{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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