let f be continuous differentiable
TRANSCRIPT
4.2 the Mean Value Theorem
Rolle's theorem
Let f be continuous on theclosed interval a b and differentiableon the open interval la b
If f a fcb then there isat least one number c in Ca bsuch that f c O
i ia andCb HD is
also 0
Make sure to check the hypotheses ofany theorem you plan to use ForRolle's Theorem check
1 f continuous on Ca b2 f differentiable on Ca b3 fCa Hb
Examples
Given Hx x23 2 on the interval
1,2 find all values of c guaranteed
by Rolle's theoremCheck conditions hypotheses1 f is continuous on 11,2because f is a polynomial2 f is differentiable on 4,2because f is a polynomial
3 f i 12 3.1 2 0f 2 22 3.2 2 0
f l f 2
f x 2x 3 03 t3
2x 32 I contained
x 32
E 1,2
I C Zz7
Therefore
f x sin x on 0 it
Same instructions
check the conditions firstl f is continuous on O IT
since trigonometricfunctions arecontinuous on their domains and thedomain of f is l n and0 it E C o o
Ta subset of
2 f is diff on 10,1T since
trig functions are diff on
their domains3 f o sin 0 0
f it sin IT O
fco flit
f x cos X Ox Az 3Iz
Tonlyone in the interval Lo a
I2
The Mean Value Theorem
If f is continuous on theclosed interval a b and differentiableon the open interval ca b then thereexists at least one number c in Ca bsuch that
f c f b ffa slopeofthe line
b a connecting
get tohffhedpf.in
on theinterval
e.s.tt rEsTopeisfCb fab a
flat1 1 11 7 Xa c b
The proof is on p 289
Examples find all values of c inthe given interval guaranteedby the MIT
f x 32X t 2 on 1,2
f is apolynomial so it iscontinuous and differentiable everywhere
Find c such that f c fetab a
3 22 2 2 341211 2 piz Let's findf 12
14 8 this value
Now set the derivative equal to thisvalue and solve the resulting equation
f x 6 1 8t th6 9
6472
X 32 or x 1.5
and 3 is in the interval 1,2so c 32
f x lax on 1,4Domain of fCx tax is CO N
and f is continuous in its domainand 1,4 c co a f continuouson 1,4J f is also differentiable on1,4 since f is diff on lo n
O
ftp.a fl44IfI ln4zlnl
Ind3
f x tf duty
x 3g a 2.16 E 1,4
C 3In 4
f x E on 0,3Hx e is an exponential function
so it is continuous and differentiableeverywhere Since the domain is C ago
ftp.afal H32of 3
e e e bzt eazI.ee3ele
f x e C 2 32e
2x
So 2 e3e
2eH II
I 2 Fee I
e6ele
In e en leftIn eff2
2
xzen eaTfs 1.79430.8972 E 0,3
c tzdnfefe.LIShow that the equation x7e o
has exactly one real root
Let f x x3teExistence of a root we can use the
Interfax to help usshow the existence of a root
IIT If f is continuous ona closed interval Ca b's and K is
any number between fca and fcbthen there exist a valued in Ca bsuch that f d K
f b o
K Mf a o
l l l la d dz b
We must choose an interval a bSuch that f't a a 0 Lf b
since f is the sum of twocontinuous functions on C o D f isalso continuous on C o o so it willbe continuous on any closed intervala b
For f x x3 exfl D C D t e I t te LO
and f l P t e I te 0
so the interval E I I suffices
By IVT there is a valued E C t Dsuch that fCd 0
Uniqueness only one root
Use proof by contradiction
suppose there exists another root.dzThen Hd O and f de 0
Since f is continuous and differentialeverywhere we can use the MVT to
say there exists CE Cd dz suchthat f Cc ftdddz ffd ff.co
0So f c 0 for some value c
F ut f x zxfte.IO so for all x
So this is a contradictionSo there cannot be another valuedz such that flak O
BB