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{PARA 0 "" 0 "" {TEXT -1 17 "This is necessary" }}{PARA 7 "" 1 "" 
{TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 58 "Here is one way to \+
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*oF^qF_gn7%$\"+E5N*z\"F4$!+LQs#3(F^qF^jl7%$\"+pBqY=F4$!+-86psF^qF[hn7%
$\"+7P0%*=F4$!+s()\\buF^qFbhn7%$\"+b]ST>F4$!+Ti)=k(F^qFihn7%$\"+(Rc())
>F4$!+5PFGyF^qF\\[m7%$\"+Sx5O?F4$!+!=hY,)F^qFein7%$\"+$3fM3#F4$!+\\'[5
?)F^qF\\jn7%$\"+E/\"38#F4$!+=hV(Q)F^qFcjn7%$\"+p<;y@F4$!+)eBQd)F^qFj[m
7%$\"+6J^DAF4$!+d5@g()F^qF_[o7%$\"+aW'GF#F4$!+E&)fY*)F^qFa\\m7%$\"+'z:
-K#F4$!+\"*f)H8*F^qF[\\o-%*AXESSTYLEG6#%$BOXG" 1 2 0 1 10 0 2 1 1 2 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Here is the derivative of r
(t), note that D(r) does not work." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "rprime:= diff(r(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'rprim
eG7%-%$cosG6#%\"tG,$-%$sinGF(!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 53 "Suppose I want to evaluate rprime at the point t = 1." }}
{PARA 0 "" 0 "" {TEXT -1 65 "Since rprime is not a function I must eva
luate by brute force ..." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "rprimeo
fone:= evalf(([cos(1), -sin(1), 1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%,rprimeofoneG7%$\"+fI-.a!#5$!+[)4ZT)F($\"\"\"\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 18 "or redefine rprime" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "rprime:= t -> [cos(t), -sin(t), 1];
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'rprimeGf*6#%\"tG6\"6$%)operator
G%&arrowGF(7%-%$cosG6#9$,$-%$sinGF/!\"\"\"\"\"F(F(F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(rprime(1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%$\"+fI-.a!#5$!+[)4ZT)F&$\"\"\"\"\"!" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 77 "Either way the result is the sam
e, now I compare it to a rough approximation." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalf(((r(1.25) - r
(1))/(1.25 - 1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"*RX0I%!\"*$
!*u(>***)F&$\"+++++5F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "A good \+
enough approximation to the derivative for some purposes." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{MARK "6 0 0" 53 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }