Section 1.5 Continuity V63.0121, Calculus I February 2–3, 2009 Announcements I Quiz 1 is this week: Tuesday (Sections 11 and 12), Thursday (Section 4), Friday (Sections 13 and 14). 15 minutes, covers Sections 1.1–1.2 I Fill your ALEKS pie by February 27, 11:59pm I Congratulations to the Super Bowl XLIII Champion Pittsburgh Steelers!
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Section 1.5Continuity
V63.0121, Calculus I
February 2–3, 2009
Announcements
I Quiz 1 is this week: Tuesday (Sections 11 and 12), Thursday(Section 4), Friday (Sections 13 and 14). 15 minutes, coversSections 1.1–1.2
I Fill your ALEKS pie by February 27, 11:59pm
I Congratulations to the Super Bowl XLIII ChampionPittsburgh Steelers!
Outline
Warmup
Grader Notes
Continuity
The Intermediate Value Theorem
Back to the Questions
Hatsumon
Here are some discussion questions to start.
I Were you ever exactly three feet tall?
I Was your height (in inches) ever equal to your weight (inpounds)?
I Is there a pair of points on opposite sides of the world at thesame temperature at the same time?
Hatsumon
Here are some discussion questions to start.
I Were you ever exactly three feet tall?
I Was your height (in inches) ever equal to your weight (inpounds)?
I Is there a pair of points on opposite sides of the world at thesame temperature at the same time?
Hatsumon
Here are some discussion questions to start.
I Were you ever exactly three feet tall?
I Was your height (in inches) ever equal to your weight (inpounds)?
I Is there a pair of points on opposite sides of the world at thesame temperature at the same time?
Outline
Warmup
Grader Notes
Continuity
The Intermediate Value Theorem
Back to the Questions
But first, a word from our friendly graders
I Please turn in neat problem sets: loose-leaf paper, stapled
I Label homework with Name, Section (10 or 4), Date, ProblemSet number
I Do not turn in scratch work
Please label your graphs
Example: Graph F (x) = |2x + 1|.
incomplete
x
y
(−1/2, 0)
(0, 1)
better
Example 6= explanation
ProblemIf f is even and g is odd, what can you say about fg?
Let f (x) = x2 (even) and g(x) = x3 (odd).Then (fg)(x) = x2 · x3 = x5, and that’s odd.So the product of an even function and an oddfunction is an odd function.
Dangerous!
The trouble with proof by example
ProblemWhich odd numbers are prime?
The numbers 3, 5, and7 are odd, and all prime.So all odd numbers areprime.
Fallacious!
Use the definitions
Let f be even and g be odd. Then
(fg)(−x) = f (−x)g(−x) = f (x)(−g(x))
= −f (x)g(x) = −(fg)(x)
So fg is odd.
Better
Outline
Warmup
Grader Notes
Continuity
The Intermediate Value Theorem
Back to the Questions
Recall: Direct Substitution Property
Theorem (The Direct Substitution Property)
If f is a polynomial or a rational function and a is in the domain off , then
limx→a
f (x) = f (a)
Definition of Continuity
DefinitionLet f be a function defined near a. We say that f is continuous ata if
limx→a
f (x) = f (a).
Free Theorems
Theorem
(a) Any polynomial is continuous everywhere; that is, it iscontinuous on R = (−∞,∞).
(b) Any rational function is continuous wherever it is defined; thatis, it is continuous on its domain.
Showing a function is continuous
Example
Let f (x) =√
4x + 1. Show that f is continuous at 2.
SolutionWe have
limx→a
f (x) = limx→2
√4x + 1
=√
limx→2
(4x + 1)
=√
9 = 3.
Each step comes from the limit laws.
In fact, f is continuous on (−1/4,∞). The function f is rightcontinuous at the point −1/4.
Showing a function is continuous
Example
Let f (x) =√
4x + 1. Show that f is continuous at 2.
SolutionWe have
limx→a
f (x) = limx→2
√4x + 1
=√
limx→2
(4x + 1)
=√
9 = 3.
Each step comes from the limit laws.
In fact, f is continuous on (−1/4,∞). The function f is rightcontinuous at the point −1/4.
Showing a function is continuous
Example
Let f (x) =√
4x + 1. Show that f is continuous at 2.
SolutionWe have
limx→a
f (x) = limx→2
√4x + 1
=√
limx→2
(4x + 1)
=√
9 = 3.
Each step comes from the limit laws.
In fact, f is continuous on (−1/4,∞).
The function f is rightcontinuous at the point −1/4.
Showing a function is continuous
Example
Let f (x) =√
4x + 1. Show that f is continuous at 2.
SolutionWe have
limx→a
f (x) = limx→2
√4x + 1
=√
limx→2
(4x + 1)
=√
9 = 3.
Each step comes from the limit laws.
In fact, f is continuous on (−1/4,∞). The function f is rightcontinuous at the point −1/4.
The Limit Laws give Continuity Laws
TheoremIf f and g are continuous at a and c is a constant, then thefollowing functions are also continuous at a:
I f + g
I f − g
I cf
I fg
If
g(if g(a) 6= 0)
Transcendental functions are continuous, too
TheoremThe following functions are continuous wherever they are defined:
1. sin, cos, tan, cot sec, csc
2. x 7→ ax , loga, ln
3. sin−1, tan−1, sec−1
What could go wrong?
In what ways could a function f fail to be continuous at a point a?Look again at the definition:
limx→a
f (x) = f (a)
Pitfall #1
: The limit does not exist
Example
Let
f (x) =
{x2 if 0 ≤ x ≤ 1
2x if 1 < x ≤ 2
At which points is f continuous?
SolutionAt any point a in [0, 2] besides 1, lim
x→af (x) = f (a) because f is
represented by a polynomial near a, and polynomials have thedirect substitution property. However,
limx→1−
f (x) = limx→1−
x2 = 12 = 1
limx→1+
f (x) = limx→1+
2x = 2(1) = 2
So f has no limit at 1. Therefore f is not continuous at 1.
Pitfall #1: The limit does not exist
Example
Let
f (x) =
{x2 if 0 ≤ x ≤ 1
2x if 1 < x ≤ 2
At which points is f continuous?
SolutionAt any point a in [0, 2] besides 1, lim
x→af (x) = f (a) because f is
represented by a polynomial near a, and polynomials have thedirect substitution property. However,
limx→1−
f (x) = limx→1−
x2 = 12 = 1
limx→1+
f (x) = limx→1+
2x = 2(1) = 2
So f has no limit at 1. Therefore f is not continuous at 1.
Graphical Illustration of Pitfall #1
x
y
−1 1 2
−1
1
2
3
4
Pitfall #2
: The function has no value
Example
Let
f (x) =x2 + 2x + 1
x + 1
At which points is f continuous?
SolutionBecause f is rational, it is continuous on its whole domain. Notethat −1 is not in the domain of f , so f is not continuous there.
Pitfall #2: The function has no value
Example
Let
f (x) =x2 + 2x + 1
x + 1
At which points is f continuous?
SolutionBecause f is rational, it is continuous on its whole domain. Notethat −1 is not in the domain of f , so f is not continuous there.
Graphical Illustration of Pitfall #2
x
y
−1
1
f cannot be continuous where it has no value.
Pitfall #3
: function value 6= limit
Example
Let
f (x) =
{7 if x 6= 1
π if x = 1
At which points is f continuous?
Solutionf is not continuous at 1 because f (1) = π but lim
x→1f (x) = 7.
Pitfall #3: function value 6= limit
Example
Let
f (x) =
{7 if x 6= 1
π if x = 1
At which points is f continuous?
Solutionf is not continuous at 1 because f (1) = π but lim
x→1f (x) = 7.
Graphical Illustration of Pitfall #3
x
y
π
7
1
Special types of discontinuites
removable discontinuity The limit limx→a
f (x) exists, but f is not
defined at a or its value at a is not equal to the limitat a.
jump discontinuity The limits limx→a−
f (x) and limx→a+
f (x) exist, but
are different. f (a) is one of these limits.
Graphical representations of discontinuities
x
y
π
7
1
removable
x
y
−1 1 2
−1
1
2
3
4
jump
The greatest integer function
x
y
−2
−2
−1
−1
1
1
2
2
3
3
The greatest integer function f (x) = [[x ]] has jump discontinuities.
The greatest integer function
x
y
−2
−2
−1
−1
1
1
2
2
3
3
The greatest integer function f (x) = [[x ]] has jump discontinuities.
Outline
Warmup
Grader Notes
Continuity
The Intermediate Value Theorem
Back to the Questions
A Big Time Theorem
Theorem (The Intermediate Value Theorem)
Suppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
cc1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b]
and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
cc1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b]
and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
cc1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b).
Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
cc1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
c
c1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
cc1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
c
c1 c2 c3
Using the IVT
Example
Prove that the square root of two exists.
Proof.Let f (x) = x2, a continuous function on [1, 2]. Note f (1) = 1 andf (2) = 4. Since 2 is between 1 and 4, there exists a point c in(1, 2) such that
f (c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the methodof bisections.
Using the IVT
Example
Prove that the square root of two exists.
Proof.Let f (x) = x2, a continuous function on [1, 2].
Note f (1) = 1 andf (2) = 4. Since 2 is between 1 and 4, there exists a point c in(1, 2) such that
f (c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the methodof bisections.
Using the IVT
Example
Prove that the square root of two exists.
Proof.Let f (x) = x2, a continuous function on [1, 2]. Note f (1) = 1 andf (2) = 4. Since 2 is between 1 and 4, there exists a point c in(1, 2) such that
f (c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the methodof bisections.
Using the IVT
Example
Prove that the square root of two exists.
Proof.Let f (x) = x2, a continuous function on [1, 2]. Note f (1) = 1 andf (2) = 4. Since 2 is between 1 and 4, there exists a point c in(1, 2) such that
f (c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the methodof bisections.
Finding√
2 by bisections
x f (x) = x2
1 1
2 4
1.5 2.25
1.25 1.5625
1.375 1.8906251.4375 2.06640625
Finding√
2 by bisections
x f (x) = x2
1 1
2 4
1.5 2.25
1.25 1.5625
1.375 1.8906251.4375 2.06640625
Finding√
2 by bisections
x f (x) = x2
1 1
2 4
1.5 2.25
1.25 1.5625
1.375 1.8906251.4375 2.06640625
Finding√
2 by bisections
x f (x) = x2
1 1
2 4
1.5 2.25
1.25 1.5625
1.375 1.8906251.4375 2.06640625
Finding√
2 by bisections
x f (x) = x2
1 1
2 4
1.5 2.25
1.25 1.5625
1.375 1.890625
1.4375 2.06640625
Finding√
2 by bisections
x f (x) = x2
1 1
2 4
1.5 2.25
1.25 1.5625
1.375 1.8906251.4375 2.06640625
Using the IVT
Example
Let f (x) = x3 − x − 1. Show that there is a zero for f .
Solutionf (1) = −1 and f (2) = 5. So there is a zero between 1 and 2.
(More careful analysis yields 1.32472.)
Using the IVT
Example
Let f (x) = x3 − x − 1. Show that there is a zero for f .
Solutionf (1) = −1 and f (2) = 5. So there is a zero between 1 and 2.(More careful analysis yields 1.32472.)
Outline
Warmup
Grader Notes
Continuity
The Intermediate Value Theorem
Back to the Questions
Back to the Questions
True or FalseAt one point in your life you were exactly three feet tall.
True or FalseAt one point in your life your height in inches equaled your weightin pounds.
True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.
Question 1: True!
Let h(t) be height, which varies continuously over time. Thenh(birth) < 3 ft and h(now) > 3 ft. So there is a point c in(birth, now) where h(c) = 3.
Back to the Questions
True or FalseAt one point in your life you were exactly three feet tall.
True or FalseAt one point in your life your height in inches equaled your weightin pounds.
True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.
Question 2: True!
Let h(t) be height in inches and w(t) be weight in pounds, bothvarying continuously over time. Let f (t) = h(t)− w(t). For mostof us (call your mom), f (birth) > 0 and f (now) < 0. So there is apoint c in (birth, now) where f (c) = 0. In other words,
h(c)− w(c) = 0 ⇐⇒ h(c) = w(c).
Back to the Questions
True or FalseAt one point in your life you were exactly three feet tall.
True or FalseAt one point in your life your height in inches equaled your weightin pounds.
True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.
Question 3
I Let T (θ) be the temperature at the point on the equator atlongitude θ.
I How can you express the statement that the temperature onopposite sides is the same?