{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 "Module 8 : Differential Calculus " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 34 "802 : Definition Of The Derivative" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 17 "O B J E C T I V E" }}{PARA 0 "" 0 "" {TEXT -1 383 "In this project well develop the concept of the derivative, ge ometrically and algebraically. We'll look at secant lines drawn betwee n two points on the graph of a function and how a family of secant lin es through one fixed point lead to the definition of the tangent lines . Finally, we'll compute the the derivative using the definition of th e derivative and the difference quotient." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 10 "S E T U P" }}{PARA 0 "" 0 "" {TEXT -1 253 "In this project we will use the following command packag es. Type and execute this line before beginning the project below. If \+ you re-enter the worksheet for this project, be sure to re-execute thi s statement before jumping to any point in the worksheet." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "resta rt; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name cha ngecoords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________ ____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 15 "A . Secant Lines" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________ ____________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 352 "A secant line is any straight line that passes thro ugh two points on the graph of a function. In this section, our task w ill be to construct and graph a secant line using Maple. In the comman ds below, we will first define a function and x values a and b which w ill determine the points (a,f(a)) and (b,f(b)) through which we will l ater draw the secant." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x -> 20 -10*cos(x) + (x^2)*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arro wGF(,(\"#?\"\"\"*&\"#5F.-%$cosG6#9$F.!\"\"*&)F4\"\"#F.-%$sinGF3F.F.F(F (F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a := 1; b:=3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "Next, we compute the slope of the line through the points (a,f(a)) and (b,f(b)) using a familiar formula for the slo pe between two points and evaluate it as a decimal number. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "This is the slo pe of the secant line!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "m := evalf( f(b) - f(a)) / (b-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"+g&yd'y!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 220 "Next we define the \+ secant line as a function L(x) by solving for y in the point-slope for m y - y1 = m(x-x1), and letting x1 = a, y1 = f(a), and m be the slope \+ we just computed. Finally, we graph f(x) and the secant line." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "L := x-> f(a) + m*(x-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG R6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%\"fG6#%\"aG\"\"\"*&%\"mGF1,&9$F1 F0!\"\"F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "displ ay( plot( \{f(x), L(x)\}, x=0..8, thickness=2),\n plot(\{[[a.0 ].[a,f(a)]], [[b,0],[b,f(b)]] \}, x=0..8, color=blue));" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)$ \"3K)\\EhOpEd(!#<7$$\"3ELLLLBxVVp?GW*)F,7$$\"3ymmm;$>5E$F0 $\"3$\\Zg[\\rP,\"!#;7$$\"3MLLLLAKn\\F0$\"3K'H[Q]&)z9\"F87$$\"3=LLLLc$ \\o'F0$\"3K$=4/<*3$G\"F87$$\"3)emmm^&Q%R)F0$\"3jhj#Gr]vT\"F87$$\"3wKLL $Qk#z**F0$\"3E*fA\\x8Aa\"F87$$\"3))*****\\YJ?;\"F,$\"3')))o&\\:&Hr;F87 $$\"3?LLL=\"\\F87$$\"3wmmm'3Q\\n\"F,$\"3iecvXhtu?F87$$\"3OLLLB6@G=F,$\"3oM&>BK (H&>#F87$$\"3&)******f-w+?F,$\"3:_MH_1-JBF87$$\"3%*********y,u@F,$\"3) >F+`F,tY#F87$$\"3)*******RP)4M#F,$\"3'4r_x!Hj)f#F87$$\"3ILLL=Zg#\\#F,$ \"3vmZ\"=h%*yr#F87$$\"3cmmmEn*Gn#F,$\"3SMN=8$3(fGF87$$\"3Tmmm1xiDGF,$ \"3+8n(*>J%)zHF87$$\"3!)*****\\9!H.IF,$\"3]yE\")f%)e>JF87$$\"3Immm1:bg JF,$\"3!e3/Xw'GVKF87$$\"3<+++X@4LLF,$\"3?\">x)4M+zLF87$$\"31+++N;R(\\$ F,$\"3]%Gj/vP#3NF87$$\"3wmmm;4#)oOF,$\"3>rAHF-3VOF87$$\"3jmmm6lCEQF,$ \"3u&*oG9x!pw$F87$$\"3ELLL$G^g*RF,$\"3(=.BvQs/!RF87$$\"3oKLL=2VsTF,$\" 3PAyMX&3#RSF87$$\"3f*****\\`pfK%F,$\"3ImUdy(y*fTF87$$\"3!HLLLm&z\"\\%F ,$\"3q![n!RRT!H%F87$$\"3s******z-6jYF,$\"3?4m%HAm^U%F87$$\"3<******4#3 2$[F,$\"3WYgRV]*pb%F87$$\"3O*****\\#y'G*\\F,$\"35KDp0ia%o%F87$$\"3G*** ***H%=H<&F,$\"3XN\\aC+qM`F,$\"35GwvC`U`\\F87$$\"3%)** *****HSu]&F,$\"3GYY$)>vH*3&F87$$\"3'HLL$ep'Rm&F,$\"3/CL:BzT7_F87$$\"3' )******R>4NeF,$\"3OI2MW5-Z`F87$$\"3#emm;@2h*fF,$\"33a:^\"4sOZ&F87$$\"3 ]*****\\c9W;'F,$\"3*=_X'G*egg&F87$$\"3Lmmmmd'*GjF,$\"3P7Z0g7\\NdF87$$ \"3j*****\\iN7]'F,$\"3;pf(e\"\\*4(eF87$$\"3aLLLt>:nmF,$\"3B_LUT5],gF87 $$\"35LLL.a#o$oF,$\"3)y\"H@/C'\\8'F87$$\"3ammm^Q40qF,$\"3-)RN&R'=tE'F8 7$$\"3y******z]rfrF,$\"3)y\"*4QFS*)Q'F87$$\"3gmmmc%GpL(F,$\"3=1Ho\"RK$ GlF87$$\"3/LLL8-V&\\(F,$\"3k0Shjj+`mF87$$\"3=+++XhUkwF,$\"3q&4<7#[$fy' F87$$\"3=+++:o\"pv\\DNb$>F87$FX$\"3e%)H=;,\"\\:#F87$Fgn$\"3#R,)3Ua(HQ#F87$F \\o$\"3Cg.thV!yd#F87$Fao$\"3)o')='4oq!y#F87$Ffo$\"3[4J?o%>l&HF87$F[p$ \"3/VV#G-G'*3$F87$F`p$\"3QM@+d@?sJF87$Fep$\"39,4^Gi)[@$F87$Fjp$\"3'=WZ V=#f)>$F87$F_q$\"3!383mp=#F87$Fhr$\"3'ztf>$py[=F87$F]s $\"3M(R\"=4ME_9F87$Fbs$\"3k>Y1;D'3-\"F87$Fgs$\"3P&\\P3C>YV'F,7$F\\t$\" 3%e7Fi;s2]#F,7$Fat$!3Md&\\%fIhD7F,7$Fft$!3/#GQxsuA nF,7$$\"3K****\\FJ*G3&F,$!3j*[u>DmNq(F,7$F`u$!39tuG!ewaT)F,7$$\"3qKLLo ,\"QD&F,$!3ZYMy')QF3))F,7$Feu$!3s=HuGe0`*)F,7$$\"3`KLL.62@aF,$!3]YQ$)Q ;J?))F,7$Fju$!3OsmUN2&)y$)F,7$$\"3'fmm\"HOq&e&F,$!3&yR$pa*3Tq(F,7$F_v$ !3Gtu=+0#Gw'F,7$Fdv$!3cgD\")*>-Tw$F,7$Fiv$\"3c2`p(z&H!H#F07$F^w$\"3WG= 6uTwnbF,7$Fcw$\"3!eHW<-hV=\"F87$Fhw$\"3cJ(yQ*f+Q>F87$F]x$\"3\"oxIr!3&z t#F87$Fbx$\"3:-ZKRq-2OF87$Fgx$\"3+BZ_l\"y@\\%F87$F\\y$\"3[IJ)QrW(*H&F8 7$Fay$\"3cN\"Q(*oc]='F87$Ffy$\"3;4K>y&\\*4pF87$F[z$\"3#H#3#ot*p!e(F87$ F`z$\"3%)4-#RmNZ4)F87$Fez$\"3Uc#)>7GRx%)F8-Fjz6&F\\[lF(F][lF(F`[l-F$6$ 7$7$$\"\"$F)F(7$Fcfl$\"3!fKaQ]+q6$F8-Fjz6&F\\[lF(F($\"*++++\"!\")-%+AX ESLABELSG6%Q\"x6\"Q!6\"%(DEFAULTG-%%VIEWG6$;F(FezFdgl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________ ______________________________________________________________________ ____" }}{PARA 4 "" 0 "" {TEXT -1 16 "B. Tangent Lines" }}{PARA 0 "" 0 "" {TEXT -1 83 "______________________________________________________ _____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 497 "While a \+ secant line passes through two distinct points on the graph of a funct ion, a tangent line passes only through a single point of the graph. A lthough there are an infinite number of straight lines which pass thro ugh this single point, there is only one which is momentarily travelli ng in the same direction as f(x) at the point of impact. One of the ma in goals of calculus is to find the slope of tangent lines because the y give us information about the direction of the curve at each point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 97 "Here is an animation which shows the tang ent line to different points on the graph of a function." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f := \+ x -> x^3 - 4*x^2 + 2*x + 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "T := (x,a) -> f(a) + (x - a) * D(f)(a) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "display(plot(f(x), x= -2..4, y= -4..8, thicknes s = 3, color = red),\n animate(T(x,t), x=-2..4, t=-1.5..3.5, vi ew = -4.. 8, color = blue));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%(ANIMATEG627$-%'CURVESG6$7S7$$!\"#\"\"!$!3+++++++ vB!#;7$$!++++v=!\"*$!3+++++]i:@F17$$!++++]FQ$!3pM\\>%pt!3CF17$$!3!******\\2<#p=FQ$!3LBz30<`CAF17$$!3 ')**\\78.K7=FQ$!3)GC,w_B:2#F17$$!31++D^NUbF17$$!36++]K 3XF;FQ$!3E0HViY(fh\"F17$$!3%)****\\F)H')\\\"FQ$!3\"\\/sv[eYL\"F17$$!3# ****\\i3@/P\"FQ$!3a2Ois#yE3\"F17$$!3;++Dr^b^7FQ$!3F!=iUCF[o$\"3u 6.\"f'eCe7FQ7$$!3B++]7YY08F[o$\"3]*356u7&o;FQ7$$\"3%z-+++XDn%!#?$\"3kc qIhyD4?FQ7$$\"3C++++y?#>\"F[o$\"3W2QFQ)*\\F [o$\"3lpCDKgAD@FQ7$$\"3_,+]isVIiF[o$\"3=-F2)z3_$>FQ7$$\"3&=++](o:;vF[o $\"3)QC$feT8o;FQ7$$\"3#>++v$)[op)F[o$\"3)\\EvSI]loJtS#>F[o7$$ \"3y******4FL(\\\"FQ$!3E+He#pxK;'F[o7$$\"3#)****\\d6.B;FQ$!3++M^Z1U:5F Q7$$\"3(****\\(o3lWFQ7$$\"3s***\\(=n#f(GFQ$!31x[ 'p^0`a\"FQ7$$\"3P+++!)RO+IFQ$!3UpdWvVz\")**F[o7$$\"30++]_!>w7$FQ$!31#[ `NTJRy#F[o7$$\"3O++v)Q?QD$FQ$\"3o;Faw@yvgF[o7$$\"3G+++5jypLFQ$\"3c>2^8 k?$e\"FQ7$$\"3<++]Ujp-NFQ$\"3R@o=eH./HFQ7$$\"3++++gEd@OFQ$\"3%3ghRNs(z UFQ7$$\"39++v3'>$[PFQ$\"3WTwK;\\bgfFQ7$$\"37++D6EjpQFQ$\"3[)o'frl8(y(F Q7$$\"\"%F.$\"#5F.-F`[l6&Fb[lFc[lF[qF[q-%*THICKNESSG6#\"\"$7$-F(6$7S7$ F,$!3>'H'H'H'H@?F17$F3$!3LjH'H'zeG=F17$F9$!3wH'H'H'zej\"F17$F>$!3P'H'H 'HrJW\"F17$FC$!3)H'H'H'HY]7F17$FH$!3wH'H'HYvd5F17$FM$!3ojH'H'HY]')FQ7$ FS$!3wH'H'H'zLs'FQ7$FX$!3%eH'H'H'H'z%FQ7$Fgn$!3!G'H'H'H@pGFQ7$F]o$!3h( H'H'H'H@%*F[o7$Fbo$\"3kTq.Pq`\\)*F[o7$Fgo$\"34Qq.Pq.7HFQ7$F\\p$\"3Cq.P q.7R[FQ7$Fap$\"3;/Pq.P?mnFQ7$Ffp$\"3IOq.PqG$p)FQ7$F[q$\"3?Pq.Pq.i5F17$ F_q$\"3Uq.Pq`ua7F17$Fdq$\"3+/Pq.PXZ9F17$Fiq$\"3QPq.P?;S;F17$F^r$\"3Cq. 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You will then see several command buttons appear in the menu bar at top. Click on the recycle (circling arrows) button and then the play \+ (triangle) buttons." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________________________ __________________________" }}{PARA 4 "" 0 "" {TEXT -1 27 "C. Families Of Secant Lines" }}{PARA 0 "" 0 "" {TEXT -1 83 "_____________________ ______________________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 632 "One of the main goals of calculus is to find t he slope of tangent lines. However, its impossible (without knowing th e methods of calculus) to compute the slope of a tangent line knowing \+ only the single point of tangency (because the slope formula m = (y2 - y1)/(x2 - x1) requires two points, and if you use the same point twic e, both the numerator and denominator are 0.) On the other hand, its \+ easy to compute the slope of secant lines. We'll use what's easy to fi nd to get what is hard to find. Our approach is to estimate the slope \+ of the tangent by computing the slope of secant lines which gradually \+ approach the tangent line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 350 "A diagram will demonstrate this idea. We're goin g to plot a family of secant lines simultaneously. All of the lines wi ll pass through one fixed point (c,f(c)) where we want to examine the \+ tangent line, and another point of the form(c+ 1/k, f(c+ 1/k)). These \+ other points are approaching 2 closer and closer as k gets larger. Let 's see how this looks." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "c := 2; left := c-1; right := c+1 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%leftG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ri ghtG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x-> 20 - \+ 10*cos(x) + (x^2)*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6# %\"xG6\"6$%)operatorG%&arrowGF(,(\"#?\"\"\"*&\"#5F.-%$cosG6#9$F.!\"\"* &)F4\"\"#F.-%$sinGF3F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "SL := (a,b) -> f(a) + ( (f(b)-f(a)) / (b-a) ) * (x-a);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SLGR6$%\"aG%\"bG6\"6$%)operatorG%&a rrowGF),&-%\"fG6#9$\"\"\"*&*&,&-F/6#9%F2F.!\"\"F2,&%\"xGF2F1F9F2F2,&F8 F2F1F9F9F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "displa y(plot(f(x), x= 0..8, y= -20..40, color = red, thickness = 3),\n \+ plot(\{SL(c, c + 1/k) $ k =1..8 \}, x= 0..8, y = -20..40,color = gol d), 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2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur ve 9" "Curve 10" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "As k gets large, the secant lines get closer and closer t o being the tangent line at x = 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________ ____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 27 "D . Definition of Derivative" }}{PARA 0 "" 0 "" {TEXT -1 83 "___________ ______________________________________________________________________ __" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 90 "Now we'll look at the slope of the tangen t line and the derivative from an algebraic view." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x-> x^3 + 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operator G%&arrowGF(,&*$)9$\"\"$\"\"\"F1F1F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 274 "Our first task is to find the \+ slope of the tangent for this function at x = 2. To do this, we will c ompute the slope of a secant line through (2,f(2)) and (2+h,f(2+h)), s implify this expression, take the limit as h approaches 0, and evaluat e this result as a decimal number." }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 58 "The slope of a secant line through (2,f( 2)) & (2+h,f(2+h))" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "(f(2+h) - f(2) )/h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$),&\"\"#\"\"\"%\"hGF)\"\"$F)F)\"\")!\"\"F)F*F-" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Simplif y this expression" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( % );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"#7\"\"\"*&\"\"'F%%\"hGF%F%*$)F(\"\"#F%F%" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Take the limit as h app roaches 0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "limit( %, h = 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " Evaluate this result as a decimal number" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "slope_of_tangent := e valf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1slope_of_tangentG$\"#7 \"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 399 "The last result was the slope of the tangent line at a particular point on the graph of f(x) - in particular at x = 2. If we follow the same steps, but leave x as an unknown, well get an expression for the slope of the tangent at any point (x,f(x)) on the graph. This express ion is called the derivative of f(x) because it is a function in its o wn right, which is derived from the original function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The slope of a secan t line through (2,f(2)) & (2+h, f(2+h))" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "(f(x + h) - f(x) )/ h; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$),&%\"xG\"\"\"%\"hGF)\"\"$F) F)*$)F(F+F)!\"\"F)F*F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Siimplify this expression" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify( %);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"\"\"$*(F)F(F&F(% \"hGF(F(*$)F+F'F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 92 "Take the limit as h approaches 0 (No need to evaluate a n algebraic expression as a decimal.)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "limit( %, h=0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"xG\"\"#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }