Unit Two Notes—The Derivative
2.1
CONTINUITY
D
A function f is continuous at x = c if the following three conditions are satisfied:
1.
f(c) is defined
2.
)
(
lim
x
f
c
x
→
exists
3.
)
(
)
(
lim
c
f
x
f
c
x
=
→
D
Continuity on a closed interval

A function is continuous on a closed interval [a, b] if
the following conditions are met:
1.
f is continuous on the open interval (a, b)
2.
f is continuous from the right at x = a, and
3.
f is continuous from the left at x = b
D
Types of discontinuities—1) Removable
2) NonRemovable
Ex 1
(removable discontinuity):
(
29
2
2
2



=
x
x
x
x
f
Ex 2
(nonremovable discontinuity):
(
29
2
2
2
+


=
x
x
x
x
f
Ex 3:
Determine if f(x) is continuous on [2, 4]:
(
29
≥
+

<

=
3
,
8
3
,
4
2
x
x
x
x
x
f
Ex 4:
Determine all values x = a where f(x) is discontinuous.
Classify what types of
discontinuities exist, if any.
(
29
(
29 (
29
6
3
1
2
2
+
+

=
x
x
x
x
f
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2.2
Amount of Change, Percentage Change, and Average Rate of Change (Over an Interval)
Describing Change is the underlying theme of the first four units.
Three ways that we may consider
describing numeric change that is important for Calculus can be done in the following ways:
Type of Change
How Change is
Calculuated
Units of
Change
Change
output units
Percentage Change
percentage %
Average Rate of
Change
units of output
(A.R.O.C.)
per
unit input
Ex 1:
The number of Facebook users in the Eastern US grew from 0.9 million in 2007 to 12.0
million in 2010.
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 Spring '12
 MICHAELDERECK
 Topology, Derivative, Continuous function

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