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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 31 "Solutions and answers to
 Quiz 1" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 53 "Q.1. Find the derivat
ives of the following functions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 257 "" 0 "" {TEXT -1 2 "a)" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "f:= x -> exp(sqrt(x));" }{TEXT -1 0 "" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#
-%%sqrtG6#9$F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 
5 "" 0 "" {TEXT -1 8 "Solution" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 10 "u=sqrt(x);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%\"uG*$%\"xG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 19 "Use the Chain Rule:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "Diff(f(x), x) = Diff(exp(u), u) * Diff(sqrt(x), x);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$expG6#*$%\"xG#\"\"\"\"
\"#F+*&-F%6$-F(6#%\"uGF4F--F%6$F*F+F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 257 7 "Answer:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "Diff(f(x), x) = diff(f(x), x);" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%D
iffG6$-%$expG6#*$%\"xG#\"\"\"\"\"#F+,$*&F,F-*&F+#!\"\"F.F'F-F-F-" }}}
{EXCHG {PARA 258 "" 0 "" {TEXT -1 20 "____________________" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "b)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "g:= x-> ln((x+1)/(x-1))^(3/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)-%#lnG6#*&,&9$\"\"\"F4F4F4
,&F3F4F4!\"\"F6#\"\"$\"\"&F4F(F(F(" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 
-1 8 "Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Use laws of logs
 first:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "g(x)=(3/5)*\{ln(
x+1) - ln(x-1)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)-%#lnG6#*&,&%
\"xG\"\"\"F,F,F,,&F+F,F,!\"\"F.#\"\"$\"\"&F,,$*&F/F,<#,&-F'6#F*F,-F'6#
F-F.F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "And now differentiate
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Diff(rhs(%), x) = diff
(rhs(%), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,$*&#\"\"$
\"\"&\"\"\"<#,&-%#lnG6#,&%\"xGF,F,F,F,-F06#,&F3F,F,!\"\"F7F,F,F3,$*&F)
F,<#,&*&F,F,F2F7F,*&F,F,F6F7F7F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
258 7 "Answer:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff(g(x)
, x)= simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$
*$)-%#lnG6#*&,&%\"xG\"\"\"F/F/F/,&F.F/F/!\"\"F1#\"\"$\"\"&F/F.,$*&F2F/
<#,$*&\"\"#F/,&*$)F.F:F/F/F/F1F1F1F/F/" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 20 "____________________" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "c)" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "h:=x-> log[3](x^2+3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-&%$logG6#\"\"$6#,&*$)9
$\"\"#\"\"\"F7F0F7F(F(F(" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Solut
ion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Convert to ln first:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "'h(x)'= h(x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG*&-%#lnG6#,&*$)F'\"\"#\"\"\"F0\"\"
$F0F0-F*6#F1!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Now differen
tiate:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Diff(h(x), x) = d
iff(h(x), x);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 7 "Answer:" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&-%#lnG6#,&*$)%\"xG\"\"#
\"\"\"F0\"\"$F0F0-F)6#F1!\"\"F.,$**F/F0F.F0F+F4F2F4F0" }}}}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 28 "Q.2. Evaluate the integrals:" }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "Val:= Int(1/(2*x+3), x=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$ValG-%$IntG6$*&\"\"\"F),&*&\"\"#F)%\"xGF)F)\"\"$F)!\"\"/F-;\"\"!
F." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "Solution" }}{SECT 1 {PARA 
20 "" 0 "" {TEXT 260 13 "First Method:" }{TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 11 "Substitute:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "u:=2*x+3;    du = diff(u, x)*dx; dx = 1/2*\{du\};" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG,&*&\"\"#\"\"\"%\"xGF(F(\"\"$F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#duG,$*&\"\"#\"\"\"%#dxGF(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#dxG,$*&#\"\"\"\"\"#F(<#%#duGF(F(" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Evaluate the new bounds of integ
ration:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "lo:=eval(u, x=0)
;   up:=eval(u, x=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#loG\"\"$" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#upG\"\"*" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 32 "Then evaluate the integral in u:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 33 "Val= 1/2*Int(1/'u', 'u'=lo..up); " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&*&\"\"#F(%\"xGF(F(\"\"
$F(!\"\"/F,;\"\"!F-,$*&#F(F+F(-F%6$*&F(F(%\"uGF./F8;F-\"\"*F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Use the 1st FTC  (the first Fundam
ental Theorem of Calculus ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 79 "1/2*Int(1/'u', 'u'=3..9)=1/2*\{eval(int(1/t, t), t=9) - eval(int
(1/t, t), t=3)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"\"#
F'-%$IntG6$*&F'F'%\"uG!\"\"/F-;\"\"$\"\"*F'F',$*&F&F'<#,&-%#lnG6#F2F'-
F86#F1F.F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Val=simplif
y(rhs(%));" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 7 "Answer:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&*&\"\"#F(%\"xGF(
F(\"\"$F(!\"\"/F,;\"\"!F-,$*&#F(F+F(<#-%#lnG6#F-F(F(" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 23 "_______________________" }}}}{SECT 1 {PARA 20 "
" 0 "" {TEXT 262 14 "Second Method:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "Integral:= Int(1/(2*x+3), x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)IntegralG-%$IntG6$*&\"\"\"F),&*&\"\"#F)%\"xGF)F)\"\"
$F)!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Since the denominat
or is linear we know it will be a ln-type antyderivative:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Integral = int(1/(2*x+3), x) + C;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&*&\"\"#F(%\"xG
F(F(\"\"$F(!\"\"F,,&*&#F(F+F(-%#lnG6#F)F(F(%\"CGF(" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 47 "So evaluate this integal at the original bounds" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Val = eval(int(1/(2*x+3),
 x), x=3) - eval(int(1/(2*x+3), x), x=0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&*&\"\"#F(%\"xGF(F(\"\"$F(!\"\"/F,
;\"\"!F-,&*&#F(F+F(-%#lnG6#\"\"*F(F(*&#F(F+F(-F66#F-F(F." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 9 "Simplify:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "Val = simplify(rhs(%));" }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 263 7 "Answer:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6
$*&\"\"\"F(,&*&\"\"#F(%\"xGF(F(\"\"$F(!\"\"/F,;\"\"!F-,$*&#F(F+F(-%#ln
G6#F-F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "____________________
___" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 2 "b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Integral:=
 Int((exp(x)-1)/exp(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Inte
gralG-%$IntG6$*&,&-%$expG6#%\"xG\"\"\"F.!\"\"F.F*F/F-" }}}{SECT 0 
{PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 52 "Split the integrand into two fractions and simplify:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Integral = Int((1-1/exp(x)),
 x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,&-%$expG6#%\"xG\"
\"\"F-!\"\"F-F)F.F,-F%6$,&F-F-*&F-F-F)F.F.F," }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "Int((1-1/exp(x)), x) = Int(1-exp(-x), x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,&\"\"\"F(*&F(F(-%$expG6#%\"
xG!\"\"F.F--F%6$,&F(F(-F+6#,$F-F.F.F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 42 "Now by integrating each summand we obtain:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 264 7 "Answer:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "Integral= int(1-exp(-x), x) + C;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&,&-%$expG6#%\"xG\"\"\"F-!\"\"F-F)F.F,,(F,F-
-F*6#,$F,F.F-%\"CGF-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "_________
______________" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{MARK "2 2 0 0
" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }