Download - Lesson 5: Continuity
. . . . . .
Section1.5Continuity
V63.0121.006/016, CalculusI
February2, 2010
Announcements
I OfficeHours: M,W 1:30–2:30, R 9–10(CIWW 726)I WrittenAssignment#2dueThursdayI FirstQuiz: FridayFebruary12inrecitation(§§1.1–1.4)
. . . . . .
Hatsumon
Herearesomediscussionquestionstostart.
TrueorFalseAtsomepointinyourlifeyouwereexactlythreefeettall.
TrueorFalseAtsomepointinyourlifeyourheight(ininches)wasequaltoyourweight(inpounds).
TrueorFalseRightnowthereareapairofpointsonoppositesidesoftheworldmeasuringtheexactsametemperature.
. . . . . .
Hatsumon
Herearesomediscussionquestionstostart.
TrueorFalseAtsomepointinyourlifeyouwereexactlythreefeettall.
TrueorFalseAtsomepointinyourlifeyourheight(ininches)wasequaltoyourweight(inpounds).
TrueorFalseRightnowthereareapairofpointsonoppositesidesoftheworldmeasuringtheexactsametemperature.
. . . . . .
Hatsumon
Herearesomediscussionquestionstostart.
TrueorFalseAtsomepointinyourlifeyouwereexactlythreefeettall.
TrueorFalseAtsomepointinyourlifeyourheight(ininches)wasequaltoyourweight(inpounds).
TrueorFalseRightnowthereareapairofpointsonoppositesidesoftheworldmeasuringtheexactsametemperature.
. . . . . .
Outline
Continuity
TheIntermediateValueTheorem
BacktotheQuestions
. . . . . .
Recall: DirectSubstitutionProperty
Theorem(TheDirectSubstitutionProperty)If f isapolynomialorarationalfunctionand a isinthedomainoff, then
limx→a
f(x) = f(a)
Thispropertyissousefulit’sworthnaming.
. . . . . .
DefinitionofContinuity
Definition
I Let f beafunctiondefinednear a. Wesaythat f is continuousat aif
limx→a
f(x) = f(a).
I A function f iscontinuous ifitiscontinuousateverypointinitsdomain.
. .x
.y
.
.a
.f(a)
. . . . . .
DefinitionofContinuity
Definition
I Let f beafunctiondefinednear a. Wesaythat f is continuousat aif
limx→a
f(x) = f(a).
I A function f iscontinuous ifitiscontinuousateverypointinitsdomain.
. .x
.y
.
.a
.f(a)
. . . . . .
Scholium
DefinitionLet f beafunctiondefinednear a. Wesaythat f is continuousata if
limx→a
f(x) = f(a).
Therearethreeimportantpartstothisdefinition.I Thefunctionhastohavealimitat a,I thefunctionhastohaveavalueat a,I andthesevalueshavetoagree.
. . . . . .
FreeTheorems
Theorem
(a) Anypolynomialiscontinuouseverywhere; thatis, itiscontinuouson R = (−∞,∞).
(b) Anyrationalfunctioniscontinuouswhereveritisdefined;thatis, itiscontinuousonitsdomain.
. . . . . .
Showingafunctioniscontinuous
ExampleLet f(x) =
√4x+ 1. Showthat f iscontinuousat 2.
SolutionWewanttoshowthat lim
x→2f(x) = f(2). Wehave
limx→a
f(x) = limx→2
√4x+ 1 =
√limx→2
(4x+ 1) =√9 = 3 = f(2).
Eachstepcomesfromthelimitlaws.
QuestionAtwhichotherpointsis f continuous?
AnswerThefunction f iscontinuouson (−1/4,∞). Itis rightcontinuousat −1/4 since lim
x→−1/4+f(x) = f(−1/4).
. . . . . .
Showingafunctioniscontinuous
ExampleLet f(x) =
√4x+ 1. Showthat f iscontinuousat 2.
SolutionWewanttoshowthat lim
x→2f(x) = f(2). Wehave
limx→a
f(x) = limx→2
√4x+ 1 =
√limx→2
(4x+ 1) =√9 = 3 = f(2).
Eachstepcomesfromthelimitlaws.
QuestionAtwhichotherpointsis f continuous?
AnswerThefunction f iscontinuouson (−1/4,∞). Itis rightcontinuousat −1/4 since lim
x→−1/4+f(x) = f(−1/4).
. . . . . .
Showingafunctioniscontinuous
ExampleLet f(x) =
√4x+ 1. Showthat f iscontinuousat 2.
SolutionWewanttoshowthat lim
x→2f(x) = f(2). Wehave
limx→a
f(x) = limx→2
√4x+ 1 =
√limx→2
(4x+ 1) =√9 = 3 = f(2).
Eachstepcomesfromthelimitlaws.
QuestionAtwhichotherpointsis f continuous?
AnswerThefunction f iscontinuouson (−1/4,∞). Itis rightcontinuousat −1/4 since lim
x→−1/4+f(x) = f(−1/4).
. . . . . .
Atwhichotherpoints?
Forreference: f(x) =√4x+ 1
I If a > −1/4, then limx→a
(4x+ 1) = 4a+ 1 > 0, so
limx→a
f(x) = limx→a
√4x+ 1 =
√limx→a
(4x+ 1) =√4a+ 1 = f(a)
and f iscontinuousat a.
I If a = −1/4, then 4x+ 1 < 0 totheleftof a, whichmeans√4x+ 1 isundefined. Still,
limx→a+
f(x) = limx→a+
√4x+ 1 =
√limx→a+
(4x+ 1) =√0 = 0 = f(a)
so f iscontinuousontherightat a = −1/4.
. . . . . .
Atwhichotherpoints?
Forreference: f(x) =√4x+ 1
I If a > −1/4, then limx→a
(4x+ 1) = 4a+ 1 > 0, so
limx→a
f(x) = limx→a
√4x+ 1 =
√limx→a
(4x+ 1) =√4a+ 1 = f(a)
and f iscontinuousat a.I If a = −1/4, then 4x+ 1 < 0 totheleftof a, whichmeans√
4x+ 1 isundefined. Still,
limx→a+
f(x) = limx→a+
√4x+ 1 =
√limx→a+
(4x+ 1) =√0 = 0 = f(a)
so f iscontinuousontherightat a = −1/4.
. . . . . .
Showingafunctioniscontinuous
ExampleLet f(x) =
√4x+ 1. Showthat f iscontinuousat 2.
SolutionWewanttoshowthat lim
x→2f(x) = f(2). Wehave
limx→a
f(x) = limx→2
√4x+ 1 =
√limx→2
(4x+ 1) =√9 = 3 = f(2).
Eachstepcomesfromthelimitlaws.
QuestionAtwhichotherpointsis f continuous?
AnswerThefunction f iscontinuouson (−1/4,∞).
Itis rightcontinuousat −1/4 since lim
x→−1/4+f(x) = f(−1/4).
. . . . . .
Showingafunctioniscontinuous
ExampleLet f(x) =
√4x+ 1. Showthat f iscontinuousat 2.
SolutionWewanttoshowthat lim
x→2f(x) = f(2). Wehave
limx→a
f(x) = limx→2
√4x+ 1 =
√limx→2
(4x+ 1) =√9 = 3 = f(2).
Eachstepcomesfromthelimitlaws.
QuestionAtwhichotherpointsis f continuous?
AnswerThefunction f iscontinuouson (−1/4,∞). Itis rightcontinuousat −1/4 since lim
x→−1/4+f(x) = f(−1/4).
. . . . . .
TheLimitLawsgiveContinuityLaws
TheoremIf f(x) and g(x) arecontinuousat a and c isaconstant, thenthefollowingfunctionsarealsocontinuousat a:
I (f+ g)(x)I (f− g)(x)I (cf)(x)I (fg)(x)
Ifg(x) (if g(a) ̸= 0)
. . . . . .
Whyasumofcontinuousfunctionsiscontinuous
Wewanttoshowthat
limx→a
(f+ g)(x) = (f+ g)(a).
Wejustfollowournose:
limx→a
(f+ g)(x) = limx→a
[f(x) + g(x)] (defof f+ g)
= limx→a
f(x) + limx→a
g(x) (iftheselimitsexist)
= f(a) + g(a) (theydo; f and g arects.)
= (f+ g)(a) (defof f+ g again)
. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan .sec
.cot .csc
. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan .sec
.cot .csc
. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan
.sec
.cot .csc
. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan .sec
.cot .csc
. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan .sec
.cot
.csc
. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan .sec
.cot .csc
. . . . . .
ExponentialandLogarithmicfunctionsarecontinuous
Foranybase a > 1,
I thefunction x 7→ ax iscontinuouson R
I thefunction loga iscontinuousonitsdomain: (0,∞)
I Inparticular ex andln = loge arecontinuousontheirdomains
.
.ax
.loga x
. . . . . .
ExponentialandLogarithmicfunctionsarecontinuous
Foranybase a > 1,
I thefunction x 7→ ax iscontinuouson R
I thefunction loga iscontinuousonitsdomain: (0,∞)
I Inparticular ex andln = loge arecontinuousontheirdomains
.
.ax
.loga x
. . . . . .
ExponentialandLogarithmicfunctionsarecontinuous
Foranybase a > 1,
I thefunction x 7→ ax iscontinuouson R
I thefunction loga iscontinuousonitsdomain: (0,∞)
I Inparticular ex andln = loge arecontinuousontheirdomains
.
.ax
.loga x
. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1
.
.
.cos−1.sec−1
.
..csc−1
.
.
.tan−1
.cot−1
. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1.
.
.cos−1
.sec−1
.
..csc−1
.
.
.tan−1
.cot−1
. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1.
.
.cos−1.sec−1
.
.
.csc−1
.
.
.tan−1
.cot−1
. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1.
.
.cos−1.sec−1
.
..csc−1
.
.
.tan−1
.cot−1
. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1.
.
.cos−1.sec−1
.
..csc−1
.
.
.tan−1
.cot−1
. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1.
.
.cos−1.sec−1
.
..csc−1
.
.
.tan−1
.cot−1
. . . . . .
Whatcouldgowrong?
Inwhatwayscouldafunction f failtobecontinuousatapoint a?Lookagainatthedefinition:
limx→a
f(x) = f(a)
. . . . . .
ContinuityFAIL
:Thelimitdoesnotexist
ExampleLet
f(x) =
{x2 if 0 ≤ x ≤ 1
2x if 1 < x ≤ 2
Atwhichpointsis f continuous?
SolutionAtanypoint a in [0, 2] besides 1, lim
x→af(x) = f(a) because f is
representedbyapolynomialnear a, andpolynomialshavethedirectsubstitutionproperty. However,
limx→1−
f(x) = limx→1−
x2 = 12 = 1
limx→1+
f(x) = limx→1+
2x = 2(1) = 2
So f hasnolimitat 1. Therefore f isnotcontinuousat 1.
. . . . . .
ContinuityFAIL:Thelimitdoesnotexist
ExampleLet
f(x) =
{x2 if 0 ≤ x ≤ 1
2x if 1 < x ≤ 2
Atwhichpointsis f continuous?
SolutionAtanypoint a in [0, 2] besides 1, lim
x→af(x) = f(a) because f is
representedbyapolynomialnear a, andpolynomialshavethedirectsubstitutionproperty. However,
limx→1−
f(x) = limx→1−
x2 = 12 = 1
limx→1+
f(x) = limx→1+
2x = 2(1) = 2
So f hasnolimitat 1. Therefore f isnotcontinuousat 1.
. . . . . .
GraphicalIllustrationofPitfall#1
. .x
.y
..−1
..1
..2
..−1
..1
..2
..3
..4
.
.
.
.
. . . . . .
ContinuityFAIL
:Thefunctionhasnovalue
ExampleLet
f(x) =x2 + 2x+ 1
x+ 1
Atwhichpointsis f continuous?
SolutionBecause f isrational, itiscontinuousonitswholedomain. Notethat −1 isnotinthedomainof f, so f isnotcontinuousthere.
. . . . . .
ContinuityFAIL:Thefunctionhasnovalue
ExampleLet
f(x) =x2 + 2x+ 1
x+ 1
Atwhichpointsis f continuous?
SolutionBecause f isrational, itiscontinuousonitswholedomain. Notethat −1 isnotinthedomainof f, so f isnotcontinuousthere.
. . . . . .
GraphicalIllustrationofPitfall#2
. .x
.y
...−1
. .1
f cannotbecontinuouswhereithasnovalue.
. . . . . .
ContinuityFAIL
:functionvalue ̸= limit
ExampleLet
f(x) =
{7 if x ̸= 1
π if x = 1
Atwhichpointsis f continuous?
Solutionf isnotcontinuousat 1 because f(1) = π but lim
x→1f(x) = 7.
. . . . . .
ContinuityFAIL:functionvalue ̸= limit
ExampleLet
f(x) =
{7 if x ̸= 1
π if x = 1
Atwhichpointsis f continuous?
Solutionf isnotcontinuousat 1 because f(1) = π but lim
x→1f(x) = 7.
. . . . . .
GraphicalIllustrationofPitfall#3
. .x
.y
..π
..7
..1
.
.
. . . . . .
Specialtypesofdiscontinuites
removablediscontinuity Thelimit limx→a
f(x) exists, but f isnot
definedat a oritsvalueat a isnotequaltothelimitat a.
Byre-defining f(a) = limx→a
f(x), f canbemade
continuousat a
jumpdiscontinuity Thelimits limx→a−
f(x) and limx→a+
f(x) exist, but
aredifferent.
Thefunctioncannotbemadecontinuousbychangingasinglevalue.
. . . . . .
Graphicalrepresentationsofdiscontinuities
. .x
.y
..π
..7
..1
.
.
..Presto! continuous!
removable
. .x
.y
..1
..1
..2
.
.
.
.
. .continuous?.
. .continuous?
. .continuous?
jump
. . . . . .
Graphicalrepresentationsofdiscontinuities
. .x
.y
..π
..7
..1
.
.
..Presto! continuous!
removable
. .x
.y
..1
..1
..2
.
.
.
.
. .continuous?.
. .continuous?
. .continuous?
jump
. . . . . .
Graphicalrepresentationsofdiscontinuities
. .x
.y
..π
..7
..1
.
.
..Presto! continuous!
removable
. .x
.y
..1
..1
..2
.
.
.
.
. .continuous?
.
. .continuous?
. .continuous?
jump
. . . . . .
Graphicalrepresentationsofdiscontinuities
. .x
.y
..π
..7
..1
.
.
..Presto! continuous!
removable
. .x
.y
..1
..1
..2
.
.
.
.
. .continuous?
.
. .continuous?
. .continuous?
jump
. . . . . .
Graphicalrepresentationsofdiscontinuities
. .x
.y
..π
..7
..1
.
.
..Presto! continuous!
removable
. .x
.y
..1
..1
..2
.
.
.
.
. .continuous?
.
. .continuous?
. .continuous?
jump
. . . . . .
Specialtypesofdiscontinuites
removablediscontinuity Thelimit limx→a
f(x) exists, but f isnot
definedat a oritsvalueat a isnotequaltothelimitat a. Byre-defining f(a) = lim
x→af(x), f canbemade
continuousat a
jumpdiscontinuity Thelimits limx→a−
f(x) and limx→a+
f(x) exist, but
aredifferent.
Thefunctioncannotbemadecontinuousbychangingasinglevalue.
. . . . . .
Specialtypesofdiscontinuites
removablediscontinuity Thelimit limx→a
f(x) exists, but f isnot
definedat a oritsvalueat a isnotequaltothelimitat a. Byre-defining f(a) = lim
x→af(x), f canbemade
continuousat a
jumpdiscontinuity Thelimits limx→a−
f(x) and limx→a+
f(x) exist, but
aredifferent. Thefunctioncannotbemadecontinuousbychangingasinglevalue.
. . . . . .
Thegreatestintegerfunction
[[x]] isthegreatestinteger ≤ x.
x [[x]]0 01 1
1.5 11.9 12.1 2
−0.5 −1−0.9 −1−1.1 −2
. .x
.y
..−2
..−2
..−1
..−1
..1
..1
..2
..2
..3
..3
. .
. .
. .
. .
. ..y = [[x]]
Thisfunctionhasajumpdiscontinuityateachinteger.
. . . . . .
Thegreatestintegerfunction
[[x]] isthegreatestinteger ≤ x.
x [[x]]0 01 1
1.5 11.9 12.1 2
−0.5 −1−0.9 −1−1.1 −2
. .x
.y
..−2
..−2
..−1
..−1
..1
..1
..2
..2
..3
..3
. .
. .
. .
. .
. ..y = [[x]]
Thisfunctionhasajumpdiscontinuityateachinteger.
. . . . . .
Outline
Continuity
TheIntermediateValueTheorem
BacktotheQuestions
. . . . . .
A BigTimeTheorem
Theorem(TheIntermediateValueTheorem)Supposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. . . . . .
IllustratingtheIVT
Supposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
..
.c1
.
.c2
.
.c3
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b]
andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
..
.c1
.
.c2
.
.c3
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b]
andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
..
.c1
.
.c2
.
.c3
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b).
Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
..
.c1
.
.c2
.
.c3
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
.
.
.c1
.
.c2
.
.c3
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
..
.c1
.
.c2
.
.c3
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
.
.
.c1
.
.c2
.
.c3
. . . . . .
WhattheIVT doesnotsay
TheIntermediateValueTheoremisan“existence”theorem.I Itdoesnotsayhowmanysuch c exist.I Italsodoesnotsayhowtofind c.
Still, itcanbeusediniterationorinconjunctionwithothertheoremstoanswerthesequestions.
. . . . . .
UsingtheIVT
ExampleSupposeweareunawareofthesquarerootfunctionandthatit’scontinuous. Provethatthesquarerootoftwoexists.
Proof.Let f(x) = x2, acontinuousfunctionon [1, 2]. Note f(1) = 1 andf(2) = 4. Since 2 isbetween 1 and 4, thereexistsapoint c in(1, 2) suchthat
f(c) = c2 = 2.
Infact, wecan“narrowin”onthesquarerootof 2 by themethodofbisections.
. . . . . .
UsingtheIVT
ExampleSupposeweareunawareofthesquarerootfunctionandthatit’scontinuous. Provethatthesquarerootoftwoexists.
Proof.Let f(x) = x2, acontinuousfunctionon [1, 2].
Note f(1) = 1 andf(2) = 4. Since 2 isbetween 1 and 4, thereexistsapoint c in(1, 2) suchthat
f(c) = c2 = 2.
Infact, wecan“narrowin”onthesquarerootof 2 by themethodofbisections.
. . . . . .
UsingtheIVT
ExampleSupposeweareunawareofthesquarerootfunctionandthatit’scontinuous. Provethatthesquarerootoftwoexists.
Proof.Let f(x) = x2, acontinuousfunctionon [1, 2]. Note f(1) = 1 andf(2) = 4. Since 2 isbetween 1 and 4, thereexistsapoint c in(1, 2) suchthat
f(c) = c2 = 2.
Infact, wecan“narrowin”onthesquarerootof 2 by themethodofbisections.
. . . . . .
UsingtheIVT
ExampleSupposeweareunawareofthesquarerootfunctionandthatit’scontinuous. Provethatthesquarerootoftwoexists.
Proof.Let f(x) = x2, acontinuousfunctionon [1, 2]. Note f(1) = 1 andf(2) = 4. Since 2 isbetween 1 and 4, thereexistsapoint c in(1, 2) suchthat
f(c) = c2 = 2.
Infact, wecan“narrowin”onthesquarerootof 2 by themethodofbisections.
. . . . . .
Finding√2 bybisections
.
.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 bybisections
.
.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 bybisections
.
.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 bybisections
.
.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 bybisections
.
.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 bybisections
.
.x .f(x) = x2
..1 .1
..2 .4
..1.5 .2.25
..1.25 .1.5625
..1.375 .1.890625
..1.4375 .2.06640625
. . . . . .
UsingtheIVT
ExampleLet f(x) = x3 − x− 1. Showthatthereisazerofor f.
Solutionf(1) = −1 and f(2) = 5. Sothereisazerobetween 1 and 2.
(Morecarefulanalysisyields 1.32472.)
. . . . . .
UsingtheIVT
ExampleLet f(x) = x3 − x− 1. Showthatthereisazerofor f.
Solutionf(1) = −1 and f(2) = 5. Sothereisazerobetween 1 and 2.(Morecarefulanalysisyields 1.32472.)
. . . . . .
Outline
Continuity
TheIntermediateValueTheorem
BacktotheQuestions
. . . . . .
BacktotheQuestions
TrueorFalseAtonepointinyourlifeyouwereexactlythreefeettall.
TrueorFalseAtonepointinyourlifeyourheightininchesequaledyourweightinpounds.
TrueorFalseRightnowtherearetwopointsonoppositesidesoftheEarthwithexactlythesametemperature.
. . . . . .
Question1: True!
I Let h(t) beheight, whichvariescontinuouslyovertime.I Then h(birth) < 3 ft and h(now) > 3 ft.I SobytheIVT thereisapoint c in (birth, now) where
h(c) = 3.
. . . . . .
BacktotheQuestions
TrueorFalseAtonepointinyourlifeyouwereexactlythreefeettall.
TrueorFalseAtonepointinyourlifeyourheightininchesequaledyourweightinpounds.
TrueorFalseRightnowtherearetwopointsonoppositesidesoftheEarthwithexactlythesametemperature.
. . . . . .
Question2: True!
I Let h(t) beheightininchesand w(t) beweightinpounds,bothvaryingcontinuouslyovertime.
I Let f(t) = h(t)−w(t).I Formostofus(callyourmom), f(birth) > 0 and f(now) < 0.I SobytheIVT thereisapoint c in (birth, now) where
f(c) = 0.I Inotherwords,
h(c)−w(c) = 0 ⇐⇒ h(c) = w(c).
. . . . . .
BacktotheQuestions
TrueorFalseAtonepointinyourlifeyouwereexactlythreefeettall.
TrueorFalseAtonepointinyourlifeyourheightininchesequaledyourweightinpounds.
TrueorFalseRightnowtherearetwopointsonoppositesidesoftheEarthwithexactlythesametemperature.
. . . . . .
Question3
I Let T(θ) bethetemperatureatthepointontheequatoratlongitude θ.
I Howcanyouexpressthestatementthatthetemperatureonoppositesidesisthesame?
I Howcanyouensurethisistrue?
. . . . . .
Question3: True!
I Let f(θ) = T(θ)− T(θ + 180◦)I Then
f(0) = T(0)− T(180)
whilef(180) = T(180)− T(360) = −f(0)
I Sosomewherebetween 0 and 180 thereisapoint θ wheref(θ) = 0!
. . . . . .
Whathavewelearnedtoday?
I Definition: afunctioniscontinuousatapointifthelimitofthefunctionatthatpointagreeswiththevalueofthefunctionatthatpoint.
I Weoftenmakeafundamentalassumptionthatfunctionswemeetinnaturearecontinuous.
I TheIntermediateValueTheoremisabasicpropertyofrealnumbersthatweneedandusealot.