DA 3: The Mean Value Theorem


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1 Differentition pplictions 3: The Men Vlue Theorem 169 D 3: The Men Vlue Theorem Model 1: Pennslvni Turnpike You re trveling est on the Pennslvni Turnpike You note the time s ou pss the Lenon/Lncster Eit () One hour lter ou pss the Willow Grove Eit () Questions on this pge refer to the one hour segment during which ou trveled the 75 miles from to Construct Your Understnding Questions (to do in clss) 1 True or Flse: It must e the cse tht ou were going 75 miles per hour (mph) during the entire onehour segment descried in Model 1 Circle one nd eplin our resoning 2 You rememer looking t our speedometer somewhere in the middle of the segment nd our speed ws 80 mph True or Flse: This mens tht t some point during our trip ou must hve een going less thn 75 mph Circle one nd eplin our resoning 3 True or Flse: It must e the cse tht t some point during this 1hour segment ou were going ectl 75 mph Circle one nd eplin our resoning 4 True or Flse: For n cr trip, there is moment when our instntneous velocit is equl to the verge velocit for the trip Circle one nd eplin our resoning
2 170 Differentition pplictions 3: The Men Vlue Theorem 5 On the grph of f( ) t right Drw the secnt line through nd (the stright line from to ) Mrk the point C, with coordintes (, c f( c )), such tht the line tngent to f t C hs the sme slope s the secnt line through nd c (Check our work) Drw the tngent line to f t C, nd confirm tht this line is prllel to the secnt line through nd f () f () 6 Which of the following descrie the slope of the stright lines ou drew in the previous question? More thn one nswer is correct Circle ll tht ppl f ( c ) f ( ) f ( ) f ( c) f ( ) f( ) f ( ) f( ) f ( ) f ( ) c 7 For ech grph, mrk one or more points C, with coordintes ( c, f( c )), in the intervl (, ) where f( c) f ( ) f( ) 8 On the es t right, tr to drw function tht strts t nd ends t ut hs no f ( ) f( ) point C in (, ) where f( c) gin ssume C hs coordintes ( c, f( c ))
3 Differentition pplictions 3: The Men Vlue Theorem Is our nswer to the previous question consistent with Summr o D31? Eplin Summr o D31: Men Vlue Theorem (MVT) If f is continuous on [, ] nd differentile on (, ) then there eists numer c in (, ) where f( c) f ( ) f( ) or, in other words there is point ( c, f( c )) such tht the slope of the line tngent to f t c is equl to the slope of the secnt line etween (, f( )) nd (, f( )) 10 For ech grph, tr to find point C, with coordintes (, c ) c, in the intervl (, ) where f( c) f ( ) f( ) 11 Eplin wh the Men Vlue Theorem does not ppl to the functions in the previous f ( ) f( ) question Tht is, eplin wh there is no point C in (, ) such tht f( c) 12 (Check our work) In the previous question ou likel noted tht the functions in Question 10 re not continuous It turns out tht it is possile to drw continuous function with no point (, c ) c in (, ) such tht f( c) f ( ) f( ) Tr to drw such function on the es t right e prepred to shre our nswer s prt of wholeclss discussion Eplin wh the Men Vlue Theorem does not ppl to the function ou drew
4 172 Differentition pplictions 3: The Men Vlue Theorem 13 True or Flse: For function f tht is continuous nd differentile on [,, ] if f ( ) f( ) then there eists numer c in (, ) where f( c) 0 If ou circled True, drw such function nd mrk point C where f( c) 0 "Flse, drw such function with no point where f( c) 0 14 The following question refers to the mp of the Pennslvni Turnpike in Model 1 showing tht the distnce from the Lenon/Lncster Eit to the Willow Grove Eit is 75 miles Ken enters the Pennslvni Turnpike t The timestmped crd he receives t the Lenon/Lncster entrnce to the turnpike ss 9:00 PM t 10:00 PM he is driving the speed limit (65 mph) when he psses police officer who is prked ner the Willow Grove Eit () The police officer pulls Ken over nd informs him tht one of his rer lights is not working She sks to see Ken s timestmped crd, nd then tells him she is going to issue him ticket for speeding Ken eclims ut officer, I ws going the speed limit when I pssed ou Eplin how the Men Vlue Theorem (MVT) could e used to support the officer s cse 15 Let the functions f nd g descrie the positions of two horses (frnk nd george) during rce from time t to t ssume f ( t) g( t) for ll t in [, ] Descrie in words wht the reltion f ( t) g( t) tells ou out the reltive speed of frnk nd george during the period from time t to t
5 Differentition pplictions 3: The Men Vlue Theorem 173 t right, sketch possile functions f nd g showing frnk nd george tied t t Tht is, frnk nd george hve the sme position t t, s indicted the point on the grph f (), g () t c t right, sketch possile pir of functions f nd g showing frnk nd george tied t t f (), g () Tht is, f ( ) g( ) (Hint: Do not ssume this is fir rce nd tht frnk nd george strted t sme plce) t 16 (Check our work) re our nswers to the previous question consistent with the fct tht (when the re not tied) frnk leds one rce, while george leds the other If not, go ck nd check our work 17 The sttement in Question 13 is corollr of the MVT clled Rolle s Theorem Write this lel net to Question 13 Other corollries re given in Summr o D32 Fill in the lnks to mke ech true Choose from: incresing, decresing, constnt,,, or Summr o D32: Corollries of the Men Vlue Theorem If f nd g re continuous on [, ] nd differentile on (, ) then If f( ) 0 for ll in (, ) then f is on [, ] If f( ) 0 for ll in (, ) then f is on [, ] If f ( ) g( ) for ll in (, ) nd f ( ) g( ), then f ( ) g( ) for ll in [, ] If f ( ) g( ) for ll in (, ) nd f ( ) g( ), then f ( ) g( ) for ll in [, ] 18 (Check our work) The nmes of the corollries in Summr o D32 re: The Constnt Function Theorem, the Rcetrck Theorem (tied t the strt), the Rcetrck Theorem (tied t the finish), nd the Incresing Function Theorem Mtch ech nme to the correct theorem, nd check tht our nswers re consistent with the grphs ou drew in Question 15
6 174 Differentition pplictions 3: The Men Vlue Theorem Notes
y = f(x) This means that there must be a point, c, where the Figure 1
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