{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 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 }{CSTYLE "" -1 256 "" 0 1 101 136 80 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 "Warni ng" 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 70 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 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 0 0 0 0 0 0 0 0 1 }3 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 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 136 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 16 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 11 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 24 "MAPLE Worksheet Number 7" }} {PARA 257 "" 0 "" {TEXT -1 23 "Derivatives in Calculus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "The MAPLE command f or computing the derivative of a function depends on which of the two ways we used to define the function, as a symbol or as an operation. \+ We illustrate by defining the " }}{PARA 0 "" 0 "" {TEXT -1 9 "functio n " }{XPPEDIT 18 0 "f(x)=(2*x+3)^5" "6#/-%\"fG6#%\"xG*$,&*&\"\"#\"\"\" F'F,F,\"\"$F,\"\"&" }{TEXT -1 97 " in each way and computing its der ivative in each case. Perform the following command sequence." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f[1]:=(2*x+3)^5;0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"\"*$),&%\"xG\"\"#\"\"$F'\"\"&F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "f[2]:=x->(2*x+3)^5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"#f*6#%\"xG6\"6$%)operatorG%&arrowGF+*$),&9$F'\"\"$ \"\"\"\"\"&F4F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 61 "We can compute th e derivative of the symbolic expression " }{XPPEDIT 18 0 "f[1] " " 6#&%\"fG6#\"\"\"" }{TEXT -1 19 " using the command" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f[1],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),&%\"xG\"\"#\"\"$\"\"\"\"\"%F*\"#5" }}}{PARA 0 "" 0 "" {TEXT 257 65 "Why do you think we need to include the ,x in this command?" }{TEXT -1 57 " To tell maple what the variable is in the expression." }}{PARA 0 "" 0 "" {TEXT -1 50 "We can compute the deriva tive of the operation " }{XPPEDIT 18 0 "f[2] " "6#&%\"fG6#\"\"#" } {TEXT -1 19 " with the command" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D(f[2])(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),&%\"xG\" \"#\"\"$\"\"\"\"\"%F*\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "Now, in each of these two derivative computation s replace the x in the MAPLE syntax with a different variable and see \+ what happens. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f[1] ,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D(f[2])(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*$),&%\"tG\"\"#\"\"$\"\"\"\"\"%F*\"#5" }}}{PARA 0 "" 0 "" {TEXT 258 162 "Explain what's going on. f[1] does not depend on t, so the der ivative is zero. In the second case we take the derivative of the fun ction and evaluate it at t." }}{PARA 0 "" 0 "" {TEXT -1 44 "We could a lso have used the diff command on " }{XPPEDIT 18 0 "f[2]" "6#&%\"fG6# \"\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dif f(f[2](x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),&%\"xG\"\"#\"\" $\"\"\"\"\"%F*\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 104 "Here I bet you ge t 0 if you replace either of the x's with another variable and leave t he other alone. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diff(f[ 2](x),y); diff(f[2](y),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT 259 188 "Why is this? In the first case, we get an expression in x, this does not depend on y, so the derivative is zero. In the second case \+ we get an expression in y, which does not depend on x" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 429 "Sometimes it will be more convenient t o think of the function as a symbolic expression and other times we wi ll prefer the operation approach. Often it will be up to you to deci de which is best for your situation. (You should recall the above fun ction is the same one as g1 in worksheet #5.) Use both approaches to \+ compute the derivatives with respect to x of the other three functions at the end of worksheet #5. They were " }{XPPEDIT 18 0 "g2=ln(x)" " 6#/%#g2G-%#lnG6#%\"xG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "g3=1/x" "6#/% #g3G*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 9 ", and " }{XPPEDIT 18 0 "g4=s in(a*x)" "6#/%#g4G-%$sinG6#*&%\"aG\"\"\"%\"xGF*" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "g2 := ln(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2G-%#lnG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(g2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\" \"F$%\"xG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f2 := x - > ln(x); D(f2)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G%#lnG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$%\"xG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g3 := 1/x; diff(g3, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g3G*&\"\"\"F&%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$)%\"xG\"\"#F%!\"\"F*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 25 "f3 := x -> 1/x; D(f3)(x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#f3Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-9$ !\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$)%\"xG\" \"#F%!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g4 := sin(a *x); diff(g4, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g4G-%$sinG6#*& %\"aG\"\"\"%\"xGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$cosG6#*&%\" aG\"\"\"%\"xGF)F)F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f4 := w -> sin(a*w); D(f4)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f4G f*6#%\"wG6\"6$%)operatorG%&arrowGF(-%$sinG6#*&%\"aG\"\"\"9$F1F(F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$cosG6#*&%\"aG\"\"\"%\"xGF)F)F(F) " }}}{PARA 0 "" 0 "" {TEXT -1 136 "Note, as far as MAPLE is concerned \+ all derivatives are partial derivatives. To illustrate perform the fo llowing MAPLE command sequence." }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "g:=a*x^3*y^2+sin(x*y)+1/x;" "6#>%\"gG,(*(%\"aG\"\"\"*$%\"xG\"\"$ F(%\"yG\"\"#F(-%$sinG6#*&F*F(F,F(F(*&F(F(F*!\"\"F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"gG,(*(%\"aG\"\"\")%\"xG\"\"$F()%\"yG\"\"#F(F(-%$s inG6#*&F*F(F-F(F(*&F(F(F*!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "dgx:=diff(g,x);" "6#>%$dgxG-%%diffG6$%\"gG%\"xG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dgxG,(*(%\"aG\"\"\")%\"xG\"\"#F()% \"yGF+F(\"\"$*&-%$cosG6#*&F*F(F-F(F(F-F(F(*&F(F(*$F)F(!\"\"F6" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "dgy:=diff(g,y);" "6#>%$dgyG-%% diffG6$%\"gG%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dgyG,&*(%\"aG \"\"\")%\"xG\"\"$F(%\"yGF(\"\"#*&-%$cosG6#*&F*F(F,F(F(F*F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(dgx,y);" "6#-%%diffG6$%$d gxG%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(%\"aG\"\"\")%\"xG\"\"# F&%\"yGF&\"\"'*(-%$sinG6#*&F(F&F*F&F&F(F&F*F&!\"\"-%$cosGF/F&" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(dgy,x);" "6#-%%diffG6$%$d gyG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(%\"aG\"\"\")%\"xG\"\"# F&%\"yGF&\"\"'*(-%$sinG6#*&F(F&F*F&F&F(F&F*F&!\"\"-%$cosGF/F&" }}} {PARA 261 "" 1 "" {TEXT 260 111 "Explain why the last two answers are \+ the same. The mixed second order partials are equal of \"nice\" func tions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 232 "Next let's use MAPLE to recall all the basic rules of differentia tion: power rule, product rule, quotient rule, and chain rule. Define each of the functions as operations. This makes substitution for the chain rule example easier." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=x->cos(x/2);" "6#>%\"fGf*6#%\"xG7 \"6$%)operatorG%&arrowG6\"-%$cosG6#*&F'\"\"\"\"\"#!\"\"F,F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(-%$cosG6#,$9$#\"\"\"\"\"#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "u:=x->x^2-3*x+1;" "6#>%\"uGf*6#%\"xG7\"6$%)operatorG%&a rrowG6\",(*$F'\"\"#\"\"\"*&\"\"$F0F'F0!\"\"F0F0F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*$)9$\" \"#\"\"\"F1*&\"\"$F1F/F1!\"\"F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "h:=x->sqrt(x^2+2*x);" "6#>%\"hGf*6#%\"xG7\"6$%)operator G%&arrowG6\"-%%sqrtG6#,&*$F'\"\"#\"\"\"*&F2F3F'F3F3F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqr tG6#,&*$)9$\"\"#\"\"\"F4*&F3F4F2F4F4F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 38 "Compute the following derivatives and " }{TEXT 261 71 "state wh ich differentiation rule, or rules, MAPLE appears to be using." }} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(u(x),x);\n" "6#-%%diffG6$ -%\"uG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"#\"\"$! \"\"" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 35 "Power, sum, constant mu ltiple rules" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(f(x)*u(x ),x);" "6#-%%diffG6$*&-%\"fG6#%\"xG\"\"\"-%\"uG6#F*F+F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#,$%\"xG#\"\"\"\"\"#F+,(*$)F)F,F+F+* &\"\"$F+F)F+!\"\"F+F+F+#F2F,*&-%$cosGF'F+,&F)F,F1F2F+F+" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 13 "Product Rule " }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(f(x)/u(x),x);" "6#-%%diffG6$*&-%\"fG6#%\"xG\" \"\"-%\"uG6#F*!\"\"F*" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*&-%$sinG6#,$%\"xG#\"\"\"\"\"#F+,(*$)F)F,F+F+*&\"\"$F+F)F+!\"\"F+F +F2#F2F,*(-%$cosGF'F+F-!\"#,&F)F,F1F2F+F2" }}}{PARA 0 "" 0 "" {TEXT -1 54 "This last one looks like the product rule used on " } {XPPEDIT 18 0 " f(x)*(1/u(x)) " "6#*&-%\"fG6#%\"xG\"\"\"*&F(F(-%\"uG6# F'!\"\"F(" }{TEXT -1 27 " . Try the next command." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,,*&-%$sinG6#,$%\"xG#\"\"\"\"\"#F-)F+F.F-F-*(\"\"$F-F'F-F+F- !\"\"F'F-*(\"\"%F--%$cosGF)F-F+F-F-*&\"\"'F-F5F-F2F-,(*$F/F-F-*&F1F-F+ F-F2F-F-!\"##F2F." }}}{PARA 0 "" 0 "" {TEXT -1 66 "This looks like the quotient rule with the numerator expanded out." }}{EXCHG {PARA 0 "> \+ " 0 "" {XPPEDIT 19 1 "diff(f(u(x)),x);" "6#-%%diffG6$-%\"fG6#-%\"uG6#% \"xGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#,(*$)%\"xG\"\"# \"\"\"#F-F,*&#\"\"$F,F-F+F-!\"\"F.F-F-,&F+F-#F1F,F2F-F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 10 "Chain Rule" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(f(u),u);" "6#-%%diffG6$-%\"fG6#%\"uGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$sinG6#,$%\"uG#\"\"\"\"\"##!\"\"F+" }}} {EXCHG {PARA 265 "" 0 "" {TEXT -1 53 "Just the derivative of f, if you call the variable u." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff (u(x),x);" "6#-%%diffG6$-%\"uG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"#\"\"$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 18 " Derivative of u(x)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%*%%; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"xG\"\"#\"\"$!\"\"\"\"\"-% $sinG6#,$%\"uG#F*F'F*#F)F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 35 "Pr oduct of the last two expressions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(u=u(x),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&,&%\"xG\"\"#\"\"$!\"\"\"\"\"-%$sinG6#,(*$)F&F'F*#F*F'*&#F(F'F*F&F*F) F1F*F*#F)F'" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 89 "substitue the ex pression for u in terms of x, leads to the same result as the chain ru le." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(u(h(x)),x);" "6#- %%diffG6$-%\"uG6#-%\"hG6#%\"xGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,( %\"xG\"\"#F%\"\"\"*&#\"\"$F%F&*&,&*$)F$F%F&F&*&F%F&F$F&F&#!\"\"F%,&F$F %F%F&F&F&F0" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 17 "Chain Rule again ." }}}{PARA 0 "" 0 "" {TEXT -1 185 "The second derivative can be compu ted either of two ways, one is simply to compute the derivative of th e derivative , try it on f(x). Also check out the symbolic version of the input." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "diff(diff(f(x ),x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$cosG6#,$%\"xG#\"\"\" \"\"##!\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 85 "The other is to use \+ a shortcut syntax as follows and check out its symbolic version.:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(f(x),x$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$cosG6#,$%\"xG#\"\"\"\"\"##!\"\"\"\"%" }}} {PARA 0 "" 0 "" {TEXT -1 83 "The latter has the advantage that it is j ust as easy to compute 15 derivatives via:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "diff(f(x),x$15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$-%$sinG6#,$%\"xG#\"\"\"\"\"##F*\"&oF$" }}}{PARA 0 "" 0 "" {TEXT -1 62 "Compute the second and third derivatives of f, u, and h above." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f(x); diff(f(x), x, x); diff (f(x),x,x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cosG6#,$%\"xG#\"\" \"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$cosG6#,$%\"xG#\"\"\"\" \"##!\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$sinG6#,$%\"xG# \"\"\"\"\"##F*\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "u(x) ; diff(u(x),x$2); diff(u(x), x$3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,(*$)%\"xG\"\"#\"\"\"F(*&\"\"$F(F&F(!\"\"F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "h(x); diff(h(x), x$2); diff( h(x), x$3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,&*$)%\"xG \"\"#\"\"\"F,*&F+F,F*F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&* $)%\"xG\"\"#\"\"\"F**&F)F*F(F*F*#!\"$F),&F(F)F)F*F)#!\"\"\"\"%*&F*F**$ -%%sqrtG6#F%F*F0F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*$)%\"xG\" \"#\"\"\"F**&F)F*F(F*F*#!\"&F),&F(F)F)F*\"\"$#F/\"\")*&#F/F)F**&F%#!\" $F)F.F*F*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 32 "Next try the following commands." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dd2h:=diff(h(x )*f(x),x$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dd2hG,**(,&*$)%\"xG \"\"#\"\"\"F,*&F+F,F*F,F,#!\"$F+-%$cosG6#,$F*#F,F+F,,&F*F+F+F,F+#!\"\" \"\"%*&#F,F+F,*(F'#F7F+-%$sinGF2F,F5F,F,F7*&F'F " 0 "" {MPLTEXT 1 0 18 "diff(f (u(x)),x$3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#,(*$)%\"x G\"\"#\"\"\"#F-F,*&#\"\"$F,F-F+F-!\"\"F.F-F-),&F+F-#F1F,F2F1F-F-*(F1F- -%$cosGF'F-F4F-F2" }}}{PARA 0 "" 0 "" {TEXT -1 199 "Here's a problem t hat used to be on every standard calculus test known to mankind. Firs t try doing it by hand to see how well you would have done on such a t est. (Thank goodness for our technology.)" }}{PARA 0 "" 0 "" {TEXT -1 9 "Compute " }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(sqrt(1+sq rt(1+sqrt(1+sqrt(x^2+1)))),x);" "6#-%%diffG6$-%%sqrtG6#,&\"\"\"F*-F'6# ,&F*F*-F'6#,&F*F*-F'6#,&*$%\"xG\"\"#F*F*F*F*F*F*F5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*,,&\"\"\"F&*$-%%sqrtG6#,&F&F&*$-F)6#,&F&F&*$-F)6#, &*$)%\"xG\"\"#F&F&F&F&F&F&F&F&F&F&#!\"\"F7F+F8F/F8F3F8F6F&#F&\"\")" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 22 "GRAPHS \+ AND DERIVATIVES" }{TEXT -1 7 ". " }}{PARA 0 "" 0 "" {TEXT 262 145 "What is the basic relation between the graph of a function and th e derivative of the same function? The derivative gives the slope of \+ the graph." }}{PARA 262 "" 0 "" {TEXT -1 183 "What is the basic relati on between the graph of a function and the second derivative of the sa me function? The sign of the second derivative tells if the graph is \+ concave up or down." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 304 "Find all intercepts, local maxima, local minima, and i nflection points of the following functions. Also comment on any asym ptotic behavior and sketch the graphs of the function, its first deriv ative, and its second derivative on the same coordinate axes. Color th e function red and its derivative green." }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p1:=2*x^4-4*x^3-11*x^2+8*x+4;" "6#>%#p1G,,*&\"\"#\"\"\" *$%\"xG\"\"%F(F(*&F+F(*$F*\"\"$F(!\"\"*&\"#6F(*$F*F'F(F/*&\"\")F(F*F(F (F+F(" }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#p1G,,*$)%\"xG\"\"%\"\"\"\"\"#*&F)F*)F(\"\"$F* !\"\"*&\"#6F*)F(F+F*F/*&\"\")F*F(F*F*F)F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "dp1 := diff(p1, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dp1G,**$)%\"xG\"\"$\"\"\"\"\")*&\"#7F*)F(\"\"#F*!\"\"*&\"#AF* F(F*F0F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "cp:=fsolve(dp 1,x,complex);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG6%$!+eH=u7!\"*$ \"+g[b(>$!#5$\"+suUaCF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "These \+ are the critical points. Use the second derivative test." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "d2p1 := diff(dp1, x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%d2p1G,(*$)%\"xG\"\"#\"\"\"\"#C*&F+F*F(F*!\"\" \"#AF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "subs(x=cp[1], d2p 1); subs(x=cp[2], d2p1); subs(x=cp[3], d2p1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+./aaZ!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+'fG ?s#!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(=)[nj!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "T" }{TEXT 267 72 "hus cp[1] is a local min , cp[2] is a local max and cp[3] is a local min." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ip:=fsolve(d2p1, x, complex);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ipG6$$!+(\\M7!e!#5$\"+]M7!e\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(d2p1, x=-1..2);" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"\"\"\" !$\"#EF*7$$!3Z*****\\P&3Y$*!#=$\"3qKH3uRWR@!#;7$$!3B++Dcx6x()F0$\"3S(* )p!o`Tb+++vGVZ=F0$!3gho`7Rqu;F37$$!3_*****\\(4J@7F 0$!3Rpi0ap3r=F37$$!3;,+]iIKFl!#>$!3ZhIP>)=J.#F37$$\"3'R,++]siL#!#?$!3% Q$RRaRf0AF37$$\"3J,+++!R5'fFcp$!3-:(p$zn`MBF37$$\"3!)***\\P/QBE\"F0$!3 0M#>W>&)\\F0$ !3W\"Qxvt%***z#F37$$\"3v***\\P>:mk&F0$!3Yz!*)4Ll**y#F37$$\"3c***\\iv&Q AiF0$!3h5+^`&QTw#F37$$\"3j++]PPBWoF0$!3n-_6Y7P=FF37$$\"3%*)*****\\Nm'[ (F0$!3xL%Q`5'f^EF37$$\"36****\\(yb^6)F0$!3TE021$*4nDF37$$\"3')***\\PMa Ks)F0$!3@A>?]qHnCF37$$\"3a****\\7TW)R*F0$!3(z8T`a)oNBF37$$\"3z*****\\@ 80+\"FC$!3Mt['=ln()>#F37$$\"30++]7,Hl5FC$!3mpL!H,tI.#F37$$\"3')**\\P4w )R7\"FC$!3VM:.7Z`l=F37$$\"3;++]x%f\")=\"FC$!31CYY!oZMm\"F37$$\"3!)**\\ P/-a[7FC$!3dg@%)\\,Db9F37$$\"3/+](=Yb;J\"FC$!3?`.!p)p\"*=7F37$$\"3')** **\\i@Ot8FC$!3Wq(>ZzCPp*FC7$$\"3')**\\PfL'zV\"FC$!3-a\"z_n$R&)oFC7$$\" 3>+++!*>=+:FC$!3+\\86&oj7*RFC7$$\"3-++DE&4Qc\"FC$!3#49Sj/IUR)F07$$\"3= +]P%>5pi\"FC$\"3v87Zo!Q#yCFC7$$\"38+++bJ*[o\"FC$\"3'f]!QIHK&p&FC7$$\"3 3++Dr\"[8v\"FC$\"3,rsd'*Q$4e*FC7$$\"3++++Ijy5=FC$\"3Z&o&*p#feB8F37$$\" 31+]P/)fT(=FC$\"3t^Q5.k&>t\"F37$$\"31+]i0j\"[$>FC$\"3d'pHTzu39#F37$$\" \"#F*F+-%'COLOURG6&%$RGBG$\"#5F)$F*F*F`[l-%+AXESLABELSG6$Q\"x6\"Q!Fe[l -%%VIEWG6$;F(Fhz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 102 "Fro m the graph, the second derivative changes sign at ip[1] and ip[2], so these are inflection points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "disp lay([plot(p1, x= -2..3, y=-30..20,color=red, legend=\"p1\"), plot(dp1, x=-2..3, color=green, legend=\"dp1\"),plot(d2p1, x=-2..3, color=blue, legend=\"d2p1\")]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7co7$$!\"#\"\"!$\"\")F*7$$!3OLLe9r]X>!#<$\" 3Jt8/ahM3\\F07$$!3smm;HU,\"*=F0$\"3$RM0RF(zf@F07$$!3&**\\P4E+O%=F0$\"3 qe\"H'e1C\"G$!#>7$$!3SL$3FH'='z\"F0$!3ykt!\\;e2'=F07$$!37+DcEV'Gu\"F0$ !3?\"\\@'3G=EPF07$$!3gmmTgBa*o\"F0$!3;hzi(e?zK&F07$$!3`mm\"H_\">#e\"F0 $!33Y&QT!*\\y\"yF07$$!3ML$3_!4Nv9F0$!3!)Rg=m%>]U*F07$$!3))**\\iSM#eU\" F0$!30#>K2K\"p3**F07$$!3km;/wfHw8F0$!3gwv!>2oU-\"!#;7$$!3>+]ilXl]8F0$! 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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "convert(r1, parfrac, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&,&%\"xG!\"#\"\"%!\"\"F$,&*$)F'\"\"#F$F$F)F$ F*F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "So y=1 is a horizontal as ymptote." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "dr1 := normal(d iff(r1, x)); d2r1 := normal(diff(r1,x,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dr1G,$*&,(%\"xG\"\"%*$)F(\"\"#\"\"\"F-F)!\"\"F-,&F*F -F)F-!\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%d2r1G,$*&,**$)%\"xG \"\"#\"\"\"\"\"'\"\")!\"\"*$)F*\"\"$F,F,*&\"#7F,F*F,F/F,,&F(F,\"\"%F,! \"$!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "cp:=solve(dr1=0, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG6$,&!\"#\"\"\"*&\"\"#F(-% %sqrtG6#F*F(F(,&F'F(*&F*F(F+F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Two critical points. 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\"&F%*&\"\"%F%F$F%F%F%,&*$)F$\"\"#F%F%\"\"$!\"\"F0F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "We have slant asymptote, y=x." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "dr2 := normal(diff(r2,x)); d2r2 := normal (diff(r2, x,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dr2G*&,**$)%\"x G\"\"%\"\"\"F+*&\"#5F+)F)\"\"#F+!\"\"\"\"$F0*&F-F+F)F+F+F+,&*$F.F+F+F1 F0!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%d2r2G,$*&,**$)%\"xG\"\"$ \"\"\"\"\"%*&\"#OF,F*F,F,*&\"#:F,)F*\"\"#F,!\"\"F1F4F,,&*$F2F,F,F+F4! \"$F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "cp := fsolve(dr2=0 , x, complex);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG$!+&*y21O!\"* " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "One critical point, apply sec ond derivative test:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sub s(x=cp, d2r2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+SAp`5!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The critical point is a local max. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ip := fsolve(d2r2, x, c omplex);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ipG$\"+d[d/^!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(d2r2, x=0.4..0.6);" }}} {EXCHG {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVES G6$7S7$$\"3A+++++++S!#=$\"3]hgC7q%eR#F*7$$\"3mLLL3VfVSF*$\"3#y,F)yp)eI #F*7$$\"3%pm;H[D:3%F*$\"32RdDU,SFAF*7$$\"3mLL$e0$=CTF*$\"3'elK!oZ))Q@F *7$$\"3FLL$3RBr;%F*$\"3%\\iY(4S]\\?F*7$$\"3±zjf)4UF*$\"3$R&)RX F*7$$\"3?LLe4;[\\UF*$\"3q))f)=(HCx=F*7$$\"37++Dmy]!H%F*$\"3gz_#p****4z \"F*7$$\"3>LLezs$HL%F*$\"33MXC2z[,h\"F *7$$\"3!omm;_M(=WF*$\"3N$f%H1fV>:F*7$$\"3BLL$3y_qX%F*$\"3)*G([-(=lP9F* 7$$\"3T+++l+>+XF*$\"3')GsA'*=@X8F*7$$\"3K+++vW]VXF*$\"3RFGP#*G)>D\"F*7 $$\"3?+++NfC&e%F*$\"3b,(Hr$oth6F*7$$\"3aLLez6:BYF*$\"3@i87![I%z5F*7$$ \"3(pmm;=C#oYF*$\"3$*p91;<16)*!#>7$$\"3$pmmm#pS1ZF*$\"3s!p!*)=#RT(*)Fg p7$$\"30++DOD#3v%F*$\"3S=fMb>%e*zFgp7$$\"3!pmmm(y8!z%F*$\"3(oyI22ya7(F gp7$$\"3#)***\\i.tK$[F*$\"3BCTHUljlhFgp7$$\"3B++v3zMu[F*$\"3\"\\6I`l6n C&Fgp7$$\"3!omm\"H_?<\\F*$\"3g#*GaEAi#G%Fgp7$$\"3)om;zihl&\\F*$\"3cq#4 'ysQ#R$Fgp7$$\"3ULL$3#G,**\\F*$\"3sY1Vj8tECFgp7$$\"3%HL$ezw5V]F*$\"37C tS#Fgp7$$\"3G++Dcp@[_F*$!38Fr(ph/XO$Fgp7$$\"3e****\\2'HKH&F*$ !378(e$GohMWFgp7$$\"33mmmwanL`F*$!3:E?7HB\"HS&Fgp7$$\"3i+++v+'oP&F*$!3 _mm\"HYt7v&F*$!3d,.8(3S;e\"F*7$$\"3<+++q(G**y&F*$! 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Obviously any given value for y puts some kind of restrictio n on the possible values x can have. But the restriction is not explic itly given. Plot the graph of rel. (Recall implicitplot.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "implicitplot(rel, x=-5..5, y=-5..5, grid=[50,50]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%'CURVESG6cu7$7$$!\"&\"\"!$!3$p6dc<+&\\@!#=7$$!3oO!oQ0t]([!#<$!3tw \\eCwnpAF-7$7$$!3)*QpMn$=fz%F1$!3mP(*RF=e`AF-F.7$F57$$!3)pHM#Q*=%fYF1$ !3dfZ+%F1$!3ti!yU\"=&*4GF-7$7$$!3'\\pMn$=fzRF1$ !3[BOJHz\"*)z#F-F\\o7$Fbo7$$!3^D\\ezm?$y$F1$!3QyX8-)HF-Fho7$7$$!3Rt&G9dG9d$F1$!3CRI%H\"G(f=$F-7$$!3=_^ #fZnxo$F1$!3T[fz*[C71$F-7$7$$!3j_^#fZnxo$F1F\\qF^p7$7$$!3&HdG9dG9d$F1F gp7$$!328qK$=3]b$F1$!3K[:\"3PGaA$F-7$7$$!3%>^v(QpMnLF1$!39z/+?nT@MF-Ff q7$7$F]r$!3pz/+?nT@MF-7$$!3]@Xi\\5v>LF1$!3u^eI\"Q$=PNF-7$7$$!3$4XAhIlK ;$F1$!3sp1+u7*Rp$F-Fer7$F[s7$$!3I>F%f8M%zIF1$!3OhKf\">O&**QF-7$7$$!3#* *QpMn$=fHF1$!3VP))fcJm7SF-Fas7$7$Fhs$!3)z$))fcJm7SF-7$$!3OR?)oURI$GF1$ !3ee%pc&pmAVF-7$7$$!3#*Gj\"3/-^v#F1$!3:`'=T_b#*Q%F-F`t7$7$Fgt$!3g_'=T_ b#*Q%F-7$$!3j)4Mn'HNzDF1$!3-a#=1B:(=[F-7$7$$!3\"zEj\"3/-^DF1$!3FKnl(\\ M&R[F-F_u7$7$$!3!p?5bxQpM#F1$!36@O`/Sux`F-7$$!3O'eOF1$!3CU6_T*e ep'F-7$7$$!3n%3/-^v(Q>F1$!3WYMpHZ*f\"oF-Ffw7$7$$!3WB5bxQpMF1F_x7$Fbx 7$$!3h2ZV,(4je\"F1$!3YL.-/*)pE')F-7$7$$!3Aiz*[C71`\"F1$!3e\\`bMa;))))F -Fay7$7$$!3)4!\\C71`E8F1$!3YI2`\"3*e:5F17$$!3P]-5bJhw9F1$!3G&y(QpMn$=* F-7$FbzFgy7$7$F^z$\"3Dcn$)**HWwYF17$$!3Yl$y9\\!*\\M\"F1$\"3yKpMn$=fz%F 17$7$$!3ol$y9\\!*\\M\"F1F_[l7$$!3lpyL_q%fzw8F1$\"3y$*************\\F1Fe[l7$7$$!3wR=fz*[C7\"F1$!38)\\9H^$4` 6F17$$!3ZJ5s%RkL;\"F1Fb\\l7$Ff\\lF]z7$7$Fb\\l$\"3MDg!Hy,()o$F17$$!3(Q+ g%e>TS6F1$\"3uF;3/-^vPF17$F^]l7$$!3'QX=J-_U9\"F1$\"3ST#3wC8tz$F17$7$$! 3!R^T([KG$=\"F1$\"3v)oMn$=fzRF1Fd]l7$Fj]l7$$!3Y^+\"zDpq>\"F1$\"3C+H0:@ @aSF17$7$$!3YEP(p+TcA\"F1$\"3u\\xQpMn$=%F1F`^l7$Ff^l7$$!3:*3(y(oe-D\"F 1$\"3$*)*HexJ[6VF17$7$$!3p1I,%o&>t1uF-$\"3qhK;3/-^DF17$Fhdl7$$! 3#*)eD3t!4uvF-$\"3G.,'pwVTf#F17$7$$!3`l(Q5B8(f#)F-$\"3qAj\"3/-^v#F1F^e l7$Fdel7$$!3&*QbD/Uc?!*F-$\"33fh&pusG%HF17$7$$!3S@\")y02J#4*F-$\"3r$Qp Mn$=fHF1Fjel7$F`fl7$Fez$\"3N*o#z2.#4)HF17$Fjcl7$$!3r\"*y&Qc=VX&F-$!3MH [9qWQ&\\\"F17$7$Fcv$!3VRQxYTc6:F1Fjfl7$7$Fcv$\"3To'=Hj3\\:#F17$$!3')3t HXty'G'F-$\"3p+-^v(QpM#F17$FgglFedl7$7$F\\q$!3Ur\"y=z_of\"F17$$!3YC\"* R<99cYF-Fhy7$FahlF`gl7$7$F\\q$\"3O0wTVu(R$>F17$$!3'p0S2Y2S5$F-$\"3oyS? 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You can use the subs co mmand with two or more variables, just make sure the expression into w hich the substitution is to be made is the last entry in the command. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(x=0, y=sqrt(3), dre l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs(x=0, y=-sqrt(3), drel);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6##!\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 356 "Plot the graphs of rel and these two tan gent lines on the same axes. (Recall that when using implicitplot you must enter the expression to be plotted as a relation, even if it is \+ an explicit function. So to plot the function f(x) using implicitplot you have to enter it as y=f(x). This how you will get the graphs of t he two tangent lines on the picture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "tl1 := y-sqrt(3) = (-1/2)*(x-0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tl1G/,&%\"yG\"\"\"*$-%%sqrtG6#\"\"$F(!\"\",$%\"xG#F. \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tl2 := y+sqrt(3)=( -1/2)*(x-0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tl2G/,&%\"yG\"\"\"* $-%%sqrtG6#\"\"$F(F(,$%\"xG#!\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "implicitplot(\{rel, tl1 , tl2\}, x=-4..4, y=-4..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6U7$7$$!3f++++++!o$!#<$\"3\\x)ov!30sNF*7$$!3E VA'[Q)*e*QF*$\"3#))***********zOF*7$F-7$$!\"%\"\"!$\"3?x)ov!30KPF*7$F' 7$$!30dx8:;5kMF*$\"3;cx8:;5kMF*7$7$$!3J++++++gLF*$\"3Nx)ov!307MF*F:7$7 $$!3#4++++++/$F*$\"3@x)ov!30_KF*7$$!3fVA'[Q)*eD$F*$\"3)*)***********fL F*7$7$FL$\"3U************fLF*F@7$FF7$$!3Pdx8:;5CGF*$\"3)evP^h,T9$F*7$7 $$!3k++++++?FF*$\"31x)ov!30#4$F*FU7$7$$!3y+++++++CF*$\"3\"y()ov!30KHF* 7$$!3\"RCi[Q)*eh#F*$\"39************RIF*7$F`o7$Ffn$\"3]x)ov!30#4$F*7$F [o7$$!3odx8:;5%=#F*$\"3[cx8:;5CGF*7$7$$!3&4++++++3#F*$\"3nx)ov!30sFF*F jo7$7$$!3!4++++++w\"F*$\"3`x)ov!307EF*7$$!3BWA'[Q)*e(>F*$\"3I********* ***>FF*7$7$$!3XWA'[Q)*e(>F*F^qF`p7$Ffp7$$!3Mdx8:;5W:F*$\"3vbx8:;5/DF*7 $7$$!3$3++++++W\"F*$\"3Qx)ov!30_CF*Feq7$7$$!3x++++++?6F*$\"3Cx)ov!30#H #F*7$$!3aWA'[Q)*eL\"F*$\"3Y*************R#F*7$FfrF[r7$Far7$$!3kmvP^h,T !*!#=$\"3Ybx8:;5%=#F*7$7$$!332++++++!)F`s$\"3bx)ov!30K@F*F]s7$7$$!3g2+ +++++[F`s$\"3&y()ov!30s>F*7$$!3o[Ci[Q)*epF`s$\"3i************z?F*7$F_t Fds7$7$F[t$\"3jx)ov!30s>F*7$$!3&fcx8:;5k#F`s$\"3ibx8:;5k=F*7$7$$!3'p++ ++++g\"F`s$\"3[x)ov!307=F*Fit7$7$$\"35$************f\"F`s$\"3cx)ov!30_ ;F*7$$!3-vWA'[Q)*e&!#>$\"3c************f6F*$\"3Nx)ov!30s6F*Fjw7$7$Ffw$\"3Wx)ov!3075F*7$F]xFax7$Fi xF`x7$Ffx7$$\"3GVA'[Q)*el\"F*$\"3WavP^h,T!*F`s7$7$F^v$\"3Av()ov!30_)F` sF\\y7$7$Fbt$\"3!Qx)ov!30#pF`s7$F\\uF_w7$7$$\"3Sbx8:;5k=F*F_w7$F^v$\"3 6u()ov!30_)F`s7$Ffy7$$\"3iVA'[Q)*eH#F*$\"3#\\bx8:;5%eF`s7$7$Fir$\"3fu( )ov!30K&F`sFbz7$7$F^q$\"3&fx)ov!30s$F`s7$FhqFiv7$F_[lFhz7$F\\[l7$$\"3' GCi[Q)*e$HF*$\"3%)fvP^h,TEF`s7$7$Fco$\"3Zw()ov!307#F`sFb[l7$7$FN$\"3%* px)ov!30_F]v7$FXFfu7$F_\\l7$Fco$\"3vw()ov!307#F`s7$F\\\\l7$$\"34UA'[Q) *ed$F*$!3w\"RCi[Q)*e&F]v7$7$F0$!3JB7JC>\\z5F`sFe\\l7$7$$\"3A)********* ****RF*$!3S@7JC>\\zEF`s7$$\"3cbx8:;5%y$F*F`u7$Fd]l7$F0$!3\\z5F`s -%'COLOURG6&%$RGBG\"\"\"F6F6-F$6U7$7$F($\"3FH7JC>\\z5F`s7$$!3*GvP^h,Ty $F*Ffu7$Ff^l7$F4$\"3\\zEF`s7$Fc^l7$$!3UZA'[Q)*ed$F*$\"3F^YA'[Q)* e&F]v7$7$FA$!3))3x)ov!30_F]vF^_l7$7$$!3[++++++SIF*$!3kq()ov!307#F`s7$$ !3@`x8:;5WJF*F`u7$F]`lFd_l7$Fh_l7$$!3uZA'[Q)*e$HF*$!3?LvP^h,TEF`s7$7$F fn$!3Hp()ov!30s$F`sFb`l7$7$F\\o$!3$zw)ov!30K&F`s7$$!3k_x8:;5/DF*$!3[1+ +++++[F`s7$7$$!34`x8:;5/DF*$!3/2++++++[F`sFh`l7$F\\al7$$!3][A'[Q)*eH#F *$!3QHvP^h,TeF`s7$7$Fap$!3Co()ov!30#pF`sF[bl7$7$$!3n++++++gF*Fbgl7$7$F _w$!3mw)ov!30K@F*7$$\"3KvCi[Q)*epF`sFap7$F_hl7$$\"3g#************z%F`s $!3_w)ov!30s>F*7$7$F_w$!35x)ov!30K@F*7$$\"3Y9vP^h,T!*F`s$!3z_x8:;5%=#F *7$7$Fax$!3!o()ov!30#H#F*F\\il7$7$Ffw$!3%p()ov!30_CF*7$$\"3?ZA'[Q)*eL \"F*F\\o7$FiilFbil7$Ffil7$$\"3z^x8:;5W:F*Ffal7$7$$\"3z************fF*Ffn7$F[[mF bjl7$Fhjl7$$\"3N_x8:;5%=#F*$!3Q`x8:;5CGF*7$7$Fir$!3#p()ov!30KHF*F`[m7$ 7$$\"3'))***********>FF*$!31x)ov!30#4$F*7$$\"3dYA'[Q)*eh#F*Fi_l7$F_\\m Ff[m7$Fj[m7$$\"3/_x8:;5CGF*F^`l7$7$Fco$!3xw)ov!30_KF*Fd\\m7$7$$\"3a)** *********fLF*$!3Yw)ov!307MF*7$$\"3DYA'[Q)*eD$F*FA7$Fa]mFh\\m7$F\\]m7$$ \"3s^x8:;5kMF*$!31`x8:;5kMF*7$7$F0$!3gw)ov!30sNF*Ff]m7$7$F`]l$!3vw)ov! 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3>++++++?FF*7$$\"3m#*************zF`s$!3\")))oAn!o,n#F*7$Fa^n7$$\"3ut! 3O5Z(35F*$\"3OD>R'*GDJ7F*7$7$Fax$\"3F4S*HJTQ:\"F*Fgan7$7$$\"3+A*Rm8ED= \"F*F4Fi^n7$7$Ffw$\"3edx$p=%*oS*F`s7$$\"3@*)*z*>b.i6F*Fax7$FhbnF]bn7$7 $F^v$\"3IKl-?nzKwF`s7$$\"350y0Iw;x;F*F_w7$F`cnFebn7$F]cn7$$\"3?-%3))>W x!=F*$\"3Snf\">,eD_(F`s7$7$Fbt$\"3=h3jsotQiF`sFecn7$7$$\"32++++++!3#F* F\\dn7$$\"3zOSZ(*Hk=BF*$\"3!*>'f_-qNh&F`s7$7$Fir$\"35$=+<\")[RA&F`sFbd n7$7$F^q$\"3$=7C\"edjgWF`s7$$\"3J&H+N_&HaDF*Fiv7$F_enFhdn7$F\\en7$$\"3 k')Q(RE`KgU$F`sF^fn7$Fdfn7$$\"3_95 &yM?W\\$F*$\"3tO)*[@lzbMF`s7$7$F0$\"3bSGA(\\i(oIF`sFhfn7$F^gn7$$\"3NC( 4KvM<&QF*$\"32IF!zY_E3$F`s7$7$Ffem$\"3'43a#zSDzFF`sFbgnF[^l-%+AXESLABE LSG6$%\"xG%\"yG" 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 -1 389 "In the above we have concentrated mostly on the use of the derivative to give information about the shape of the graph of the original functio n. However, this last bit about tangent lines leads to another, and p robably more important, use of derivatives: to approximate functions w ith lines. To begin we approximate a differentialble function by its t angent line. Recall the function f :" }{XPPEDIT 18 0 "x->cos(x/2)" "6 #f*6#%\"xG7\"6$%)operatorG%&arrowG6\"-%$cosG6#*&F%\"\"\"\"\"#!\"\"F*F* F*" }{TEXT -1 58 " defined above and perform the following command seq uence." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x->cos(x/2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(-%$cosG6#,$9$#\"\"\"\"\"#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "dqf:=(f(x)-(f(Pi)+D(f)(Pi)*(x-Pi)))/(x-Pi);" "6#>%$dqfG *&,&-%\"fG6#%\"xG\"\"\",&-F(6#%#PiGF+*&--%\"DG6#F(6#F/F+,&F*F+F/!\"\"F +F+F7F+,&F*F+F/F7F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dqfG*&,(-%$c osG6#,$%\"xG#\"\"\"\"\"#F-*&F,F-F+F-F-*&#F-F.F-%#PiGF-!\"\"F-,&F+F-F2F 3F3" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "limit(dqf,x=Pi);" "6#- %&limitG6$%$dqfG/%\"xG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "numer(dqf);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(-%$cosG6#,$%\"xG#\"\"\"\"\"#F+F(F*%#PiG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 152 "Notice that the numerator of dqf is \+ f(x) minus the value of a line at x. This line is the tange nt line to the graph of f(x) at the point ( " }{XPPEDIT 18 0 "Pi,f(Pi )" "6$%#PiG-%\"fG6#F#" }{TEXT -1 12 " ) . That " }{XPPEDIT 18 0 "lim it(dqf,x=Pi)=0" "6#/-%&limitG6$%$dqfG/%\"xG%#PiG\"\"!" }{TEXT -1 190 " is equivalent to the limit of the difference quotient for f equallin g the derivative of f. It also shows that the values of f are so clos e to those of the tangent line when x is close to " }{XPPEDIT 18 0 "Pi " "6#%#PiG" }{TEXT -1 165 " the quotient is small even though the den ominator is going to 0! This is exactly what we mean when we say the \+ tangent line approximates the graph of f(x) near ( " }{XPPEDIT 18 0 " Pi,f(Pi)" "6$%#PiG-%\"fG6#F#" }{TEXT -1 87 " ). The equation of the t angent line is called the \"linearization of f\" in Calculus I." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Sketch th e graph of f along with this tangent line on the same axes." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "tlin := f(Pi) + D(f)(Pi)*(x-Pi);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tlinG,&%\"xG#!\"\"\"\"#*&#\"\"\"F) F,%#PiGF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "tlin;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG#! \"\"\"\"#*&#\"\"\"F'F*%#PiGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(\{f(x), tlin\}, x=0..6);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)$\"3b'*[zEjzq:!#<7$ $\"3%*******\\#HyI\"!#=$\"3]'*[Hk[S0:F,7$$\"33++]([kdW#F0$\"3Z'*)>C53& [9F,7$$\"3++++v;\\DPF0$\"3g'*[/V<_%Q\"F,7$$\"3W+++D*po+c7F,7$$\"3d****\\(G[W[(F0$\"3i' *)>C\"Rd'>\"F,7$$\"3i****\\()fB:()F0$\"3_'*)>u_M]8\"F,7$$\"39++](Q=\") 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It is defined mu ch the same way and approximates the graph for much the same reason. \+ The tangent plane to the graph of " }{XPPEDIT 18 0 "z=f(x,y)" "6#/% \"zG-%\"fG6$%\"xG%\"yG" }{TEXT -1 17 " at the point ( " }{XPPEDIT 18 0 "x[0],y[0]" "6$&%\"xG6#\"\"!&%\"yG6#F&" }{TEXT -1 2 " )" }}{PARA 0 " " 0 "" {TEXT -1 24 "is given by the equation" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "z=f(x[0],y[0])+diff(f,x)(x[0],y[0])*(x-x[0])+diff(f,y)( x[0],y[0])*(y-y[0])" "6#/%\"zG,(-%\"fG6$&%\"xG6#\"\"!&%\"yG6#F,\"\"\"* &--%%diffG6$F'F*6$&F*6#F,&F.6#F,F0,&F*F0&F*6#F,!\"\"F0F0*&--F46$F'F.6$ &F*6#F,&F.6#F,F0,&F.F0&F.6#F,F>F0F0" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT -1 19 "Plot the graph of " }{XPPEDIT 18 0 "z=-x^2+y^2+3" "6#/ %\"zG,(*$%\"xG\"\"#!\"\"*$%\"yGF(\"\"\"\"\"$F," }{TEXT -1 51 " and it s tangent plane at (0,1) on the same axes." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "f := (x,y) -> -x^2+y^2+3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(*$)9$\" 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1.000000 67.000000 -137.000000 0 0 "Curve 1" "Curve 2" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 170 "A better approximation to a curved graph would probably be a polynomial approx imation, such is called the Taylor Polynomial approximation to the gra ph of f at the point (" }{XPPEDIT 18 0 "x[0],y[0]" "6$&%\"xG6#\"\"!&% \"yG6#F&" }{TEXT -1 39 "). Try the following command sequence." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "taylor(sin(x),x=0,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"xG\"\"\"F%-%\"OG6#F%\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "t1:=convert(%,polynom);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "taylor(sin(x),x=0,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"xG\"\"\"F%-%\"OG6#F%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "taylor(sin(x),x=0,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"xG\"\"\"F%#!\"\"\"\"'\"\"$-%\"OG6#F%\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "t3:=convert(%,polynom);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3G,&%\"xG\"\"\"*&#F'\"\"'F'*$)F&\" \"$F'F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "taylor(sin( x),x=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG\"\"\"F%#!\"\"\" \"'\"\"$#F%\"$?\"\"\"&#F'\"%S]\"\"(#F%\"'!)GO\"\"*-%\"OG6#F%\"#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "t9:=convert(%,polynom);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t9G,,%\"xG\"\"\"*&#F'\"\"'F'*$)F&\" \"$F'F'!\"\"*&#F'\"$?\"F')F&\"\"&F'F'*&#F'\"%S]F'*$)F&\"\"(F'F'F.*&#F' \"'!)GOF')F&\"\"*F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 99 "On the same axe s sketch the graphs of y=sin(x), and its taylor polynomials of orders \+ 1, 3, and 9. " }}{PARA 0 "" 0 "" {TEXT -1 14 "Use the range " } {XPPEDIT 18 0 "x=-2*Pi..2*Pi" "6#/%\"xG;,$*&\"\"#\"\"\"%#PiGF)!\"\"*&F (F)F*F)" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y=-2..2" "6#/%\"yG;,$\" \"#!\"\"F'" }{TEXT -1 61 ". Use colors and a legend to distinguish on e from the other." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl ots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords h as been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "dis play([plot(sin(x), x=-2*Pi..2*Pi, y=-2..2, color=red, legend=\"sin\"), plot(t1, x=-2*Pi..2*Pi, color=green, legend= \"1st order taylor polyn omial\"), plot(t3, x=-2*Pi..2*Pi, color=blue, legend=\"3rd order taylo r polynomial\"), plot(t9, x=-2*Pi..2*Pi, color=black, legend=\"9th ord er taylor polynomial\")]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%7[s7$$!3(****>YH&=$G'!#<$\"3(=k#[]'efD\"!#D 7$$!3@Na4O(Hi9'F*$\"3BE$)3/$y_O\"!#=7$$!3dp3dxTF4gF*$\"30pS$))>))\\q#F 37$$!3]PgRy)4,*eF*$\"345eCY5JIQF37$$!3V07Azb%4x&F*$\"3>D%=3J28!\\F37$$ !3,:TzhG$pj&F*$\"3u$3Ba&z)>-'F37$$!3[DqOW,#H]&F*$\"3^r<2B!zY.(F37$$!3; (*zl.\">!o`F*$\"3Omto_#Rm#zF37$$!3')o*[H1=JB&F*$\"31M8:Cocu')F37$$!3GX $QL=e))4&F*$\"3?$=^+gcCE*F37$$!3gAxs.$)fk\\F*$\"3wR0I@Yj$o*F37$$!3tfxH c-O-\\F*$\"3(yt9^f'4?)*F37$$!3'ozn)3A7S[F*$\"3'GZ$)*p>`=**F37$$!3W:G:& =.!4[F*$\"3#p)>upZO`**F37$$!3,MyVhT)yx%F*$\"3]Q*4Adf&y**F37$$!3Z`GsP^w YZF*$\"3a7)p3)>4%***F37$$!3/sy+9hk:ZF*$\"3#*)p\\a&p%*****F37$$!3F4_VnT U$o%F*$\"3-H&=8\\0e***F37$$!3kXD'3A-7l%F*$\"3x>TkTnG\")**F37$$!3+#))*G u-)*=YF*$\"3y'*)p#zdSc**F37$$!3O=srF$ene%F*$\"3'4h:ZV)=@**F37$$!3(>*=d MWJAXF*$\"3sg\"*[P:!*>)*F37$$!3dllUT0(yX%F*$\"32BhW]l%yn*F37$$!3u)))o% 3YdCVF*$\"3?&HDa+ytD*F37$$!3!>@6bny7>%F*$\"3O(\\(Qk&pEn)F37$$!3.U5IC-T eSF*$\"35\"RXh9'pOzF37$$!3/t34t_OF*$\"31YSxe31()[F37$$!39T&yWM==`$F* $\"31$4k)*[rR!QF37$$!35AGA$)zV6MF*$\"3E2r+nO#em#F37$$!3*)**)f%3#=fF$F* $\"3tJ,q=(>#R8F37$$!3oxppL%)RSJF*$!3SNs&*3'>U>\"!#?7$$!3.2%=Rtn^%))pF37$$!3H;D'G%*3Y3#F*$!3iM=Am$ p(3()F37$$!3s.k\">JaY'>F*$!3se^f!>aVB*F37$$!38\"Hq4o*pW=F*$!3UK]h'4Csi *F37$$!3L&*Qs/=$\\x\"F*$!3]@'f#pgO#z*F37$$!3a*\\x%GR;0)*F37$$!3kGdA2Eg=8F*$!3&[E-%eNn$o*F 37$$!3/&=Qq]*3$=\"F*$!3N(z\"exYyd#*F37$$!3YT1&oSwv/\"F*$!3>5#fXpZ@m)F3 7$$!3;cW:%*)e`=*F3$!3v*Rg\"*zPr%zF37$$!3q(\\-)>P&\\*yF3$!3w^I>>+)**4(F 37$$!336%eFH^&[lF3$!3gp!)4vMW!4'F37$$!3WCVrl)[@?&F3$!3HzTe!4m1(\\F37$$ !3T<$Qj8Jd'RF3$!3y:/\\0pfiQF37$$!3S5B'pS8$HFF3$!3G\\n9vUb&p#F37$$!3#z# pf^)pcR\"F3$!3y-(QNBV6R\"F37$$!3.VX:B'HE?'Fiu$!3x>\\L^)*e-iFiu7$$\"31$ =+TaaKK\"F3$\"3NDU1?iR>8F37$$\"3b6>V%QN&3FF3$\"32yyQ\\&Rbn#F37$$\"3U2O AGkU9RF3$\"3m2R\\CLA:QF37$$\"3H.`,suJ?^F3$\"3MnKxUV\\**[F37$$\"3<:]xNG rAkF3$\"3eAUMjo:!*fF37$$\"30FZ`*>3^s(F3$\"3&)\\*[(oeNzpF37$$\"3%Hbw(\\ +hq!*F3$\"3pmZTo`'p(yF37$$\"3)y$=+!>6;/\"F*$\"3o%))f`C(=K')F37$$\"3!=$ *\\hHUK<\"F*$\"3Gi#)RJu5?#*F37$$\"3tD!)H-M([I\"F*$\"3^AD3uO]['*F37$$\" 3Y`)pSL`&o8F*$\"3S`w')*)\\='z*F37$$\"3?\"oTeELAV\"F*$\"3()[,)z)[:/**F3 7$$\"32&fF8f\\\"F*$\"3!Q>#)Rov> (**F37$$\"3!GU*\\jJvF:F*$\"3mYzggyt!***F37$$\"3nO`QHJff:F*$\"3q0GS[CP* ***F37$$\"3/pg$H)f%\\f\"F*$\"3qB!fA6%3(***F37$$\"3S,o[O))HI;F*$\"3S5T? 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "for i from 1 to 4 do t||i:=convert( taylor(exp(x),x=1, i+1), polynom); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G,&-%$expG6#\"\"\"F)*&F&F),&%\"xGF)F)!\"\"F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t2G,(-%$expG6#\"\"\"F)*&F&F),&%\"xGF)F)!\"\" F)F)*(#F)\"\"#F)F&F))F+F0F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3 G,*-%$expG6#\"\"\"F)*&F&F),&%\"xGF)F)!\"\"F)F)*(#F)\"\"#F)F&F))F+F0F)F )*(#F)\"\"'F)F&F))F+\"\"$F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t4 G,,-%$expG6#\"\"\"F)*&F&F),&%\"xGF)F)!\"\"F)F)*(#F)\"\"#F)F&F))F+F0F)F )*(#F)\"\"'F)F&F))F+\"\"$F)F)*(#F)\"#CF)F&F))F+\"\"%F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "myrange:=-2..4;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(myrangeG;!\"#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 134 "Sketch the graphs of all these on the same axes. Note that the Ta ylor Polynomial of order 1 is just the equation for the tangent line. 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}}{PARA 0 "" 0 "" {TEXT -1 216 "Actually MAPLE has certa in common general constructs built into it. To see some examples cons ider the following commands. Note that here k(x) has NOT BEEN EXPLICI TELY DEFINED, it is just general funcation notation." }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "limit((k(x)-k(a))/(x-a),x=a);" "6#-%&limitG 6$*&,&-%\"kG6#%\"xG\"\"\"-F)6#%\"aG!\"\"F,,&F+F,F/F0F0/F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#--%\"DG6#%\"kG6#%\"aG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {XPPEDIT 19 1 "limit((k(x)-(k(a)+D(k)(a)*(x-a)))/(x-a),x=a);" " 6#-%&limitG6$*&,&-%\"kG6#%\"xG\"\"\",&-F)6#%\"aGF,*&--%\"DG6#F)6#F0F,, &F+F,F0!\"\"F,F,F8F,,&F+F,F0F8F8/F+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "taylor(k(x),x=a,7 );" "6#-%'taylorG6%-%\"kG6#%\"xG/F)%\"aG\"\"(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+3,&%\"xG\"\"\"%\"aG!\"\"-%\"kG6#F'\"\"!--%\"DG6#F*F+F& ,$---%#@@G6$F/\"\"#F0F+#F&F7F7,$---F56$F/\"\"$F0F+#F&\"\"'F>,$---F56$F /\"\"%F0F+#F&\"#CFF,$---F56$F/\"\"&F0F+#F&\"$?\"FN,$---F56$F/F@F0F+#F& \"$?(F@-%\"OG6#F&\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "What does the expression " }{XPPEDIT 18 0 "O((x-a)^7 )" "6#-%\"OG6#*$,&%\"xG\"\"\"%\"aG!\"\"\"\"(" }{TEXT -1 271 " represen t? (To answer this question look at the Taylor expansions for sin(x), taylor(sin(x),x=0,n), for several different values of n. Then if you 're nice in class I might show you a fancy \"general\" limit calculati on that gives some indication of what it represents.) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "tp := taylor(sin(x), x, 10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#tpG+/%\"xG\"\"\"F'#!\"\"\"\"'\"\"$# F'\"$?\"\"\"&#F)\"%S]\"\"(#F'\"'!)GO\"\"*-%\"OG6#F'\"#5" }}}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "tp : = convert(tp, polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#tpG,,%\" xG\"\"\"*&#F'\"\"'F'*$)F&\"\"$F'F'!\"\"*&#F'\"$?\"F')F&\"\"&F'F'*&#F' \"%S]F'*$)F&\"\"(F'F'F.*&#F'\"'!)GOF')F&\"\"*F'F'" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 61 "MAPLE can comput e multivariable taylor approximations also. " }}{PARA 0 "" 0 "" {TEXT -1 57 "Now let's compute the 1st order Taylor approximation to \+ " }{XPPEDIT 18 0 "z=-x^2+y^2+3" "6#/%\"zG,(*$%\"xG\"\"#!\"\"*$%\"yGF( \"\"\"\"\"$F," }{TEXT -1 161 " . Of course this should be the same a s the equation of the tangent plane, so try it at the point (0,1) and \+ compare with your previous result. Is it the same?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "mtaylor(-x^2+y^2+3,[x=0,y=1],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"#\"\"\"*&F$F%%\"yGF%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 10 "Next try, " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "mtaylor(-x^2+y^2+3,[x=0,y=1],3);" }{TEXT -1 113 "but \+ first try to guess the answer. Think! You want a degree 2 polynomial \+ in x and y that approximates the graph." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"#\"\"\"*&F$F%%\"yGF%F%*$),&F'F%F%!\"\"F$F%F%*$)%\"xGF$F%F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"!\"\"*$)%\"yGF'F(F(\"\" $F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "C ompute the degree 1, 2, and 3 Taylor approximations to " }{XPPEDIT 18 0 "z=exp(x^2+y^2)" "6#/%\"zG-%$expG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F ," }{TEXT -1 12 " at (0,0). " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for i from 1 to 3 do t||i:= mtaylor(exp(x^2+y^2), [x=0,y=0], i+1 ); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t2G,(\"\"\"F&*$)%\"xG\"\"#F&F&*$)%\"yGF*F&F&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3G,(\"\"\"F&*$)%\"xG\"\"#F&F&*$)% \"yGF*F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 72 " Plot the graphs of z with each of these approximations on three graphs." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot3d(\{exp(x^2+y^2), 1\}, x=-1..1, y=-1..1);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%%GRIDG6%;$! \"\"\"\"!$\"\"\"F)F&X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":3FF0000000 0000003FF00000000000003FF00000000000003FF00000000000003FF0000000000000 3FF00000000000003FF00000000000003FF00000000000003FF00000000000003FF000 00000000003FF00000000000003FF00000000000003FF00000000000003FF000000000 00003FF00000000000003FF00000000000003FF00000000000003FF00000000000003F 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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:='f';" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGF$" }}}{PARA 0 "" 0 "" {TEXT -1 163 "Experiment to see the extent to whcih it appears that MAPLE ca n derive a \"general\" formula for the Taylor Polynomial for a given f unction of two or more variables." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "mtaylor(f(x,y), [x=a, y=b], 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"fG6$%\"aG%\"bG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "mtaylor(f(x,y), [x=a, y=b], 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%\"fG6$%\"aG%\"bG\"\"\"*&--&%\"DG6#F)6#F%F&F),&%\"xG F)F'!\"\"F)F)*&--&F.6#\"\"#F0F&F),&%\"yGF)F(F3F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "mtaylor(f(x,y), [x=a, y=b], 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.-%\"fG6$%\"aG%\"bG\"\"\"*&--&%\"DG6#F)6#F% F&F),&%\"xGF)F'!\"\"F)F)*&--&F.6#\"\"#F0F&F),&%\"yGF)F(F3F)F)*(#F)F9F) --&F.6$F)F)F0F&F))F1F9F)F)*(F1F)--&F.6$F)F9F0F&F)F:F)F)*(F=F)--&F.6$F9 F9F0F&F))F:F9F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "mtaylo r(f(x,y), [x=a, y=b], 4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6-%\"fG6 $%\"aG%\"bG\"\"\"*&--&%\"DG6#F)6#F%F&F),&%\"xGF)F'!\"\"F)F)*&--&F.6#\" \"#F0F&F),&%\"yGF)F(F3F)F)*(#F)F9F)--&F.6$F)F)F0F&F))F1F9F)F)*(F1F)--& F.6$F)F9F0F&F)F:F)F)*(F=F)--&F.6$F9F9F0F&F))F:F9F)F)*(#F)\"\"'F))F1\" \"$F)--&F.6%F)F)F)F0F&F)F)**F=F)FBF)--&F.6%F)F)F9F0F&F)F:F)F)**F=F)FMF )F1F)--&F.6%F)F9F9F0F&F)F)*(FOF))F:FRF)--&F.6%F9F9F9F0F&F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "To have a Taylor Approximation at a point the function must be differentiable a t the point." }}{PARA 0 "" 0 "" {TEXT -1 134 "Let's look at some funct ions that are continuous at x=a but not differentiable at x=a. The fi rst is gotten with the MAPLE command " }{XPPEDIT 18 0 "abs " "6#%$a bsG" }{TEXT -1 91 " . Plot the graph of y=abs(x) with x= -10..10 and \+ see if it is a familiar function to you." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(abs(x),x=-10..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$!#5\"\"!$\"#5F*7 $$!3!pmmm\"p0k&*!#<$\"3!pmmm\"p0k&*F07$$!3uKL$3s%HaF0$\"3;mmm\">s%HaF07$$!3]******\\$*4)*\\F0$\"3]******\\$ *4)*\\F07$$!3o******\\_&\\c%F0$\"3o******\\_&\\c%F07$$!3$)******\\1aZT F0$\"3$)******\\1aZTF07$$!3Imm;/#)[oPF0$\"3Imm;/#)[oPF07$$!3%HLLL=exJ$ F0$\"3%HLLL=exJ$F07$$!3lKLLL2$f$HF0$\"3lKLLL2$f$HF07$$!3%)****\\PYx\" \\#F0$\"3%)****\\PYx\"\\#F07$$!3gLLLL7i)4#F0$\"3gLLLL7i)4#F07$$!3n)*** \\P'psm\"F0$\"3n)***\\P'psm\"F07$$!3?****\\74_c7F0$\"3?****\\74_c7F07$ $!3L:LL$3x%z#)!#=$\"3L:LL$3x%z#)Fdr7$$!3')HLL3s$QM%Fdr$\"3')HLL3s$QM%F dr7$$!3>****\\ivF@AFdr$\"3>****\\ivF@AFdr7$$!3\\^omm;zr)*!#?$\"3\\^omm ;zr)*Fds7$$\"3aPL$3-Dg5#FdrFhs7$$\"3eVLLezw5VFdrF[t7$$\"3-.++v$Q#\\\") FdrF^t7$$\"3%\\LL$e\"*[H7F0Fat7$$\"3=++++dxd;F0Fdt7$$\"3e+++D0xw?F0Fgt 7$$\"34,+]i&p@[#F0Fjt7$$\"3++++vgHKHF0F]u7$$\"3ElmmmZvOLF0F`u7$$\"3%4+ ++v+'oPF0Fcu7$$\"3UKL$eR<*fTF0Ffu7$$\"3K-++])Hxe%F0Fiu7$$\"3!fmm\"H!o- *\\F0F\\v7$$\"3X,+]7k.6aF0F_v7$$\"3#emmmT9C#eF0Fbv7$$\"32****\\i!*3`iF 0Fev7$$\"3;NLLL*zym'F0Fhv7$$\"3'eLL$3N1#4(F0F[w7$$\"3,pm;HYt7vF0F^w7$$ \"37-+++xG**yF0Faw7$$\"3gpmmT6KU$)F0Fdw7$$\"3qNLLLbdQ()F0Fgw7$$\"3Z++] i`1h\"*F0Fjw7$$\"3@-+]P?Wl&*F0F]x7$F+F+-%'COLOURG6&%$RGBG$F,!\"\"$F*F* Ffx-%+AXESLABELSG6$Q\"x6\"Q!F[y-%%VIEWG6$;F(F+%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "To see if it is cont inuous at x=0 compute the limit as x->0 and compare the result with ab s(0)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "limit(abs(x),x=0);a bs(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "To see if abs is differentiable at 0 compute the limit of its difference quotient at x=0. " }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "limit((abs(x)-abs(0))/(x-0),x=0);" "6#-%&limitG6$*&,&-% $absG6#%\"xG\"\"\"-F)6#\"\"!!\"\"F,,&F+F,F/F0F0/F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "Try approaching 0 from the left and from \+ the right. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "limit((abs(x )-abs(0))/(x-0), x=0, right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "limit((abs(x)-abs(0)) /(x-0), x=0, left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}} {PARA 0 "" 0 "" {TEXT 263 28 "What happens in each case? " }{TEXT -1 121 "Is this consistent with the above graph near 0? Of course abs(x) is the absolute value of x and is usually denoted by " }{XPPEDIT 18 0 " abs(x)" "6#-%$absG6#%\"xG" }{TEXT -1 5 " . " }}{PARA 0 "" 0 " " {TEXT -1 163 "The slope of the graph computed with 0 and a point to \+ the right of 0 is 1, and computed with 0 and a point to the left is -1 . The one-sided limits are not equal," }}{PARA 0 "" 0 "" {TEXT -1 76 "so the limit does not exist. Thus, the function is not differenti able at 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "For each of these functions, show the function is continuous a t x=0, then test to see which are, and which are not, differentiable \+ at x=0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "abs(x^3) " "6#-%$absG6#*$%\"xG\"\"$" }{TEXT -1 12 " , \+ " }{XPPEDIT 18 0 " abs(sin(x))" "6#-%$absG6#-%$sinG6#%\"xG" }{TEXT -1 9 " , " }{XPPEDIT 18 0 "piecewise(x=0,0,x*sin(1/x))" "6#-%*piece wiseG6%/%\"xG\"\"!F(*&F'\"\"\"-%$sinG6#*&F*F*F'!\"\"F*" }{TEXT -1 7 " \+ , " }{XPPEDIT 18 0 "piecewise(x=0,0,x^2*sin(1/x))" "6#-%*piecewise G6%/%\"xG\"\"!F(*&F'\"\"#-%$sinG6#*&\"\"\"F/F'!\"\"F/" }{TEXT -1 2 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x->abs(x^3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(-%$absG6#*$)9$\"\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(f(x), x=0); f(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "So the function is continuous at 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit((f(x)-f(0))/(x-0),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The function is differentiable at 0 with derivative f'(0)=0." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x->abs(sin(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(-%$absG6#-%$sinG6#9$F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(f(x), x=0); f(0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Again, continuous at \+ 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit((f(x)-f(0))/(x- 0),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Not differentiable at 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f := x-> piecewise(x=0,0,x*sin(1/x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6%/9$\"\"!F1*&F0\"\" \"-%$sinG6#*&F3F3F0!\"\"F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG 6$7$\"\"!/%\"xGF'7$*&F)\"\"\"-%$sinG6#*&F,F,F)!\"\"F,%*otherwiseG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(f(x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "which agrees with the function value at x=0, so its continuous at \+ x =0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "limit((f(x)-0)/(x- 0), x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#;!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "We saw this before. Limit does not exist , so it's not differentiable at 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f := x -> \npiecewise(x=0 ,0,x^2*sin(1/x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(-%*piecewiseG6%/9$\"\"!F1*&)F0\"\"#\"\"\"-%$s inG6#*&F5F5F0!\"\"F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$\"\"!/%\" xGF'7$*&)F)\"\"#\"\"\"-%$sinG6#*&F.F.F)!\"\"F.%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(f(x),x=0);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "So, it' s continuous at 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit ((f(x)-0)/(x-0),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "So it's differentiable at 0 with f '(0)=0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 120 "We finish by looking at a function that is differentiable but not twice differentiable at x=0. Consider the function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 260 "" 0 "" {XPPEDIT 18 0 "piecewise(x <= 0,x^2,x^3); " "6#-%*piecewiseG6%1%\"xG\"\"!*$F'\"\"#*$F'\"\"$" }{TEXT -1 6 " . \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f := x -> piecewise(x <= 0,x^2,x^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%) operatorG%&arrowGF(-%*piecewiseG6%19$\"\"!*$)F0\"\"#\"\"\"*$)F0\"\"$F5 F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$*$)%\"xG\"\"#\"\"\"1F)\"\"!7$* $)F)\"\"$F+%*otherwiseG" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Plot it for x =-2..2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(f, -2..2); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$ 7W7$$!\"#\"\"!$\"\"%F*7$$!3MLLL$Q6G\">!#<$\"3A!e4#)QZ)eOF07$$!3amm;M! \\p$=F0$\"3&*[e7a'***!#=$ \"3mq\"))\\%))R#***Fbo7$$!3E++++0\"*H\"*Fbo$\"3ND5!QdEbL)Fbo7$$!35++++ 83&H)Fbo$\"3yo4Oxt$3)oFbo7$$!3[LLL3k(p`(Fbo$\"3ZBt(zL,1o&Fbo7$$!3Anmmm j^NmFbo$\"3%37I_u2IS%Fbo7$$!3(zmmmYh=(eFbo$\"38$[s$3d(yW$Fbo7$$!3+,++v #\\N)\\Fbo$\"3W.`jPjd$[#Fbo7$$!3commmCC(>%Fbo$\"3&*4!*RKWoh6\"Fbo7$$!3t*****\\#=/8DFbo$\"3CH$*>9#z`J'! #>7$$!3u#Fbr7$$!3jomm;Wn(o)Fbr$\"37a/+x'ova (!#?7$$!3IqLLL$eV(>F]s$\"3gBHSG34)*Q!#B7$$\"3)Qjmm\"f`@')Fbr$\"351\\2^ OY3k!#@7$$\"3%z****\\nZ)H;Fbo$\"3hd6ewH`HVF]s7$$\"3bkmm;$y*eCFbo$\"3FK peC&Ro[\"Fbr7$$\"3f)******R^bJ$Fbo$\"3=*)f[FiuWOFbr7$$\"3&e*****\\5a`T Fbo$\"3A;jiN)[c;(Fbr7$$\"3&o****\\7RV'\\Fbo$\"3EEA&\\kWMA\"Fbo7$$\"3X' *****\\@fkeFbo$\"3$)Q^.Y].$)Fbo$\"3F#[#H.1'* edFbo7$$\"3L*******pfa<*Fbo$\"3-,t\"pLLZs(Fbo7$$\"38HLLeg`!)**Fbo$\"3q t$pZv@<%**Fbo7$$\"3v****\\#G2A3\"F0$\"3:W#p#>:Xn7F07$$\"3:LLL$)G[k6F0$ \"3J$oa%GD1z:F07$$\"3\")****\\7yh]7F0$\"3$GjvwUAg&>F07$$\"3wmmm')fdL8F 0$\"3BI$=qvk;P#F07$$\"3bmmm,FT=9F0$\"3gJ41Wop`GF07$$\"3FLL$e#pa-:F0$\" 3%*zojc4A#R$F07$$\"3*)******Rv&)z:F0$\"3O/Fw%=XK%RF07$$\"3HLLLGUYo;F0$ \"31WK8saiWYF07$$\"3\"*****\\n'*33UY(zu]hF07$$\"33+++S2ls=F0$\"3P7P(\\b]qc'F07$$\"34++]2%)38>F 0$\"3ik^*=9E<+(F07$$\"3/++v.Uac>F0$\"39g+ " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 58 "Show that it's differentiable at x=0 with derivative =0. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "li mit((f(x)-f(0))/(x-0), x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{PARA 0 "" 0 "" {TEXT -1 20 "Plot its derivative." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fp := x->piecewise(x<0,2*x, x=0,0, \+ 3*x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fpGf*6#%\"xG6\"6$%)opera torG%&arrowGF(-%*piecewiseG6'29$\"\"!,$F0\"\"#/F0F1F1,$*$)F0F3\"\"\"\" \"$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(fp(x), x= -2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CU RVESG6$7U7$$!\"#\"\"!$!\"%F*7$$!3MLLL$Q6G\">!#<$!3ommmmFiDQF07$$!3amm; M!\\p$=F0$!34LLLo!)*Qn$F07$$!3LLLL))Qj^1DBF07$$!3KLLLQW*e3\"F0$!3jmmmw))yr@F07$$!3w++++()>'***!#=$!3;+++S( R#**>F07$$!3E++++0\"*H\"*Fbo$!30++++@)f#=F07$$!35++++83&H)Fbo$!3-+++gi ,f;F07$$!3[LLL3k(p`(Fbo$!3qmmm\"G&R2:F07$$!3Anmmmj^NmFbo$!3WLLLtK5F8F0 7$$!3(zmmmYh=(eFbo$!3eLLL$HsV<\"F07$$!3+,++v#\\N)\\Fbo$!3+-++]&)4n**Fb o7$$!3commmCC(>%Fbo$!37PLLL\\[%R)Fbo7$$!39*****\\FRXL$Fbo$!3F)*****\\& y!pmFbo7$$!3t*****\\#=/8DFbo$!3Y******\\O3E]Fbo7$$!3$!3sLLL$)[`P!#?$!3gSnmmmr[RFes7$$\"3V[mmT+07UFjr$\"3%R?^g m4CK&Fes7$$\"3)Qjmm\"f`@')Fjr$\"3C$=roWE*HAFjr7$$\"3%z****\\nZ)H;Fbo$ \"3f(p36L5#pzFjr7$$\"3bkmm;$y*eCFbo$\"3-Z5b3B(R\"=Fbo7$$\"3f)******R^b J$Fbo$\"3/&e7eKkyH$Fbo7$$\"3&e*****\\5a`TFbo$\"3`>0@w4dv^Fbo7$$\"3&o** **\\7RV'\\Fbo$\"3m?# G/$)Fbo$\"3(H\")=)G&*ew?F07$$\"3L*******pfa<*Fbo$\"3 WA(*=@=nDDF07$$\"38HLLeg`!)**Fbo$\"3b`2N+IL))HF07$$\"3v****\\#G2A3\"F0 $\"3y-\"))o!y^8NF07$$\"3:LLL$)G[k6F0$\"3>S*Gn:h!oSF07$$\"3\")****\\7yh ]7F0$\"3;ao#)QZ8#p%F07$$\"3wmmm')fdL8F0$\"3'3#>kOZFN`F07$$\"3bmmm,FT=9 F0$\"3G'*zuwPoNgF07$$\"3FLL$e#pa-:F0$\"3)3g$*HzTHx'F07$$\"3*)******Rv& )z:F0$\"3s`X3S&\\y[(F07$$\"3HLLLGUYo;F0$\"3-OQoV'=8N)F07$$\"3_mmm1^rZ< F0$\"3fKE@#GCN;*F07$$\"34++]sI@K=F0$\"3Dn>\"HU,r+\"!#;7$$\"34++]2%)38> F0$\"3!fEtk " 0 "" {MPLTEXT 1 0 17 "limit(fp(x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "which shows that f' is continous \+ at 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit((fp(x)-fp(0) )/(x-0),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "limit((fp(x)-fp(0))/(x-0),x= 0, left); limit((fp(x)-fp(0))/(x-0),x=0, right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "The two-sided limit does not exist , so fp is not differentiable at 0." }}}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{MARK "307 0 0" 69 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }