Definition of the Derivative MTH445 Real Analysis Lab LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9
<Text-field style="Heading 1" layout="Heading 1">Interpreting the Definition Geometrically</Text-field> Definition of Derivative: Let f be a real-valued function defined on an interval I containing the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==. Then if 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 exists and is finite, we define the limit to be the derivative of f at the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==, and is denoted by f '(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==). Its geometric interpretation is the slope of the tangent line to graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at the point (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==, 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). The derivative represents the instantaneous rate of change of the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. If f is differentiable at every point in a set S, the derivative defines a function on S, denoted by f ' where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ' (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) = 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 . In Maple when we use D(f) to denote this function. Suppose we are given a function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and a point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== in the domain of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. We find the tangent line to the graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at the point (1,f(1)) on the graph, plotting both the function and the tangent line. 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 We next compute the equation of the secant line joining the two points (1,f(1)) and (2,f(2)) and add it to the graph (in blue). 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiYtRiw2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Ji1GLDYlUSJ4RidGNkY5LUY9Ni1RJyZyYXJyO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1GIzYnRistRiM2JS1GLDYlUSJmRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GT0ZobkZqbi1JKG1mZW5jZWRHRiQ2JC1GIzYjLUkjbW5HRiQ2JFEiMUYnRkBGQC1GPTYtUSIrRidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnL0ZVRmJwLUYjNiUtRiw2JVEibUYnRjZGOS1GPTYtUTEmSW52aXNpYmxlVGltZXM7RidGQEZCRkVGR0ZJRktGTUZPRmhuRmpuLUZmbzYkLUYjNiVGWC1GPTYtUSgmbWludXM7RidGQEZCRkVGR0ZJRktGTUZPRmFwRmNwRmpvRkBGK0YrRitGK0Yr 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 Observe what happens as the second point on the secant line is chosen closer and closer to the starting point x=1. Do this by replacing the WHAT with successive values getting closer to 1: i.e. 2, 1.5, 1.1. Rexecute the statements to plot the updated secant line each time. 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 The following procedure animateSecant will create an animation of the tangent line to f at the point a as the limit of of secant lines starting with the secant line through (a, f(a)) and (a, f(a+h)). 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 To display the animation, execute the following statement, then click on the resulting plot and use the animation toolbar or the Animation menu. 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
<Text-field style="Heading 1" layout="Heading 1">Exercises</Text-field> Exercise 1: For each of the following functions, draw the graph of f(x) along with a secant and tangent line at the specified point x = a. a) f(x) = cos(x), a = 1. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSIxRidGQC8lK2V4ZWN1dGFibGVHRkRGQEYrRlpGQEYrRlpGQA== 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSQyLjJGJ0ZALyUrZXhlY3V0YWJsZUdGREZARitGWkZARitGWkZA 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 JSFH 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 b) 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 and a = 3. c) 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 and a = 5. JSFH Exercise 2 For each of the following functions, compute D(f)(x), plot f(x) and D(f)(x) on the same axes (as above). a) 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 . 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 b) f(x) = |x|. 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 JSFH Exercise 3: For the following plot of a function and its derivative, which one is the derivative and which is the orginal function. 6&-%'CURVESG6%7jq7$$!""!""$!2AE>CK.]:"!#;7$$"1LLLe%zfZ&!#<$!1RgD'*frq()!#;7$$"2,](o%Hw3A"!#<$!1R@!G1J5!p!#;7$$"1n;aVY:%*=!#;$!2duA<!HwiY!#<7$$"/D"GcE8l#!#9$!1$fgEKg#f<!#;7$$"1LL3#[)\3M!#;$"2XfTwuXQZ"!#<7$$"2MLL$G%*oqT!#<$"16iYv!z)f\!#;7$$"1MLeu.)G$\!#;$"1,3OL,Ui&)!#;7$$"1,DJg([9p&!#;$"12]Bot.97!#:7$$"1n;/Yr,]k!#;$"1lnJP(4'e:!#:7$$"1,]7VrI`r!#;$"2Q1>'4Ltb=!#;7$$"1L$3-9(fcy!#;$"1#3[&)zWA7#!#:7$$"1m;/'>8[e)!#;$"2;:xYquwN#!#;7$$"1**\(=DHIJ*!#;$"2M'>3s2'\a#!#;7$$"2n;aQ8:m+"!#;$"2wM%4#R^Eo#!#;7$$"2L$3_Ut#>3"!#;$"1zwGajIgF!#:7$$"2nT&)G,&p+6!#;$"23"*yw^$>qF!#;7$$"-D$oi%>6!#6$"0d*>-UKwF!#97$$"2Ne9ONI#Q6!#;$"2kB'=()fsyF!#;7$$"2m;zR-)*p:"!#;$"1)Q6R[Oux#!#:7$$"2)\PM%pld<"!#;$"2;&yX+K]sF!#;7$$"2KL3ZOLX>"!#;$"10H'eP$)Rw#!#:7$$"2l"H2N5I87!#;$"1`I1%yV>v#!#:7$$"/vV0(o?B"!#8$"0)fE30YOF!#97$$"2K3x'G")G48!#;$"10*4k5"\QE!#:7$$"2nm;>b2lQ"!#;$"1oUH\BE"\#!#:7$$"2LLe>PPD_"!#;$"1BWYdJ%>9#!#:7$$"-vIZnv;!#6$"2W6$o()4az;!#;7$$"2)***\i)3WH=!#;$"2l9qcG#4L7!#;7$$"2****\#pIix>!#;$"0$o'Q5z\(*)!#:7$$"2nT&Qy[!\/#!#;$"1>SCAm,/z!#;7$$"1L3_(o'=7@!#:$"1-^)GI,T:(!#;7$$"2:zpo(*)=_@!#;$"1u%*H^C!R'o!#;7$$"0v=iE">#>#!#9$"1x()zH)[')o'!#;7$$"1HK*3T#>7A!#:$"1z!oTb#=Vm!#;7$$"2%3xcbN>KA!#;$"10>K`Z2Dm!#;7$$"2w=U-q%>_A!#;$"1.Nj!o9Oj'!#;7$$"1nm"\%e>sA!#:$"11))Qwa(zm'!#;7$$"2lm"H<-(*RB!#;$"0)o"=h$)H'p!#:7$$"2nmm'*eWxS#!#;$"1wG&**\ar\(!#;7$$"/vo.+UlD!#8$"1,)4gr%oD$*!#;7$$"2mmm@Y*)\q#!#;$"2ja[_]hP5"!#;7$$"2N3F/Pa:y#!#;$"2NTxG>Ze<"!#;7$$"/voy#>"eG!#8$"22UGvh^!>7!#;7$$"1v$fLDYj(G!#:$"2j*pr/50C7!#;7$$"0DJ!GKd%*G!#9$"22cQ,Y%yE7!#;7$$"2_7.F?+G"H!#;$"1,Q;9,:F7!#:7$$"2.+vt<F5$H!#;$"2:EkMu_]A"!#;7$$"2/v=n7"[nH!#;$"2()R849BJ@"!#;7$$"/D1w]$R+$!#8$"2-LFFFy.>"!#;7$$"2O3-)>o+!3$!#;$"2U&*G'\Y216!#;7$$"1n;aj&yg:$!#:$"1r(fza]3q*!#;7$$"2n"HKrh$fA$!#;$"115Fy4r'*z!#;7$$"2n;/"zPz&H$!#;$"1#pyR#>7xe!#;7$$"0DJlkW6P$!#9$"2X$4)4b#=qJ!#<7$$"2LLeR^&\YM!#;$"1-"=kM#[?'*!#=7$$"2LeR#))QwCN!#;$!2%)*4&effzQ$!#<7$$"2L$3_iA..O!#;$!12#>z-m20(!#;7$$"2km;C6l6n$!#;$!2&QeO4*[#G5!#;7$$"/DJizHRP!#8$!1QI7@p)fM"!#:7$$"2kTgnI$)G"Q!#;$!2[UCGN9Fn"!#;7$$"2LL37lok)Q!#;$!2xt"RN1!>(>!#;7$$"1nTN7&*[iR!#:$!1V">%*Rx<C#!#:7$$",NP5&QS!#5$!2C#z^lg*GY#!#;7$$"/v$\&>)G6%!#8$!1')RE=.TDE!#:7$$"20+vj``s=%!#;$!2vU=(o9]IF!#;7$$"1EJ&4qKKA%!#:$!2kJEaqD,w#!#;7$$"1^7`l=@fU!#:$!0M!\$o()ex#!#97$$"18.#yW,sF%!#:$!2lmi]D8'yF!#;7$$"1v$4,.">&H%!#:$!2G"oBH=%zx#!#;7$$"1Q%)R71=8V!#:$!2x>_"fQ"Rx#!#;7$$"/vo%>q6L%!#8$!1.oQS!ylw#!#:7$$"0Dc1ln5T%!#9$!1&>g`:vcp#!#:7$$"20+Dm5l4\%!#;$!2(=Y5`slnD!#;7$$"1nm;UzaMY!#:$!2-c)er-NDA!#;7$$"1,+DmK&yy%!#:$!2nJ8!\R%)p<!#;7$$"1L3_v1xE\!#:$!2t%38*4,XN"!#;7$$"1,+v'4W'y]!#:$!1MzSrP.n(*!#;7$$"1%3_Ni%4]^!#:$!14Jn*="pW%)!#;7$$"1mTN]^a@_!#:$!1%>C-u2DZ(!#;7$$".vVM)))e_!#7$!1-:!3.r06(!#;7$$"1LeRQ:B'H&!#:$!1,s)=N)))\o!#;7$$"2lm;CtuNL&!#;$!1HZg;6F*o'!#;7$$"/vVEz"4P&!#8$!1"e9G*o,Em!#;7$$"1#H2VsFuS&!#:$!0"fP))GIam!#:7$$"1%3x@_PRW&!#:$!1Kv')G"ekw'!#;7$$"1wo/?tW![&!#:$!1Gp=7%\g&p!#;7$$"1nm"z6dp^&!#:$!1yA*p>/b@(!#;7$$"1%3_v'=S$f&!#:$!1K)=XXUe%z!#;7$$"/v=<m%)pc!#8$!1Er!))eAA&))!#;7$$"1MLLwt4<e!#:$!2ti+\tn/2"!#;7$$"1Me*3'**Q#*e!#:$!1&3`YY,)\6!#:7$$"2NLeaa#onf!#;$!2=">([m3^?"!#;7$$"1#HKu5<]+'!#:$!2&)oOxGM2A"!#;7$$"20D1%p;NUg!#;$!2o,q8*)yrA"!#;7$$"0B$R]*=51'!#9$!2kHS!*QxmA"!#;7$$"14-QJiozg!#:$!17!43%zbB7!#:7$$"1)=nB^`$)4'!#:$!2Q#4uF'Gx@"!#;7$$"1nTN$z?q6'!#:$!1*G^[j0"47!#:7$$"1%3FM'Rj&='!#:$!2A3B'o\F_6!#;7$$"1,+]LrCai!#:$!2curbD-T0"!#;7$$"2vm"H0S_6k!#;$!1^g&\E!>3n!#;7$$"1,D")f"f=['!#:$!2dJ&QreOGV!#<7$$"1MLL9V>_l!#:$!2t$e"HTi4g"!#<7$$"1</,fi=Fm!#:$"2:H,H!*=Th"!#<7$$"1,vo.#y@q'!#:$"1Lq0SF*)\]!#;7$$"1,+]j]&Rx'!#:$"10;N2%>GW)!#;7$$"1,DJB>tXo!#:$"2b)e[]?E$="!#;7$$"20Dc;'f'G#p!#;$"2c6&**4y7N:!#;7$$"#q!""$"2m%=mdCqh=!#;7gs7$$!""!""$"2E.s0rZM5"!#;7$$!1ML$3F5?E#!#<$"0#)3:kjgz"!#97$$"1LLLe%zfZ&!#<$"2(*=m?\Ew[#!#;7$$"2,](o%Hw3A"!#<$"1Bo]E#*\fI!#:7$$"1n;aVY:%*=!#;$"1zvL$\$pyN!#:7$$"/D"GcE8l#!#9$"2bi2n@UD2%!#;7$$"1LL3#[)\3M!#;$"11;>61EXW!#:7$$"2NL3_&Rf*y$!#<$"2b%\y$[w+e%!#;7$$"2MLL$G%*oqT!#<$"0M)zvPlwY!#97$$"2P$e*[;P7O%!#<$"0*oxgG35Z!#97$$"2MLe9!\y^X!#<$"1t0i'p)QLZ!#:7$$"2O$3-QELUZ!#<$"1/cSYuWYZ!#:7$$"1MLeu.)G$\!#;$"1>C@bo;\Z!#:7$$"1Dc,rC_A^!#;$"1h%>isZ:u%!#:7$$"1<zWnX;7`!#;$"1.nR(e*fBZ!#:7$$"14-)Qm1=]&!#;$"1pR+[hK&p%!#:7$$"1,DJg([9p&!#;$"1'pK_uinl%!#:7$$"1%3xJ&Htqg!#;$"1=Elr[1\X!#:7$$"1n;/Yr,]k!#;$"1Xd<:^T,W!#:7$$"1,]7VrI`r!#;$"1EG#)R,,FS!#:7$$"1L$3-9(fcy!#;$"2t<'H=))\LN!#;7$$"1m;/'>8[e)!#;$"2$3*pJ=Kj"H!#;7$$"1**\(=DHIJ*!#;$"1i%RBhTd@#!#:7$$"2n;aQ8:m+"!#;$"2cb:ZH)yL9!#;7$$"2L$3_Ut#>3"!#;$"1$ovG"R_si!#;7$$"2m;zR-)*p:"!#;$!2tn#4n;]i;!#<7$$"/vV0(o?B"!#8$!1la.Zmp_"*!#;7$$"2K3x'G")G48!#;$!2&oUp?r41;!#;7$$"2nm;>b2lQ"!#;$!1eL+'ptl=#!#:7$$"2,]P>YAXX"!#;$!19"*=3VI(e#!#:7$$"2LLe>PPD_"!#;$!1'[/x"R8sG!#:7$$"0Dc;r@3c"!#9$!22Yef/3(yH!#;7$$"2m;a801"*f"!#;$!2ul\:%*)zXI!#;7$$"1DJ?@#[#=;!#:$!1&[-jw@X1$!#:7$$"2L3_5R!RP;!#;$!1;['e=>M2$!#:7$$"2:/,4cKll"!#;$!1NX-uQdsI!#:7$$"-vIZnv;!#6$!2jR_C[+@1$!#;7$$"2)****\3yb_<!#;$!2Q(*z*=SYEH!#;7$$"2)***\i)3WH=!#;$!/T#H5VGl#!#87$$"2(***\x(>`.>!#;$!1F'>&4/pyA!#:7$$"2****\#pIix>!#;$!2@;ssIy4#=!#;7$$"2nT&Qy[!\/#!#;$!224qIUooN"!#;7$$"1L3_(o'=7@!#:$!1RLN$*y21()!#;7$$"0v=iE">#>#!#9$!1HAFm2ZnH!#;7$$"1nm"\%e>sA!#:$"2l!QBUpA[B!#<7$$"2lm"H<-(*RB!#;$"1P+,`N9Xi!#;7$$"2nmm'*eWxS#!#;$"1h;9edav$*!#;7$$"1v=<VM;ZC!#:$"2D$>6.'Qx2"!#;7$$"1$3xmH#e'[#!#:$"1$=6X2*y$="!#:7$$"2toHMs"H1D!#;$"2&=>(ofRLA"!#;7$$"29H#=]6+ED!#;$"2*R)G8G8OD"!#;7$$"1'*[$pd5da#!#:$"2.pL>iKWF"!#;7$$"/vo.+UlD!#8$"2$R!pu7bcG"!#;7$$"2jR(*f=mGe#!#;$"2#>j#)3_\(G"!#;7$$"1#H2$oBJ+E!#:$"2M2Q2$4o"G"!#;7$$"2x=<1bexh#!#;$"2)[n9?Q=o7!#;7$$"2L3FHt/_j#!#;$"2Eci\r(*pC"!#;7$$"2Z(oa(4(4qE!#;$"2ArKdaX;="!#;7$$"2mmm@Y*)\q#!#;$"1R^?Z4-'3"!#:7$$"2N3F/Pa:y#!#;$"1-(y-D"pUx!#;7$$"/voy#>"eG!#8$"2laSrHL\M$!#<7$$"2.+vt<F5$H!#;$!2uo(Gou"G%=!#<7$$"/D1w]$R+$!#8$!1<U5Nayqx!#;7$$"2O3-)>o+!3$!#;$!2FV\9)Q3X9!#;7$$"1n;aj&yg:$!#:$!2/\[*=B7I@!#;7$$"2n"HKrh$fA$!#;$!2x1+^'y!Ru#!#;7$$"2n;/"zPz&H$!#;$!2;IQu<2^J$!#;7$$"0DJlkW6P$!#9$!2E;zb(p.aQ!#;7$$"2LLeR^&\YM!#;$!0>bIy1\G%!#97$$"2LeR#))QwCN!#;$!0(=iVHm#f%!#97$$"2L$3_iA..O!#;$!1$>rc5:)QZ!#:7$$"29z%*\Zl+i$!#;$!1C%4jU\zu%!#:7$$"0vouo)4PO!#9$!1D%\g]3)[Z!#:7$$"2%3F%***=8aO!#;$!1$3d&4JNTZ!#:7$$"2km;C6l6n$!#;$!1)\)zAXcDZ!#:7$$"2Mekt`J_q$!#;$!1`^/Ne,pY!#:7$$"/DJizHRP!#8$!1&*=:*4C%zX!#:7$$"2kTgnI$)G"Q!#;$!1X`\H6NwU!#:7$$"2LL37lok)Q!#;$!2y]b2ZbK$Q!#;7$$"1nTN7&*[iR!#:$!1AYI)=ZpC$!#:7$$",NP5&QS!#5$!18J*3z1_b#!#:7$$"/v$\&>)G6%!#8$!2WmJ?(ev0=!#;7$$"20+vj``s=%!#;$!01dFP!\;5!#97$$"1^7`l=@fU!#:$!2_DbPUvbY#!#<7$$"/vo%>q6L%!#8$"2mrcn?Uq)\!#<7$$"0Dc1ln5T%!#9$"2bBcTXN>E"!#;7$$"20+Dm5l4\%!#;$"2$y'R\<gG#>!#;7$$"1MeRClviX!#:$"1C)=W2HYS#!#:7$$"1nm;UzaMY!#:$"20_IPeYIw#!#;7$$"/v=tU(Gn%!#8$"2x>p.wO***G!#;7$$"1M$3Ug+7r%!#:$"2:w9I=/w*H!#;7$$"1^(=(pPOIZ!#:$"20'p\VpeJI!#;7$$"1n"H_$p_\Z!#:$"2lhec^ec0$!#;7$$"1%eR25!poZ!#:$"2;")>1)=&)pI!#;7$$"1,+DmK&yy%!#:$"1,ZJyDBuI!#:7$$"1c)3C%z@0[!#:$"1p6<$))3)pI!#:7$$"0rn&=EeA[!#9$"1G&p5O.v0$!#:7$$"1kls%HZ*R[!#:$"2XKi=sKu.$!#;7$$"2wT&)3(>Jd[!#;$"1f=e$eM(4I!#:7$$"2b7.KKT?*[!#;$"1lW\w$3@$H!#:7$$"1L3_v1xE\!#:$"1H)ylKFh#G!#:7$$"1<a8'Q2F+&!#:$"2D!R(35.[]#!#;7$$"1,+v'4W'y]!#:$"2tne-@J.3#!#;7$$"1%3_Ni%4]^!#:$"2oH^(3oB7;!#;7$$"1mTN]^a@_!#:$"2'=H6NS#[5"!#;7$$"1LeRQ:B'H&!#:$"1_6B!34Aj&!#;7$$"/vVEz"4P&!#8$"1RbZ'\]HC%!#<7$$"1%3x@_PRW&!#:$!28V)[2fXkT!#<7$$"1nm"z6dp^&!#:$!1LAh*G0o)z!#;7$$"1%3_v'=S$f&!#:$!2@!ed+q0#4"!#;7$$"/v=<m%)pc!#8$!1uUQ8XZc7!#:7$$"1H23iHD)o&!#:$!284A<Z%>v7!#;7$$"1eR(pIfmq&!#:$!2c'\H*3RbG"!#;7$$"1)=n=ll]s&!#:$!1,-"=p@uG"!#:7$$"1</w'*>ZVd!#:$!1xMZRHy!G"!#:7$$"1voa'o%G!y&!#:$!2tY!R$zT=C"!#;7$$"1MLLwt4<e!#:$!2>rBC2$zo6!#;7$$"1Me*3'**Q#*e!#:$!13kup!R#e"*!#;7$$"2NLeaa#onf!#;$!1:tGnWdG`!#;7$$"20D1%p;NUg!#;$!1()4Ztn%4:%!#<7$$"1nTN$z?q6'!#:$"1&H!)ykR9Q&!#;7$$"1%3FM'Rj&='!#:$"2B%R"p+3]7"!#;7$$"1,+]LrCai!#:$"1#o:V5Y!R<!#:7$$"1NeRpb)GL'!#:$"1ViTgs<VC!#:7$$"2vm"H0S_6k!#;$"0z,A$RF5J!#97$$"1,D")f"f=['!#:$"2CS4@v@Zk$!#;7$$"1MLL9V>_l!#:$"1nhcT=x%4%!#:7$$"1</,fi=Fm!#:$"1E">c%3"yX%!#:7$$"1,vo.#y@q'!#:$"1a'otN%[!o%!#:7$$"1E1k=C7?n!#:$"1o&)R`bG6Z!#:7$$"1^PfLm1Qn!#:$"1@G07O5LZ!#:7$$"1voa[3,cn!#:$"1ujmha$eu%!#:7$$"1,+]j]&Rx'!#:$"1oZ'e'GS\Z!#:7$$"1EJXy#**=z'!#:$"1a&[Nm_Pu%!#:7$$"1^iS$\V)4o!#:$"1"p.n$p&)GZ!#:7$$"1v$f$3xyFo!#:$"1xsa0Jr/Z!#:7$$"1,DJB>tXo!#:$"1\$G2&RMrY!#:7$$"20Dc;'f'G#p!#;$"1*)pAg8\CW!#:7$$"#q!""$"1yZHC9x<S!#:-%&COLORG6)%$RGBG$"#5!""$""!!""$"#5!""$"1_MmX%)eqk!#;$"2wmoV()eqk"!#<$"2wmoV()eqk"!#<-%%VIEWG6$;$!""!""$"#q!"";$!13x[LFxQ\!#:$"1_II$4n$R\!#:-%+AXESLABELSG6'Q"x6"Q!6"-%%FONTG6%%(DEFAULTG%(DEFAULTG"#5%+HORIZONTALG%+HORIZONTALG-%%ROOTG6'-%)BOUNDS_XG6#$"$!Q!""-%)BOUNDS_YG6#$"#]!""-%-BOUNDS_WIDTHG6#$"%SN!""-%.BOUNDS_HEIGHTG6#$"%+R!""-%)CHILDRENG6" Looking ahead to Section 26: When f(x) has a max or min at x = a, what is D(f)(a)? Answer: When f(x) is increasing (decreasing) what is the sign of D(f)? Answer: Exercise 4: A familiar example: Let f(x) = 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 if x > 0 and 0 if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. Plot f(x). Use the graph to make a conjecture whether D(f)(0) exists, Check your answer by computing D(f)(x). Conjecture: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Zooming in: 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 Exercise 5: A variation on the function in the exercise above: Has anything chaged? Let f(x) = 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 for x > 0 and 0 for x <= 0. Plot f(x). Use the graph to make a conjecture to whether D(f)(0) exists. Then compute D(f)(x). Conjecture: 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 Zooming in: 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 Exercise 6: Let f(x) = |cos(x)|. Determine values for x where f(x) is not differentaible. Answer: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=