Transcript 1 4 Continuity

1.4: Continuity
Objectives:
Assignment:
1. To determine one-sided
limits
• P. 79: 7-23 odd
2. To apply the conditions
of continuity
• P. 78-81: 1, 3, 5, 25-28,
29, 31, 33-45 eoo, 69-72,
87, 91
3. To use the Intermediate
Value Theorem
• P. 80-81: 75, 77, 83-86
Warm-Up
The points (2, −3)
and (3, 5) are on
the graph of a
polynomial
function 𝑓(𝑥).
What conclusion
can you draw
about the graph
of 𝑓(𝑥)?
There must
be a zero
between 2
and 3 since
the graph
must cross
the 𝑥-axis to
connect
these points.
This is because
the function is
continuous.
Objective 1
You will be able to determine
one-sided limits
Example 1a
Find the limit.
𝑥
lim
𝑥→0 𝑥
2
As 𝑥 → 0 from
the right,
𝑓(𝑥) → 1
1
0.5
1
1
2
0.5
As 𝑥 → 0 from
the left,
𝑓 𝑥 → −1
1
Since 𝑓(𝑥) does not
approach a unique value
from the left and right of 0,
𝑥
lim 𝑥 does not exist due to
𝑥→0
behavior that differs from
the left and the right.
One-Sided Limits
Until now, when we’ve found a limit, we looked on
both sides of 𝑐. Now we’ll look at just the left
side or the right side. This is a one-sided limit.
Limit from the Left
As 𝑥 approaches 𝑐 from values
less than 𝑐 (from the left of 𝑐),
𝑓(𝑥) approaches 𝐿.
lim− 𝑓(𝑥) = 𝐿
𝑥→𝑐
Limit from the Right
As 𝑥 approaches 𝑐 from values
greater than 𝑐 (from the right of 𝑐),
𝑓(𝑥) approaches 𝐿.
lim+ 𝑓(𝑥) = 𝐿
𝑥→𝑐
Example 1b
Find each limit.
1.
𝑥
lim−
𝑥→0 𝑥
1
0.5
2.
𝑥
lim+
𝑥→0 𝑥
2
1
1
0.5
1
2
Example 2
Find the limit of 𝑓(𝑥) = 4 − 𝑥 2 as 𝑥
approaches −2 from the right.
Example 3a
Find each limit.
1.
lim− 𝑥
𝑥→0
2.
lim+ 𝑥
𝑥→0
Existence of a Limit
Let 𝑓 be a function and let 𝑐 and 𝐿 be
real numbers. The limit of 𝑓(𝑥) as 𝑥
approaches 𝑐 is 𝐿 if and only if
lim− 𝑓(𝑥) = 𝐿
𝑥→𝑐
and
lim+ 𝑓(𝑥) = 𝐿
𝑥→𝑐
Example 3b
Find the limit.
lim 𝑥
𝑥→0
Example 4
On the Kelvin scale, absolute zero is the
temperature 0 K. Although temperatures of
approximately 0.0001 K have been produced in
laboratories, absolute zero has never been
attained. In fact, evidence suggests that absolute
zero cannot be attained. Physicist Jacque Charles
discovered that the volume of gas at a constant
pressure increases linearly with the temperature of
the gas. Use the values in the table to determine
the “lower limit” of absolute value in °C.
Example 4
Physicist Jacque Charles discovered that the
volume of gas at a constant pressure increases
linearly with the temperature of the gas. Use the
values in the table to determine the “lower limit” of
absolute value in °C.
T (°C)
V (L)
-40
-20
0
20
40
60
80
19.1482 20.7908 22.4334 24.0760 25.7186 27.3612 29.0038
You will be able to apply
the conditions of continuity
A Beautiful Function
Use Ross’s Beautiful function to investigate and
discover the three conditions for continuity.
8
6
4
2
10
5
5
2
4
10
Conditions for Continuity
Continuity at a Point
A function 𝑓 is continuous
at 𝒄 if all three conditions
are met:
1. 𝑓(𝑐) is defined.
2. lim 𝑓(𝑥) exists.
𝑥→𝑐
3. lim 𝑓(𝑥) = 𝑓(𝑐).
𝑥→𝑐
Continuity on an Open
Interval
A function is continuous on
an open interval (𝑎, 𝑏) if it
is continuous at each point
in the interval.
A function that is continuous
on the entire real line is
continuous everywhere.
Example 5
Discuss the continuity of 𝑓 𝑥 =
𝑥 = 0.
sin 𝑥
𝑥
at
Types of Discontinuities
If a function 𝑓 is defined on an open interval
(except possibly at 𝑐), and 𝑓 is not continuous at
𝑐, then 𝑓 has a discontinuity at 𝑐.
Nonremovable
Jump
Asymptote
Removable
π
2
2
2
1
1
1
π
π
2
2
π
3
2
2
2
2
1
1
1
2
2
2
Example 6a
Discuss the continuity of each function.
1. 𝑦 =
1
𝑥
2. 𝑦 =
𝑥 2 −1
𝑥−1
Example 6b
Discuss the continuity of each function.
𝑥 + 1, 𝑥 ≤ 0
3. ℎ 𝑥 = 2
𝑥 − 1, 𝑥 > 0
4. 𝑓 𝑥 = sin 𝑥
Example 7
Using the definition of continuity, draw a
function that satisfies each of the following:
1, not 2
2, not 1
not 1, not 2
1 and 2, not 3
1. 𝑓(𝑐) is defined.
2. lim 𝑓(𝑥) exists.
𝑥→𝑐
3. lim 𝑓(𝑥) = 𝑓(𝑐).
𝑥→𝑐
Continuity on a Closed Interval
A function is continuous on the
closed interval [𝑎, 𝑏] if it is continuous
on the open interval (𝑎, 𝑏) and
π
2
lim+ 𝑓 𝑥 = 𝑓(𝑎)
𝑥→𝑎
and
lim− 𝑓 𝑥 = 𝑓(𝑏)
2
2
𝑥→𝑏
π
2
Example 8
Discuss the continuity of 𝑓 𝑥 = 1 − 𝑥 2 .
Properties of Continuity
If 𝑏 is a real number, 𝑓 and 𝑔 are continuous at 𝑥 = 𝑐,
and 𝑓 is continuous at 𝑔(𝑐), then the following are
also continuous at 𝑐.
Scalar Multiple
Quotient
𝑏∙𝑓
Sum or Difference
𝑓±𝑔
Product
𝑓∙𝑔
𝑓
,
𝑔
if 𝑔(𝑐) ≠ 0
Composite Function
𝑓 𝑔 𝑥
This implies that
polynomial,
rational, radical,
and trig
functions are
continuous at
every point in
their domain.
Example 9a
Determine the intervals on which each function is
continuous.
1. 𝑦 = tan 𝑥
Example 9b
Determine the intervals on which each function is
continuous.
1
2.
sin
𝑥
𝑓 𝑥 =
0
𝑥≠0
𝑥=0
1
2.
𝑥 ∙ sin
𝑥
𝑔 𝑥 =
0
𝑥≠0
𝑥=0
Objective 3
You will be
able to use the
Intermediate
Value
Theorem
Example 10
Assume 𝑓(𝑥) is a
continuous function.
What can you
conclude about the
location of one of the
zeros of 𝑓(𝑥)?
𝑥
0
1
2
3
4
5
𝑓(𝑥)
−6
−12
28
150
390
784
There must be a zero here
Intermediate Value Theorem
If 𝑓 is continuous on the
closed interval [𝑎, 𝑏] with
𝑓(𝑎) ≠ 𝑓 𝑏 and 𝑘 is any
number between 𝑓(𝑎)
and 𝑓(𝑏), then there
exists at least one number
𝑐 in [𝑎, 𝑏] such that
𝑓 𝑐 =𝑘
Intermediate Value Theorem
If 𝑓 is continuous on the
closed interval [𝑎, 𝑏] with
𝑓(𝑎) ≠ 𝑓 𝑏 and 𝑘 is any
number between 𝑓(𝑎)
and 𝑓(𝑏), then there
exists at least one number
𝑐 in [𝑎, 𝑏] such that
𝑓 𝑐 =𝑘
For continuous
functions, does
IVT tell us how
to find the
value of 𝑐?
Example 11
On February 14, 2014,
Venom was 39.5”. A
year and a week later,
Venom measured 42.5”.
Was there ever a time
when Rowan, I mean
Venom, was exactly 42”
tall? Explain.
Example 12
Use the Intermediate Value Theorem to
show that the polynomial function
𝑓 𝑥 = 𝑥 3 + 2𝑥 − 1 has a zero in the
interval [0,1].
1.4: Continuity
Objectives:
Assignment:
1. To determine one-sided
limits
• P. 79: 7-23 odd
2. To apply the conditions
of continuity
• P. 78-81: 1, 3, 5, 25-28,
29, 31, 33-45 eoo, 69-72,
87, 91
3. To use the Intermediate
Value Theorem
• P. 80-81: 75, 77, 83-86