2.1 Functions and Their Graphs

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2.2 The Limit of a Function
State Standard 1.0 Students demonstrate
knowledge of limit of values of functions. This
includes one-sided limits, infinite limits, and
limits at infinity. Objective To be able to
find the limit of a function.
2
2.2 The Limit of a Function
Definition And say the limit of f(x), as x
approaches a, equals L This says that the
values of f(x) get closer and closer to the
number L as x gets closer to the number a (from
either side)
3
Example 1 Find the Limit of f(x) for these three
cases
2
2
2
4
Example 2 Find the Limit of f(x) for these four
cases
2
2
2
DNE
5
Example 3 Find the Limit of f(x) for these four
cases
1
3
DNE
1.5
6
Example 4 Find the Limit of f(x) for these three
cases
- 8
8
DNE
7
Example 5 Find the Limit of f(x) for these three
cases
8
8
8
8
Example 6 Find the Limit of f(x) for these two
cases
-1
4
9
Example 7 Find the Limit of f(x) for these two
cases
8
-2
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Checking for Understanding
WS on Limits
11
2.2 Homework
Pg. 102 4 9
12
2.2 The Limit of a Function "Day 2"
State Standard 1.0 Students demonstrate
knowledge of limit of values of functions. This
includes one-sided limits, infinite limits, and
limits at infinity. Objective To be able to
find the limit of a function.
13
Example 1
8
8
DNE
8
0
-1
14
Example 2
8
8
1
8
2
2
15
Example 3
0
0
0
0
DNE
DNE
16
2.2c Warm - Up
8
8
DNE
DNE
0
0
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2.2c The Limit of a Function "Day 3"
State Standard 1.0 Students demonstrate
knowledge of limit of values of functions. This
includes one-sided limits, infinite limits, and
limits at infinity. Objective To be able to
find the limit of a function.
18
Example 1 Use a table of values to estimate the
value of the limit.
1
0.75
0.90
0.99
?
0.999
1.001
1.01
1.1
1.25
x approaches 1 From the RIGHT
x approaches 1 from the LEFT
19
Example 2 Use a table of values to estimate the
value of the limit.
8
100
400
1x106
?
1x106
400
100
x approaches 8 From the RIGHT
x approaches 8 from the LEFT
20
Example 3 Use a table of values to estimate the
value of the limit.
64
63.20
63.92
63.998
?
64.001
64.24
65
x approaches 64 From the RIGHT
x approaches 64 from the LEFT
21
2.2c Homework
Pg. 103 15 20
22
2.3a Warm - Up
Use a table of values to estimate the value of
the limit.
DNE
-9
-19
-999
?
1001
21
6
x approaches 8 From the RIGHT
x approaches 8 from the LEFT
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2.3a Calculating Limits Using the Limit Laws
State Standard 1.0 Students demonstrate
knowledge of limit of values of functions. This
includes one-sided limits, infinite limits, and
limits at infinity. Objective To be able to
find the limit of a function.
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Example for case 1
Ex for case 3
16
32
Ex for case 2
9
8
2
-4
25
Limit Laws Suppose that c is a constant
and
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Example 1a Find the Limit
3(4) 10 4
6
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Example 1b Find the Limit
28
Example 2 Find the Limit
29
2.3a Homework
Pg. 111-112 1 9
30
2.3b Warm - Up
Evaluate the limit.
3/2
9
31
2.3b Calculating Limits Using the Limit Laws
State Standard 1.0 Students demonstrate
knowledge of limit of values of functions. This
includes one-sided limits, infinite limits, and
limits at infinity. Objective To be able to
find the limit of a function.
32
Functions with the Direct Substitution Property
are called continuous at a. However, not all
limits can be evaluated by direct substitution,
as the following example shows
Example 1 Find
11
2
33
Example 2 Find the Limit
34
Example 3 Find the Limit
35
Example 4 Find the Limit
36
Example 5 Find the Limit
37
2.3b Homework
Pg. 112 11 29 odd
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2.5 Warm - Up
Evaluate the limit.
DNE
16
39
2.5 Continuity
State Standard 1.0 Students demonstrate
knowledge of limit of values of functions. This
includes one-sided limits, infinite limits, and
limits at infinity. Objective To be able to
find the limit of a function.
40
If a function f is not continuous at a point c,
we say that f is discontinuous at c or c is a
point of discontinuity of f.
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Most of the techniques of calculus require that
functions be continuous. A function is
continuous if you can draw it in one motion
without picking up your pencil.
A function is continuous at a point if the limit
is the same as the value of the function.
This function has discontinuities at x1 and x2.
It is continuous at x0, x3, and x4, because
the one-sided limits match the value of the
function
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Removable Discontinuities
(You can fill the hole.)
Essential Discontinuities
oscillating
infinite
jump
43
Removing a discontinuity
44
Removing a discontinuity
45
Example Find the value of x which f is not
continuous, which of the discontinuities are
removable?
Removable discontinuity is at
Where as x 1 is NOT a removable discontinuity.
46
Continuous functions can be added, subtracted,
multiplied, divided and multiplied by a constant,
and the new function remains continuous.
47
2.5 Homework
WS 1 10, 13 17 odd Pg. 133 1-6, 10-12, and
15 20
48
2.6 Warm - Up
Describe the continuity of the graph.
49
2.6 Limits at Infinity Horizontal Asymptotes
State Standard 1.0 Students demonstrate
knowledge of limit of values of functions. This
includes one-sided limits, infinite limits, and
limits at infinity. Objective To be able to
find the limit of a function.
50
2.6 Limits at Infinity Horizontal Asymptotes
Definition The line y L is called a horizontal
asymptote of the curve y f(x) if either
or
51
2.6 Limits at Infinity Horizontal Asymptotes
52
Example 1 Case 1 Numerator and Denominator of
Same Degree
Divide numerator and denominator by x2
53
Example 2 Case 2 Degree of Numerator Less than
Degree of Denominator
Divide numerator and denominator by x3
54
Example 3 Case 3 Degree of Numerator Greater
Than Degree of Denominator
Divide numerator and denominator by x
55
Checking for Understanding
Example 4 a) b)
56
Example 5 Find the Limit
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Checking for Understanding
Example 6 Find the Limit
58
2.6 Homework
WS 1 8 and Pg. 147 11 18, and 20 22
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2.7 Warm - Up
Solve and show work!

60
2.7 Tangent Lines
State Standard 4.1 Students demonstrate an
understanding of the derivative of a function as
the slope of the tangent line to the graph of the
function. Objective To be able to find the
tangent line.
61
2.7 Tangent Lines
Definition of a Tangent Line
Tangent Line
P
Q
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Slope
Q (x,f(x))
f(x) f(a)
P(a,f(a))
x a
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Definition The tangent line to the curve y
f(x) at the point P(a,f(a)) is the line through P
with the slope Provided that this limit
exists.
64
Example 1 Find an equation of the tangent line to
the parabola yx2 at the point (2,4).
Use Point Slope y y1 m (x x1)
y 4 4(x 2)
y 4 4x 8
4
4
y 4x 4
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Provided that this limit exists.
For many purposes it is desirable to rewrite this
expression in an alternative form by letting
h x a Then x a h
66
y y1 m (x x1)
y 1 -1/3(x 3)
y 1 -1/3x 1
1
1
y -1/3x 2
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Example 3 Find an equation of the tangent line to
the parabola y x2 at the point (3,9).
y y1 m (x x1)
y 9 6(x 3)
y 9 6x 18
9
9
y 6x 9
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Example 4 Find an equation of the tangent line to
the parabola y x24 at the point (1,-3).
y y1 m (x x1)
y -3 2(x 1)
y 3 2x 2
-3
-3
y 2x 5
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2.7 Homework
Pg. 156 5a, 5b, 6a, 6b, 7 10, 11a, 12a, 13b,
and 14b
70
2.8 Derivatives
State Standard 4.0 Students demonstrate an
understanding of the derivative of a function as
the slope of the tangent line to the graph of the
function. Objective To be able to find the
derivative of a function.
71
2.8 Derivatives
is called the derivative of at .
The derivative of f with respect to a is
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the derivative of f with respect to x
f prime x
or
y prime
the derivative of y with respect to x
or
dee why dee ecks
the derivative of f with respect to x
or
dee eff dee ecks
the derivative of f of x
dee dee ecks of eff of ecks
or
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Example 1 Find the derivative of the function
f(x) x2 8x 9 at the number a.
74
Example 2 Find the derivative of the function
(x h 9)
(x 9)
(x 9)
(x h 9)
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2.8 Homework
Pg. 163 13 17