objectives: 1.be able to define continuity by determining if a graph is continuous. 2.be able to...
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![Page 1: Objectives: 1.Be able to define continuity by determining if a graph is continuous. 2.Be able to identify and find the different types of discontinuities](https://reader036.vdocument.in/reader036/viewer/2022081603/56649f335503460f94c50152/html5/thumbnails/1.jpg)
Objectives:1. Be able to define continuity by determining if a graph is
continuous.2. Be able to identify and find the different types of
discontinuities that functions may contain.3. Be able to determine if a function is continuous on a
closed interval.4. Be able to determine one-sided limits and continuity on
a closed interval.
Critical Vocabulary:Limit, Continuous, Continuity, Composite Function
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I. Continuity
Continuous: To say that a function f is continuous at x = c there is no interruption in the graph of f at c
This means a graph will contain no _____, _____, or _____
Simple Terms: If you ever have to lift your pencil to sketch a graph, then it is not continuous.
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I. Continuity
What Causes discontinuity?
1. The function is not defined at c.
This is an example of a ____ in the graph at ___
Concept: The function is not defined at c.
__________________
Let’s look at at f(x) = ½x - 2
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I. Continuity
What Causes discontinuity?
2. The limit of f(x) does not exist at x = c
This is an example of a ____ in the graph at ______
Concept: The limit does not exist at x = c
Let’s look at at
37
313
1)(
xx
xxxf
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I. Continuity
What Causes discontinuity?
3. The limit of f(x) exists at x = c but is not equal to f(c).
This is an example of a _____ in the graph
Concept: The behavior (limit) and where its defined (f(c)) are __________________
Let’s look at the first graph again
)()(lim cfxfcx
What is the limit as x approaches -2?
What is f(-2)?
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A function f is continuous at c if the following three conditions are met:
1. ___________________________
2. ___________________________
3. ___________________________
I. Continuity
Continuous: To say that a function f is continuous at x = c there is no interruption in the graph of f at c
This means a graph will contain no HOLES, JUMPS, or GAPS
Simple Terms: If you ever have to lift your pencil to sketch a graph, then it is not continuous.
![Page 7: Objectives: 1.Be able to define continuity by determining if a graph is continuous. 2.Be able to identify and find the different types of discontinuities](https://reader036.vdocument.in/reader036/viewer/2022081603/56649f335503460f94c50152/html5/thumbnails/7.jpg)
Objectives:1. Be able to define continuity by determine if a graph is
continuous.2. Be able to identify and find the different types of
discontinuities that functions may contain.3. Be able to determine if a function is continuous on a
closed interval.4. Be able to determine one-sided limits and continuity on
a closed interval.
Critical Vocabulary:Limit, Continuous, Continuity, Composite Function
![Page 8: Objectives: 1.Be able to define continuity by determining if a graph is continuous. 2.Be able to identify and find the different types of discontinuities](https://reader036.vdocument.in/reader036/viewer/2022081603/56649f335503460f94c50152/html5/thumbnails/8.jpg)
II. Discontinuities
When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable:A discontinuity is removable if you COULD define f(c).
![Page 9: Objectives: 1.Be able to define continuity by determining if a graph is continuous. 2.Be able to identify and find the different types of discontinuities](https://reader036.vdocument.in/reader036/viewer/2022081603/56649f335503460f94c50152/html5/thumbnails/9.jpg)
II. Discontinuities
When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable:A discontinuity is removable if you COULD define f(c).
2. Non-Removable:
A discontinuity is non-removable if you CANNOT define f(c).
![Page 10: Objectives: 1.Be able to define continuity by determining if a graph is continuous. 2.Be able to identify and find the different types of discontinuities](https://reader036.vdocument.in/reader036/viewer/2022081603/56649f335503460f94c50152/html5/thumbnails/10.jpg)
II. Discontinuities
When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable:A discontinuity is removable if you COULD define f(c).
2. Non-Removable:
A discontinuity is non-removable if you CANNOT define f(c).
Example 1:1
1)(
2
x
xxf What is the Domain?
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II. Discontinuities
When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
A discontinuity is removable if you COULD define f(c).
A discontinuity is non-removable if you CANNOT define f(c).
Example 2: 9
3)(
2
x
xxf What is the Domain?
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II. Discontinuities
When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable:A discontinuity is removable if you COULD define f(c).
2. Non-Removable:
A discontinuity is non-removable if you CANNOT define f(c).
Example 3: Discuss the continuity of the composite function f(g(x))
xxf
1)( 1)( xxg
![Page 13: Objectives: 1.Be able to define continuity by determining if a graph is continuous. 2.Be able to identify and find the different types of discontinuities](https://reader036.vdocument.in/reader036/viewer/2022081603/56649f335503460f94c50152/html5/thumbnails/13.jpg)
II. Discontinuities
Example 5: Graph the piecewise function, then determine on which intervals the graph is continuous.
02
01)( 2 xx
xxxf
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III. Closed Intervals
Example 5: Discuss the continuity on the closed interval.
2,14
1)(
2
x
xf
Closed Interval: Focusing on specific portion (domian) of a graph. [a, b]
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1. Page 237 #23-43 odd, 49-55 odd, 61, 63, 77
2. Page 236 #1-17 odd, 79, 88
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Objectives:1. Be able to define continuity by determine if a graph is
continuous.2. Be able to identify and find the different types of
discontinuities that functions may contain.3. Be able to determine if a function is continuous on a
closed interval.4. Be able to determine one-sided limits and continuity on
a closed interval.
Critical Vocabulary:Limit, Continuous, Continuity, Composite Function
![Page 17: Objectives: 1.Be able to define continuity by determining if a graph is continuous. 2.Be able to identify and find the different types of discontinuities](https://reader036.vdocument.in/reader036/viewer/2022081603/56649f335503460f94c50152/html5/thumbnails/17.jpg)
IV. One-Sided Limits
What does a One-Sided look like?
Lxfcx
)(lim c
Lxfcx
)(lim c Approach from the right only
Lxfcx
)(lim c Approach from the left only
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IV. One-Sided Limits
Example 1: Graph then find the limits
24)( xxf
What’s the domain?
x
f(x)
____)(lim2
xfx
____)(lim2
xfx
____)(lim0
xfx
____)(lim2
xfx
____)(lim2
xfx
____)(lim2
xfx
____)(lim2
xfx
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IV. One-Sided Limits
Example 1: Graph then find the limits
14
14)( 2 xxx
xxxf
x
f(x)
____)(lim1
xfx
____)(lim1
xfx
____)(lim1
xfx
Is this graph continuous?
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1. Page 237 #23-43 odd, 49-55 odd, 61, 63, 77
2. Page 236 #1-17 odd, 79, 88