h) β f(x)
h
Using the difference
quotient, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) β f(2)=
Slopes and the Difference Quotient
25. Example A.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h, f(x+h))
with x = 2 and h = 0.2.
f(x+h) β f(x)
h
Using the difference
quotient, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) β f(2)
0.2
=
Slopes and the Difference Quotient
26. Example A.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h, f(x+h))
with x = 2 and h = 0.2.
f(x+h) β f(x)
h
Using the difference
quotient, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) β f(2)
0.2
=
=
2.44 β 2
0.2
Slopes and the Difference Quotient
27. Example A.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h, f(x+h))
with x = 2 and h = 0.2.
f(x+h) β f(x)
h
Using the difference
quotient, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) β f(2)
0.2
=
=
2.44 β 2
0.2 =
0.44
0.2
Slopes and the Difference Quotient
28. Example A.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h, f(x+h))
with x = 2 and h = 0.2.
f(x+h) β f(x)
h
Using the difference
quotient, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) β f(2)
0.2
=
=
2.44 β 2
0.2
= 2.2
=
0.44
0.2
Slopes and the Difference Quotient
29. Example A.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h, f(x+h))
with x = 2 and h = 0.2.
f(x+h) β f(x)
h
Using the difference
quotient, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) β f(2)
0.2
=
=
2.44 β 2
0.2
= 2.2
(2.2, 2.44)
(2, 2)
2 2.2
=
0.44
0.2
Slopes and the Difference Quotient
30. Example A.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h, f(x+h))
with x = 2 and h = 0.2.
f(x+h) β f(x)
h
Using the difference
quotient, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) β f(2)
0.2
=
=
2.44 β 2
0.2
= 2.2
(2.2, 2.44)
(2, 2)
2 2.2
=
0.44
0.2
0.44
0.2
slope m = 2.2
Slopes and the Difference Quotient
31. The algebra of calculating the slopes of some basic
types of functions are given below.
Slope Algebra
32. The algebra of calculating the slopes of some basic
types of functions are given below.
Slope Algebra
Example B. Simplify the difference quotient.
Make sure the h is cancelled.
a. (2nd degree polynomials) f(x) = β2x2 + 3x + 1
33. The algebra of calculating the slopes of some basic
types of functions are given below.
Slope Algebra
Example B. Simplify the difference quotient.
Make sure the h is cancelled.
f(x+h) β f(x)
h
=
a. (2nd degree polynomials) f(x) = β2x2 + 3x + 1
34. The algebra of calculating the slopes of some basic
types of functions are given below.
Slope Algebra
Example B. Simplify the difference quotient.
Make sure the h is cancelled.
f(x+h) β f(x)
h
=
β2(x+h)2 + 3(x+h) + 1 β [β2x2 +3x +1]
h
a. (2nd degree polynomials) f(x) = β2x2 + 3x + 1
35. The algebra of calculating the slopes of some basic
types of functions are given below.
Slope Algebra
Example B. Simplify the difference quotient.
Make sure the h is cancelled.
f(x+h) β f(x)
h
=
β2(x+h)2 + 3(x+h) + 1 β [β2x2 +3x +1]
h
a. (2nd degree polynomials) f(x) = β2x2 + 3x + 1
β2x2 β4xh β2h2 +3x +3h +1 β [β2x2 +3x +1]
h
=
36. The algebra of calculating the slopes of some basic
types of functions are given below.
Slope Algebra
Example B. Simplify the difference quotient.
Make sure the h is cancelled.
f(x+h) β f(x)
h
=
β2(x+h)2 + 3(x+h) + 1 β [β2x2 +3x +1]
h
a. (2nd degree polynomials) f(x) = β2x2 + 3x + 1
β2x2 β4xh β2h2 +3x +3h +1 β [β2x2 +3x +1]
h
=
β4xh β2h2 +3h
h
=
h(β4x β2h +3)
h
=
37. The algebra of calculating the slopes of some basic
types of functions are given below.
Slope Algebra
Example B. Simplify the difference quotient.
Make sure the h is cancelled.
f(x+h) β f(x)
h
=
β2(x+h)2 + 3(x+h) + 1 β [β2x2 +3x +1]
h
a. (2nd degree polynomials) f(x) = β2x2 + 3x + 1
β2x2 β4xh β2h2 +3x +3h +1 β [β2x2 +3x +1]
h
=
β4xh β2h2 +3h
h
=
h(β4x β2h +3)
h
= = β4x β2h +3
40. Rational Expressions
βx + h β 1
2
x β 1
2
h
b. (Simple rational function) f(x) = x β 1
2
Simplify the difference quotient.
f(x+h) β f(x)
h =
41. Rational Expressions
βx + h β 1
2
x β 1
2
h
Multiply the top and bottom by (x + h β 1)(x β 1)
to remove fractions in the numerator.
b. (Simple rational function) f(x) = x β 1
2
Simplify the difference quotient.
f(x+h) β f(x)
h =
42. Rational Expressions
βx + h β 1
2
x β 1
2
h
Multiply the top and bottom by (x + h β 1)(x β 1)
to remove fractions in the numerator.
b. (Simple rational function) f(x) = x β 1
2
Simplify the difference quotient.
f(x+h) β f(x)
h =
βx + h β 1
2
x β 1
2
h
[ ]
43. Rational Expressions
βx + h β 1
2
x β 1
2
h
Multiply the top and bottom by (x + h β 1)(x β 1)
to remove fractions in the numerator.
(x + h β1)(x β 1)
(x + h β1)(x β 1)*
b. (Simple rational function) f(x) = x β 1
2
Simplify the difference quotient.
f(x+h) β f(x)
h =
βx + h β 1
2
x β 1
2
h
[ ]
44. Rational Expressions
βx + h β 1
2
x β 1
2
h
Multiply the top and bottom by (x + h β 1)(x β 1)
to remove fractions in the numerator.
(x + h β1)(x β 1)
(x + h β1)(x β 1)*
=
β2(x β 1) 2(x + h β1)
h
b. (Simple rational function) f(x) = x β 1
2
Simplify the difference quotient.
f(x+h) β f(x)
h =
βx + h β 1
2
x β 1
2
h
(x + h β1)(x β 1)
[ ]
45. Rational Expressions
βx + h β 1
2
x β 1
2
h
Multiply the top and bottom by (x + h β 1)(x β 1)
to remove fractions in the numerator.
(x + h β1)(x β 1)
(x + h β1)(x β 1)*
=
β2(x β 1) 2(x + h β1)
h
b. (Simple rational function) f(x) = x β 1
2
Simplify the difference quotient.
f(x+h) β f(x)
h =
βx + h β 1
2
x β 1
2
h
(x + h β1)(x β 1) =
β 2h
h(x + h β1)(x β 1)
[ ]
46. Rational Expressions
βx + h β 1
2
x β 1
2
h
Multiply the top and bottom by (x + h β 1)(x β 1)
to remove fractions in the numerator.
(x + h β1)(x β 1)
(x + h β1)(x β 1)*
=
β2(x β 1) 2(x + h β1)
h
b. (Simple rational function) f(x) = x β 1
2
Simplify the difference quotient.
f(x+h) β f(x)
h =
βx + h β 1
2
x β 1
2
h
(x + h β1)(x β 1) =
β 2h
h(x + h β1)(x β 1)
=
β2
(x + h β1)(x β 1)
[ ]
48. Rational Expressions
h
c. (Simple root function) f(x) = β2x β 3
Simplify the difference quotient.
f(x+h) β f(x)
h =
β2(x + h) β 3 β β2x β 3
49. Rational Expressions
h
c. (Simple root function) f(x) = β2x β 3
Simplify the difference quotient.
f(x+h) β f(x)
h =
β2(x + h) β 3 β β2x β 3
Rationalize the numerator to cancel the h in the
denominator so we may take the limits.
h
β2x + 2h β 3 β β2x β 3
*
β2x + 2h β 3 +β2x β 3
β2x + 2h β 3 +β2x β 3
50. Rational Expressions
h
c. (Simple root function) f(x) = β2x β 3
Simplify the difference quotient.
f(x+h) β f(x)
h =
β2(x + h) β 3 β β2x β 3
Rationalize the numerator to cancel the h in the
denominator so we may take the limits.
h
β2x + 2h β 3 β β2x β 3
*
=
β2x + 2h β 3 +β2x β 3
β2x + 2h β 3 +β2x β 3
2x + 2h β 3 β (2x β 3)
h β2x β 3β2x + h β 3 +
1*
51. Rational Expressions
h
c. (Simple root function) f(x) = β2x β 3
Simplify the difference quotient.
f(x+h) β f(x)
h =
β2(x + h) β 3 β β2x β 3
Rationalize the numerator to cancel the h in the
denominator so we may take the limits.
h
β2x + 2h β 3 β β2x β 3
*
=
β2x + 2h β 3 +β2x β 3
β2x + 2h β 3 +β2x β 3
2x + 2h β 3 β (2x β 3)
h β2x β 3β2x + h β 3 +
1
= 2h
h β2x β 3β2x + h β 3 +
1
*
*
52. Rational Expressions
h
c. (Simple root function) f(x) = β2x β 3
Simplify the difference quotient.
f(x+h) β f(x)
h =
β2(x + h) β 3 β β2x β 3
Rationalize the numerator to cancel the h in the
denominator so we may take the limits.
h
β2x + 2h β 3 β β2x β 3
*
=
β2x + 2h β 3 +β2x β 3
β2x + 2h β 3 +β2x β 3
2x + 2h β 3 β (2x β 3)
h β2x β 3β2x + h β 3 +
1
= 2h
h β2x β 3β2x + h β 3 +
1
=
β2x β 3β2x + h β 3 +
2
*
*
53. Another version of the difference
quotient formula is to use points
P = (a, f(a)) and Q= (b, f(b)). We get
a
P=(a, f(a))
b
Q=(b, f(b))
Ξy
m =
f(b) β f(a)
= b β aΞx
b-a=Ξx
f(b)βf(a) = Ξy
Slopes and the Difference Quotient
54. Example B.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
Slopes and the Difference Quotient
55. Example B.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
Slopes and the Difference Quotient
56. Example B.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
f(b) β f(a)
b β a
Using the formula, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
Slopes and the Difference Quotient
57. Example B.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
f(b) β f(a)
b β a
Using the formula, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
f(5) β f(3)
5 β 3
=
Slopes and the Difference Quotient
58. Example B.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
f(b) β f(a)
b β a
Using the formula, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
f(5) β f(3)
5 β 3
=
=
17 β 5
2
Slopes and the Difference Quotient
59. Example B.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
f(b) β f(a)
b β a
Using the formula, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
f(5) β f(3)
5 β 3
=
=
17 β 5
2
= 6=
12
2
Slopes and the Difference Quotient
60. Example B.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
f(b) β f(a)
b β a
Using the formula, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
f(5) β f(3)
5 β 3
=
=
17 β 5
2
= 6
(5, 17)
(3, 5)
3 5=
12
2
Slopes and the Difference Quotient
61. Example B.
a. Given f(x) = x2 β 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
f(b) β f(a)
b β a
Using the formula, the slope is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
f(5) β f(3)
5 β 3
=
=
17 β 5
2
= 6
(5, 17)
(3, 5)
3 5=
12
2
12
2
slope m = 6
Slopes and the Difference Quotient
62. b. Given f(x) = x2 β 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient
63. b. Given f(x) = x2 β 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) β f(a)
b β a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 β 2x + 2
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient
64. b. Given f(x) = x2 β 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) β f(a)
b β a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 β 2x + 2
=
b2 β 2b + 2 β [ a2 β 2a + 2]
b β a
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient
65. b. Given f(x) = x2 β 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) β f(a)
b β a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 β 2x + 2
=
b2 β 2b + 2 β [ a2 β 2a + 2]
b β a
=
b2 β a2 β 2b + 2a
b β a
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient
66. b. Given f(x) = x2 β 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) β f(a)
b β a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 β 2x + 2
=
b2 β 2b + 2 β [ a2 β 2a + 2]
b β a
=
b2 β a2 β 2b + 2a
b β a
= (b β a)(b + a) β 2(b β a)
b β a
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient
67. b. Given f(x) = x2 β 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) β f(a)
b β a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 β 2x + 2
=
b2 β 2b + 2 β [ a2 β 2a + 2]
b β a
=
b2 β a2 β 2b + 2a
b β a
= (b β a)(b + a) β 2(b β a)
b β a
= (b β a) [(b + a) β 2]
b β a
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient
68. b. Given f(x) = x2 β 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) β f(a)
b β a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 β 2x + 2
=
b2 β 2b + 2 β [ a2 β 2a + 2]
b β a
=
b2 β a2 β 2b + 2a
b β a
= (b β a)(b + a) β 2(b β a)
b β a
= (b β a) [(b + a) β 2]
b β a
= b + a β 2
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient
69. Exercise A.
Given the following f(x), x, and h find f(x+h) β f(x)
1. y = 3x+2, x = 2, h = 0.1 2. y = β2x + 3, x= β 4, h = 0.05
3. y = 2x2 + 1, x = 1, h = 0.1 4. y = βx2 + 3, x= β2, h = β0.2
B. Given the following f(x), simplify Ξy = f(x+h) β f(x)
1. y = 3x+2 2. y = β2x + 3
3. y = 2x2 + 1 4. y = βx2 + 3
C. Simplify the difference quotient f(x+h) β f(x)
h
of the following functions that show up in calculus.
βSimplifyβ in this case means to transform the difference
quotient so the denominator βhβ is canceled.
5. y = x2 β x +2 6. y = βx2 + 3x β 1
Slopes and the Difference Quotient
1. y = 2x + 3 2. y = βΒ½ x + 5
3. y = β4x β 3 4. y = mx + b
7. y = β 3x2 β 2x β 4 8. y = ax2 + bx + c
70. Slopes and the Difference Quotient
9. y = 2
x + 3
β1
2 β 3x
16. y = (3 β x)1/2
15. y = x1/2
10. y =
β4
β5 β 3x11. y = 12. y =ax + b
cx + d
3x β 4
x β 513. y = 14. y =
c
ax + b
17. y = (4 β 3x)1/2 18. y = (ax + b)1/2
19. y = 2/(4 + x)1/2 20. y = 3(2x β 3)β1/2
21. y = β 7/(4 β 3x)1/2
22. y = c(ax + b)β1/2
cx + d
3x β 4
x β 5
23. y = 24. y = ax + b
β β
(Do Long division first)