Applying Theorems in Calculus 11ter111ediate Value Theorem, Ettreme Value Theorem, Rolle 's Theorem, and l\ea11 Value Theorem Before we begin. let's remember what each of these theorems says about a function. ntermediate Value Theorem ) i ec,..._ "'v..bc.a. O'o'\. [o..l b1 o-j.. ; ') <.f('o) er (b) < '), \c; o± lust-- (a.., 'o; s (c.):. v.j11 q Extreme Value Theorem.f(.,..)\} W"\Vc,..v,:)._ (o._l b] ) MVS1" b,_ o.- o\v\.--e. >'Y\JW\ aw.cl <>.\, a\l..)t r,1 wo(..lr.i L"'J b]. oh) -c<i., <JJ,;,&..., " a.. ) x :. \,, &-r CJJ-\-\, u..1.. v t; A.,\:>]. Rolle's Theorem.f<.,. ') is c.c,-.,.+;"'u " C:: ct., 'o] q:-t.r <tl? c ( o..., lo +(cu:: ), l '-\tq / o c.. CV\ (o..., 'o) '?1Jc1_... '(c.') -= 0. Mean Value Theorem tr+ +&..) i'> 't\uow).,...,. (o.. J \,} --ti \(' m'\...\,,.q.))... '> o.j-e.,j... "t?>.,u.st \JJ_, 1,L <!. :o.) 'o) '$1.lek... -t'(c.') = +co..)-+c'oj_. o..-'t>,\
The rate at which waterloows oui)ofa pipe. in allons per hourl is given by a differentiable function R of timer. The table below shows the rate as measured every 3 hours for a 2-1-hour period. 0 3 6 9 12 15 18 21 24 <hours) R(t) (gallons per 9.6 10.4 10.8 l l.2 11.4 11.3 10.7 10.2 9.6 hour) a. Estimate the value of R'(S). indicating correct units of measure. Explain what this value means about R(t}. 10.4-10.S?: 0.\";3 Q pvhr..j 'R.'(s') {<.(' )- ( ) 3-'- -3 c.l 'R'(s') :::i, o l '("Q.t Q..+ v..>h.t wo4.u- is ft.-wi P 't) f> L \ * LV\c.. "u J.. t.:.s. b. Using correct units of measure. find the average rate of change of R(t} from 1 = 3 tor = 18. Co;) -'Rl.\V _ - \O."t - \0,1 -S c. s there some time,. 0 < t < 24. such that R'(t)= O? Justify your answer. Slf'\ca. (2.(.lrj i':. f e--bjo\e. 9"Y\ ro, 2 4:\'] o,.,.,.d C.t>') -:: R(24J = C\. L. 1 -tt...c. ceol.4's '-- : f) t " O<'...t<- '-1' ck 'R.(-t"') c.
.,Jc: The total order and transportation cost C(x). m ofbottles of Pepsi c>t- Cola is approximated,,\ C(x) = 10.oonf'.!+ "f: 1opoo -t,-. 'l.. l '\x x+3,k G.-t\)J where xis the order size in number of bottles of Pepsi Cola in hundreds. Answer the following questions. by the function a s chere guaranteed a value of r on the interval O S r S 3 such that the average rate of chan ii of cost is equal to C'(r)? Give a reason for your answer. Slv\c.A..tt:,.'), u o.:k- :.O, C.. ) 't\ 't O'V\. [0 3], e/ N\,\) :r d,l)._\.i>'\uhv.) 8" ea. o... v O r c C: re::. Sv -tt..jt C.. 1 (r') -: C (o) -CC: ). 0-3 b. s chcre a value of r on the inter v al 3 Sr S 6 such that C'(r) = 0. Give a reason for your answer and if such a value of r exists. then find t value of r. J St C. )\ W'U t<.:,.):. 10 1 000 [x '1. :x.+3 3)1.. [3,ol cl.,;. ' c ):. 10)000 [ (x "" 1'1')(1,,_;.'l)- (2.)(. j)("'\>o.-\-351 b'y\ (3 1 b) f>-a C.u)-=. ( ) : 3 3 3 3. ; 3 3 c. (i.) = to 3. 3!».3 -t 3 ) j C.C.lo)-: <g 33. 3 3 J \e. e.-,... eu ;""voj '- b'\ '('" o. V\ (., '=>J sl.lek C.. ( r) :0. +'c C.. 1 ()0.') fw,. (:;,b)) Q.. 1 (r) =a,=- 4.0C\.
A c a r company introduces a new car for which th where tis the time in months. umber of cars so. S. is modeled b S(1) =30rfs-- 9 -.- '!00 ( 6't-t\O-j \T 1+2 - -eo\',._ the function a Find the value of S'(2.5). Using correct units. explain what this value represents in the context of this problem. S'c.s):: \33.'333 wrs -solcl fu- Met\ StV\.C.L s 1 c2.. c;) '> o 1 o..,:. s N"O"'- 't\ljm\.o.,w- <;o \ t \. b. Find the average rate of change of cars sold over the first 12 months. indicate correct units of measure and explain what this value represents in the context of this problem. S(o')- (\'2) -,;o - \'301. \"t3 - a, ll'o. _u 0-\ - _,, F.r \ ""' ", 011... WQX"e.. o\.[_ e.r- V"t"O"' ' -.,.,,..,.. -, ;o f\""o\, e.yo 'ifo o-\"2-100 = 9,'+'-g (_+..Jc ').'j- \ C\t..1.r'- (-l-ti) ::. 700 j 't.. -t-\' 2. :.! 5. l"i2. t.= - 1. s.,..c;2.. -?
X 2 3 4 /(x) f'(x) g(x) g'(x) - 6 -t 2 5 9 2 3 JO -4 3 4 2 6 7 Z' \?0 The functio r all real numbers. andlg is strictlv increasmjb The table above --- - file i omj iii{1mffllrt'ir.ust derivatives at selected values ofx. The function, is given by a Find the equation of the tangent line drawn to the graph of { ' when x = 3. \ V\ (-,.') ':. ( (: )) 0,.) : ( ;-Co h'( l (5C.\)). f(1) 1-\ l ) (3<: J-(. : : 1,- " :. :2. -r --+-7-= -, < :- -_ -::, ) - c;.. b. io (,) t o Z ) : r or the interval Z,) - h(;') _ ::f( )- :C\-C. = \-3 c. Explain why there must be a value of r for < r < 3 such that h(r) h():: 3 ----- ---- - h(;) = \_ -,?,. 1'.Uo...- Stw:.a. nuv ' o,.j.jjt:.. --, '+'l _,._ L'Jol t-'is. o c:..-...\,;.\'\u. SiV\ \..J (.:"\ - 3 ' \) :-7 h(r J = -;l J -' h11- :[:'.'J.,. o-. ' Vo-..',,. -u r 6-n lr):. - d. Explai; why ere is a value of c for < c < 3 such that h'(c) = -5. 0...J... tl.r 3Sc.JC" ""(?J2.. v-.(>c.') \... J, lc J,+-i -+ir.v t{\. \. T. 9i'-1e-.Y"tL1 1 o... vo.l ca. C.. '- C. '- 3 sucj.-. 't--. '(c.,") = -? s,... -5 t -ft...t. t, <..3.