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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "1.4  #8" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "de:=diff(y(x),x)=1-sin(y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dsolve(\{de,y(0)=0\},y(x));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(y(x)=rhs(\"),de);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(\",trig);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "2*sin(arctan(x/(x+2)))*cos(arctan(x
/(x+2)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(1-\")
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "1.4  #12  Refer to Example 2
.   It refers to the IVP " }{XPPEDIT 18 0 "diff(y(t),t)=y(t), y(0)=1" 
"6$/-%%diffG6$-%\"yG6#%\"tGF*-F(6#F*/-F(6#\"\"!\"\"\"" }{TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 32 "The approximation with stepsize " }
{XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 7 " gives " }
{XPPEDIT 18 0 "y[n]=(1+1/n)^n" "/&%\"yG6#%\"nG),&\"\"\"\"\"\"*&\"\"\"F
*F&!\"\"F*F&" }{TEXT -1 36 ".  (I'll do this on the whiteboard.)" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "E:=exp(
1): limit((1+1/n)^n,n=infinity);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "\"-E;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "1.4, #13
  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "limit((E-(1+1/n)^n)/(
1/n),n=infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Problems for
 you:  " }}{PARA 0 "" 0 "" {TEXT -1 19 "1.4           5, 12" }}{PARA 
0 "" 0 "" {TEXT -1 33 "2.1, 2.2    3, 12, 23, 27, 29, 32" }}}}{MARK "0
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