First derivative analysis

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1 Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points Frmat s thorm on critical valus What you can larn hr: How to us algbraic mthods, without a calculator, to idntiy th ntir pattrn o incras/dcras o a unction and to classiy all cut points o th drivativ Frmat s thorm givs us a mthod or idntiying trm points, but mor inormation is ndd to dtrmin th pattrn o th graph btwn critical valus Idntiying such pattrn can also allow us to classiy th critical valus, and any othr spcial valus, without th us o a calculator Fortunatly, idntiying this pattrn is vry simpl, sinc w only nd to rmmbr th graphical maning o th drivativ Knot on your ingr Givn a dirntiabl unction : At any point c, ( c ) whr ( c) 0, th slop o th graph is positiv and hnc th unction is incrasing At any point c, ( c ) whr ( c) 0, th slop o th graph is ngativ and hnc th unction is dcrasing Now, i w think about it, thr ar only our options or what can happn to th drivativ at a valu c: 1) '( c) 0, corrsponding to a critical numbr ) '( c) DNE ) '( c) 0, corrsponding to an intrval o incras 4) '( c) 0, corrsponding to an intrval o dcras, corrsponding to a cut point that may b a critical numbr This small numbr o options allows us to dvlop a simpl mthod to analyz th up-down pattrn o th graph o a unction This stratgy is basd on th stratgy or solving an inquality, sinc w ar rally trying to solv/analyz th inquality: 0 by using its cut points Thror, you may want to rviw that stratgy, in cas you ar rusty on it Dirntial Calculus Chaptr 8: Graphical analysis Sction : First drivativ analysis Pag 1

2 Stratgy or prorming a First drivativ analysis To dtrmin th pattrn o incras/dcras in th graph o a unction y () and to idntiy and classiy th cut points o : 1 Us standard algbraic mthods to ind th cut points o Plac ths valus on a numbr lin, as in th stratgy to solv an inquality Tst ach rsulting intrval, to s i th unction is incrasing or dcrasing thr, and indicat this inormation on th numbr lin 4 Follow th incras/dcras pattrn and us any othr atur o th unction to classiy ach critical numbr as a maimum, minimum or othr atur Th last stp is also known as th First Drivativ Tst Eampl: ( ) 1 Stp 1: w irst comput th drivativ: 1 '( ) Thn w idntiy th cut points Ths occur whn th numrator is 0, which is at 0 and 15, and whn th dnominator is 0, that is, at 1 Notic that th last cut point is not a critical numbr, sinc it is not in th domain o th original unction Stp : W plac all th cut points on a numbr lin: Notic that whn placing th cut points on th numbr lin it is important to kp thir ordr, but thr is no nd to kp thm in scal Stp : W tst th drivativ in ach intrval to dtrmin i it is positiv (incrasing) or ngativ (dcrasing) Rmmbr that all w nd to know is whthr th drivativ is positiv or ngativ Sinc our drivativ includs two squars, which ar always positiv, w only nd to ocus on th actor This is ngativ bor 15 and positiv atr that Thror, th pattrn is as ollows: Stp 4: What do w hav at th thr critical valus? At 0 th unction is dirntiabl, but th unction is always dcrasing, so it is nithr a ma, nor a min At 1 th original unction is undind and o th orm #/0, so w hav a vrtical asymptot At 15 th unction is dirntiabl and it changs rom going down to going up Thror w hav a minimum at 15, 15 W did vrything without using th calculator, howvr, w can still rly on it to chck that our conclusions ar rasonabl Th graph w obtain in this way conirms that Eampl: y ln I you l tmptd to rwrit th unction as y ln, think again: why Dirntial Calculus Chaptr 8: Graphical analysis Sction : First drivativ analysis Pag

3 is this NOT th sam unction? 1) Th drivativ is y ln, so that its cut points occur whn: ln 0 ln Thror w nd to look at But wait! Thr is anothr option to considr, namly, whn th drivativ is undind This occurs at 0, whr th logarithm is undind 1 ) W plac all ths valus on a numbr lin: 1/ 0 1/ ) W tst ach intrval, but in this cas w do not vn nd to pick spciic numbrs, as it is suicint to considr thir rlativ positions 4) To classiy th cut points, w notic that at 1/ th unction is dirntiabl and th pattrn tlls us that w hav a maimum at 1/ and a minimum at 1/ What do w gt at 0? At this point w cannot dtrmin that by hand, as w nd to rsolv th limit: lim ln 0 This limit rquirs a mthod calld L Hospital s rul, which w hav not sn yt Thr is always mor to larn! Plas notic that you may b usd, rom your high school days, to prorm only th First Drivativ Tst This, as I mntiond, consists o only stp 4 o th irst drivativ analysis and its purpos is only to classiy critical points Although you ar pctd to know what this tst is and dos, you should always prorm a irst drivativ analysis, with all its our stps, and not just th tst portion W notic that in th irst and last intrval th valu o logarithm is positiv and so is th drivativ is larg, so its In th two middl intrvals th valu o is clos to 0, so that is vry small and its logarithm is ngativ nough to mak th whol drivativ ngativ Thror th pattrn is as ollows: 1/ 0 1/ Summary W analyz th irst drivativ by computing it, analyzing it as in th mthod or solving inqualitis and thn drawing conclusions basd on th pattrn w s on th numbr lin graph Common rrors to avoid Whn askd to analyz th irst drivativ, prorm all our stps, not just th irst drivativ tst stp that coms at th nd Dirntial Calculus Chaptr 8: Graphical analysis Sction : First drivativ analysis Pag

4 Larning qustions or Sction D 8- Rviw qustions: 1 Dscrib how to prorm a irst drivativ analysis Dscrib how to classiy th cut points o th irst drivativ by using th inormation about th rst o th pattrn Eplain th dirnc btwn a irst drivativ tst and a irst drivativ analysis Mmory qustions: 1 How is th drivativ o a unction at a point whr th unction is incrasing? How is th drivativ o a unction at a point whr th unction is dcrasing? What graphical atur occurs at a point whr th unction is continuous and its drivativ changs rom positiv to ngativ? 4 What graphical atur occurs at a point whr th unction is continuous and its drivativ changs rom ngativ to positiv? 5 What is th purpos o th irst drivativ analysis? 6 What is th purpos o th irst drivativ tst? Computation qustions: Prorm a ull irst drivativ analysis on th ach o th unctions prsntd in qustions y 10 5 y y y 1 6 y 1 7 y 1 8 ( ) 4 Dirntial Calculus Chaptr 8: Graphical analysis Sction : First drivativ analysis Pag 4

5 9 1 y Us th act th on o th critical 1 valus is at y, / / 14 y y 1/ / y (Rmmbr that ) 0 y / y 1 1/5 y 4 y on [-1, 1] 5 y 6 y y 4 1 y 1 y 1 sinh You can assum that cosh(1) cosh 4 5 y ln 6 y ln 7 8 y on [1, 4] ln y ln 9 y tan, 0 40 y sin, y tan or Dirntial Calculus Chaptr 8: Graphical analysis Sction : First drivativ analysis Pag 5

6 Thory qustions: 1 I th drivativ o a unction changs at c rom positiv to ngativ, which two possibl graphical aturs may b occurring thr? Can w know whthr an trm valu is absolut or local by using only th irst drivativ? How do w idntiy th prsnc o a maimum rom th irst drivativ analysis? Application qustions: 1 Th managr o a company that producs a crtain typ o gadgts notics that th cost pr itm o producing n gadgts in a month is givn by th unction C is positiv and thos or which it is ngativ and plain what ths valus rprsnt in th problm C n 0n Dtrmin th valus o n or which ' Tmplatd qustions: 1 Prorm a irst drivativ analysis on ach o th unctions listd in th documnt Sampl unctions to analyz Construct a rasonably simpl unction and prorm a ull irst drivativ analysis on it What qustions do you hav or your instructor? Dirntial Calculus Chaptr 8: Graphical analysis Sction : First drivativ analysis Pag 6

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