The Derivate as a Rate of Ch�an::.,o-­Mea11 Value Theorem a11d Roi/e's ,eorem

Consider the values of a differentiable function,.l{x), in the table below to answer the questions that follow. Plot the points and connect them on the grid below.

X 0 2 4 6 8 10 12 14 16

j{x) I 5 8 10 11 10 8 5 I

ll +-.,--.,--.,--.,--.,--.,--.,--�,--,--,--,--,--,--,--,--,

nt-t-+-+-+-+-+-:;�....,t-t-t-t-t-t-t-t--i

10+-+-+-+-+--!7:111 -+-+-+-1�+-+-+-+-+-;-....;

�\..:,l!L-6+-+-+1"+-�...-,;::-+-+-+-+-+-+-n--+-+-+--i

1,)\...oSe -.¥ � v .. .o.� �

� ,-.b-a,..· Qt. �,fC,.')

�(�,\�· I 2 3 4 5 6 1 i � 10 II 12 l.l 14 U 16 11 <

In calculus, the derivative has many inteipretations. One of the most important inteipretations is that the derivative represents the Rate of Change of a Function. When speaking of rate of change, there are two rates of change that can be found that are associated with a function-average rate of change and instantaneous rate of change.

S \ opt, � Sar• ,,,k �\-:-\<I.IL....;;;.._...,Average Rate bf Chlfuge of/(x)';in an Interval\

0¥.. 0...,.. U\i� �� t:0...1 b , � (Ml��� � c.,- 's

u� 1'� ... +(�)- +(b)

o..-b

Find the average rate of change of.l{x) on the interval [2, 12]. .f-('l.)- �( \� -

�-,�

s-8 ti1 -1� :: LJ£J

S\c,p < d... o.. '� U'4..lnstantaneou� Rat.Hir Ch�gp;;f j(x) �ta Pointj

M o. �toe.. 1 ?C.=o.., � ��.:b+a��e �� i ':, �"4- '9j ...

-t\(Cl.J.

Is the instantaneous rate of change off at x = 4

m�_.,"l(�)�,{t"�·�Ul'IA � � ).:.4\ lS S U­

� � \,.,� � � .x.-�1o.

Rolle·s Theorem

I+ �<:,..')'is CAv\�Y"IIJOVS � (Q., b] , J-:..�-..,\;oJo\c. O\"\. ( a. J b)," J +C.o..) :. �(b) J � � t <; ��� -+t> ox.\S't

<" vol,.. •Q.. ct C. OV'\. ( a.. .> b) S.v� � � '(c.) = 0.

Consider the fuoction,.l{x), presented on the previous page. Does Rolle·s Theorem apply on the following iotetTals? Explain why or why not?

loten'lll [2. 14]

�o�\t'c. ,C,,1/'\ \c. Off>U�\e +.r .C °"' t:,.,,i) � �\t. c.-.tnu�

� (o.J \:)) o-J.

Interval [2, 8]

For each of the functions below. determine whether Rolle·s Theorem is applicable or not. Then. apply the theorem to find the values of c guaranteed to exist.

Rolle's Theorem guarantees that if a function is continuous on the closed interval [a, b] and differentiableon the open interval (a, b), then there is guaranteed to exist a value of c where the instantaneous rate ofchange is equal to zero. �

1(c..") -:. O The Mean Value Theorem is similar. lo fact. Roue·s Theorem is a specific case of what is known incalculus as the �1ean Value Theorem.

lw�

l

�Value Theorem

+G.) i � �M"'u� � (o..J 'o1 � �.J,a..h\<..

(o...J b) ) � � i < �.u.d,.. "T1) t.�(-t-vcJ,., 1..Q, � � � (Q. .> o) s0c.k. � ...

�'(c.)-:: +(o.')-+('o) . <l-b

Consider the function h(x) = 3- 5 . The graph of h(x) is pictured below. Does the M.V.T. apply on the

interval (-1. 5]? Explain why or �:·by not. ""'-0<-) ·, '> t\t)'t C..-V... 't\l'\Uh,.I) aJt X: 0 1 �

� M,\J., , .. l'\01° �u�� Ot"\ t:.-IJ s].

-

Does the M.V.T. apply on the interval (1. S]? Why or why not?\.-.6<� '1<- ��"'u� or\ (1 1 SJ �"'-i,W.__,+;cJolr t7't-,. (I 1 '5) �o M. \J. T\<:> o..R�Uc.o k>l�. Graphicahy. what does the M.V.T. guarantee for the function

L on the interYal (1, 5]? Draw this op the gral'� to the left_ .}. 1 , .• o... "°"t., 11.. -x � C.. � � t A.¥tj" �l

• •

-i\ � i d � � � \.'I,�) i � � � � �CO•,! L:_>'\Q.. ?°'"'� �". (IJ-l-)o..,...l l '5J 1).

I

Apply the M.V.T. io find the \·alue(s} of c guaranteed for h(x) on the interval [I. 5)

�l-,_):- 3 - S X-1 5 _ hCQ- �(is)

'5 ...,_-.:2... c2. - \-s h '(,-) = "

.:s... = - '2. - 2.c�

c� - -�- �c.. '2..:. S'

Explain why you cannot apply the Mean Value Theorem for J(x) = xl - 2 on the interval [-1. l ] .

.f'(-,.'): � x-'/3

\ :2. I

-+ c_..,_):. T . x''3

+'(.,..':): , �c.k.,._ u� � 'X.:.O.

�V\.c.t... .f'()C..J ',s v�n.ul o1 �:.C 1 � ..t;..e,..) \s ncrt­��....._�Jo\e OV'\ (-\J 1). H�, � V\.\J .T. \ ':, V'\.t)-\ �f> u c.v�\ � fo.r. �) C1Y'\ [:- l J , 1.

Find the equation of the tangent line to the graph of J(x) = 2x+sinx+ Ion the interva@u the point which is guaranteed by the mean value theorem.

� . 'W' ..t '(.,..): � k t,o.<:) ')(. 'P. 0 .T. : f Qt)� .l (�J�S'in-'i"" \

+'(c..)-:. �(o)- +(-cr') : ,,--+ 2. 0-"Cf" (Tl'/2., 1i-\-�

�� � t -:: 1- (2\\-T\) S. O.T -= �\(iV�:: 2.-�CGo'V, -1'

�-t �c.. ':- �

The Mean Value Theorem guarantees that if a function is continuous on the closed interval [a. b] and differentiable on the open interval (a. b). lhen there is guaranteed to exist a value of c where the instantaneous rate of change at x = c is equal to the average rate of change of Jon the ioten al [a, bj.

Date Class ------- ----

Day #40 Homework

For the exercises l - 5. deteonine whether Roue·s Theorem can be applied to the function on the indicated interval. If Rolle's Theorem can be applied. find all values of c that satisfy the theorem.

l. f(x)=x1 -4x onthe interval O::,x::,4 . {6')\., ��V'I"� ev-.C.o,�"]�O.:.�lncJJ.c. � (. 0 / "f°) � �(o)::. fL�)-:. 0 so �ollA:� t�

Of� \A�'-<., � 1(x): 2x- 4

lc...-"i- -: 0

� c.. -=--4\

\��;>.\ 3. /(x) = 4-�Y- � on the interYal -3 ::,x S 7

-rt... � :'b {C,..)\ ... �.\ �t:V'"� .... "<>\< � '>C,:.�) w .\� W'\.. t: � Ii] <;,l".Le. �

�'+ � ;c,..) � ��f>a.,.\- � '!:- :l. . \-\ t.M.c:.e.., R.c� �

� ... � \� ru:::J: o...f�Uu...lo\t..

4. f (x) = sinx on the interval OS x S 1;r

• f<,..) \ s. �no� 8"l'\ [D 11-n-J� c:L.14..v---.:b cJ:;, \e.. OY\..

(c, 1 'LilJ'JJ� -fLo) �f{-i:liJ=o

-f \().) =- � "X­

e.co c...::: 0

,c..� %:��J

5. :;�;

c

.�:2

:::::al

�Y

li�J

�UJJ �

o.,.J__ �(,v,,) :::: �l 2.--uJ� :: -i�• + I (,x:) ::,. -- ;). $1 VL .'.2.X

-;)__.:;\ .-. �c_=- 0

Sl Y\ d-C.... -==- 0

?-c_. =- .,,... I Z°IT} 51\" J .<\if� -=.. -cf/ 2.. � " ... - - L.-�

For exercises 6- 9. detennine whether the Mean Value Theorem can be applied to the function on the indicated interval. If the Mean Value Theorem can be applied, find all values of c that satisfy the theorem.

6. f(x) =x3 -x2

-2x on-I �x� 1

'+ 16<. J : 3� '). - :2. X - .2.

! c:."" - 'l.c. -;).. = �(-1:> -f ( 1)

-I - I

-'3c. � -�� - .::>. = - \ 3c..'2.-�-\ - 0

(�c. ;- \)(.c. \) :..o

l<:=-Ys\ �

8 f( ·)-x+2 l< 1 . .t ---on -:,-xs; _ X

-

+'(..,_)::. X (. i) - (,v\ i;)(.1) _ X - X -2.

)' � 'A 2..

- 2.'Ai.

- � - -�-

c� - \

-2c.�:: -2

C.. �:. I

C.: "T \

\e.: 11

7. f(x)=-lx-3 on3�x�7

Pct� +&)�� ��

a*. -x.� 3) +u.; (S. M-t'�\-iY'l"Ou.D a.:,\. 'X.-=- 3 b\c

�- �"L i.<:> [3_, .oe).

�,�� '\{���I.':> l'U)�

��uJ:,\e..-

9. h(x)=2cosx+cos2xonO�x�n

h '(�):.. -��i"\X - � s,n 2.x

- .:2 -;11"\C - � si"" �c.. :: \., (o') - h C.'tl)0-11

-��11'"\C -�Sin.�: -1.�'13t c..: 0. 'l.1 "1 o-tl \ .'l '+8�

Using the graph of the function,.Jtx), pictured below. and given the intervals in the table below. determineifRolle's or Mean Value Theorem, whiche,·er is indicated. can be applied or not Give reasons for youranswers.

IO.

[-5. -l]Rolle's

Theorem

l l.

[-2, 8] Rolle's

Theorem

12.

[-l. 8]Mean

ValueTheorem

-li -5 � � -2 -i

i i i � i :

�vtc.a.. .+(�) � - ft>\.J �'> �+;�u ·, � � ?<.,:. -3

+ I. c., 11\:b.\ CAY\-hV'I l)O� � � \."""�� [-s J - �.

�s, �� ��-� �* 6.f{' \.;..�e..

�:: -\ , � 6,.") is t'U)1SiV\c..c. ft�; h.6.':, �C...,Sf �

��h.�Cl)o\t � � \� {_-2 , B) . ""-us,

���«NV' ' \.', �* °"f � ��\c:..

I

+G<.)\s ��\"\I.) o--tv) � (- l, �1 � �-f'-e.-�tia.Yc

� (-, ) �) ?O � � \J ciJ I� �e.,..-

' LS �pU"'-'b\e..

I