intermediate value theorem version 2
DESCRIPTION
Intermediate Value Theorem Version 2TRANSCRIPT
The Intermediate Value Theorem
Andrea Cohen
Math 101 Pomona College
December 3, 2008
What is the IVT?
Suppose that f: [a,b] → ℝ is continuous. Then…
What is the IVT?
Suppose that f: [a,b] → ℝ is continuous. Then…
if k is any value between f(a) and f(b)...
What is the IVT?
Suppose that f: [a,b] → ℝ is continuous. Then…
…then there exists some c in the
open interval (a, b) such that f(c) = k.
if k is any value between f(a) and f(b)...
Example of the IVT
x
f(x)
a c b
f(a)
f(b)
Example of the IVT
x
f(x)
a c b
f(a)
k
f(b)
Example of the IVT
x
f(x)
a c b
f(a)
k
f(b)
Proof of the IVT
{ }{ } .sup],[
.)(:],[
).()(
).()(
(S)cbacSSa
kxfbaxS
bfkaf
bfafk
=∈∃⇒∅≠⇒∈≤∈=
<<ℜ∈
that such Then
Let
that generality of loss withoutAssume
and between be Let
From Lay, 2005.
Proof of the IVT
{ }{ } .sup],[
.)(:],[
).()(
).()(
(S)cbacSSa
kxfbaxS
bfkaf
bfafk
=∈∃⇒∅≠⇒∈≤∈=
<<ℜ∈
that such Then
Let
that generality of loss withoutAssume
and between be Let
f(c) = ?
From Lay, 2005.
Proof of the IVT
{ }{ } .sup],[
.)(:],[
).()(
).()(
(S)cbacSSa
kxfbaxS
bfkaf
bfafk
=∈∃⇒∅≠⇒∈≤∈=
<<ℜ∈
that such Then
Let
that generality of loss withoutAssume
and between be Let
One of f(c) < k, f(c) > k, and f(c) = k must be true.
From Lay, 2005.
Proof of the IVT
{ }{ } .sup],[
.)(:],[
).()(
).()(
(S)cbacSSa
kxfbaxS
bfkaf
bfafk
=∈∃⇒∅≠⇒∈≤∈=
<<ℜ∈
that such Then
Let
that generality of loss withoutAssume
and between be Let
.!
.)(
.
).()(
.)(],[
.)(
ScSppc
SkpfUp
bpcUp
bfkcfbc
kxfbaUxcU
kcf
of bound upper an be cannot then , and If
p so , since However,
that such So
because know We
that such of nbhd a , Then
Suppose
∈<⇒⇐∈<∈
<<∈∃<<≠
<⇒∩∈∃<
From Lay, 2005.
Proof of the IVT
{ }{ } .sup],[
.)(:],[
).()(
).()(
(S)cbacSSa
kxfbaxS
bfkaf
bfafk
=∈∃⇒∅≠⇒∈≤∈=
<<ℜ∈
that such Then
Let
that generality of loss withoutAssume
and between be Let
.
!
.
.],[)(
.
).()(
.)(],[
.)(
Sc
Spcp
Sp
ScpkpfUp
cpaUp
cfkafac
kxfbaUxcU
kcf
of bound upper the be cannot then
, of bound upper an is and If
of bound upper an is implies This
in are in points no so , , Since
that such So
because know We
that such of nbhd a , Then
supposeNow
least
<⇒⇐
>∈<<∈∃
<<≠>⇒∩∈∃
>
From Lay, 2005.
More Examples of the IVT• Interval of ℝ:
How much catalyst should I add to get a 50% reaction?
f(x) = percent reaction with x grams of catalyst
0
50
100
0 0.5 1 1.5 2 2.5 3 3.5 4grams of catalyst
per
cen
t re
acti
on
More Examples of the IVT• In ℝn:
How loud is it 10 feet in front of the speakers?
f(x,y,z) = decibel level at position (x,y,z)
f(x,y,z): ℝ3 → ℝ(x,y,z)
More Examples of the IVT
• Function of functions:How much taller than my brother am I?
.)(
)(
)()()(
xxB
xxA
x
xBxAxf
time at height sB'
, time at height s A'
years,3 age sB' age s A' where
==
+≈=−=
Bolzano’s Theorem
Proof: Let k = 0. Since f(a) and f(b) have opposite signs, either f(a) < 0 < f(b) or f(b) < 0 < f(a). Use the IVT.
.0)(
),(
.)()(
],[
=cf
bac
bfaf
baf
that such
interval open the in one least at is there Then
signs opposite have and that assume and
interval closed a of point each at continuous be Let
From Apostol, 1967.
Bolzano’s Theorem
From Apostol, 1967 and Morscher, 2007.
• Proven by Bernard Bolzano in the early 1800s.
• First published in 1817 as:Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege.
In English: Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation.
• First analytical proof of IVT– For any continuous f(x), consider g(x) = f(x) - k
Balancing a wobbly table
Requirements:
• Rectangular table• Table legs are same length
From Baritompa et al., 2005 and Devlin, 2007.
Balancing a wobbly table
From Baritompa et al., 2005 and Devlin, 2007.
Requirements:
• Rectangular table• Table legs are same length
Balancing a wobbly table
From Baritompa et al., 2005 and Devlin, 2007.
Requirements:
• Rectangular table• Table legs are same length
Baritompa et al. (2005) showed this works for…• a “mathematical table” [2D rectangle] as long as
ground is continuous • a real[ish] table if the slope is less than 35° and
leg length ≥ ½ diagonal
Balancing a wobbly table
Sources• T. M. Apostol. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with
an Introduction to Linear Algebra. Waltham, MA: Blaisdell, p. 144, 1967.
• S.R. Lay. Analysis with an Introduction to Proof. Upper Saddle River, NJ: Pearson Prentice Hall. pp. 210-211, 2005.
• E. Morscher. “Bernard Bolzano.” Stanford Encyclopedia of Philosophy. Nov 2007. Accessed Dec 2, 2008. Accessed Nov 13, 2008. <<http://plato.stanford.edu/entries/bolzano/>>
• B. Baritompa, R. Lowen, B. Polster, and M. Ross. “Mathematical Table Turning Revisited.” Nov 2005. <<arXiv:math/0511490v1>>
• K. Devlin. “How to stabilize a wobbly table.” Devlin's Angle, MAA Online. Feb 2007. Accessed Nov 13, 2008. <<http://www.maa.org/devlin/devlin_02_07.html>>