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{SECT 0 {PARA 256 "" 0 "" {TEXT 256 21 "Calulus Worksheet II\n" }}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 "Limits and rate of change:" }}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 69 "Find an equation of the tangent l
ine to parabola at the point P(1,1):" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "f:=(x)->x^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "s:=(x
)->(x^2-1)/(x-1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "seq(s(
1+0.1*(10-k)),k=0..9):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "s
(1+0.00006):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m:=limit(s(
x),x=1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot([x^2,m*(x-
1)+f(1),s(1.5)*(x-1)+f(1)],x=0..2,\ncolor=[red,blue,magenta]);" }}}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "The limit of a function:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=(x)->x^2-x+2:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(f(x),x=2):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "limit((x-1)/(x^2-1),x=1):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(x=1.0001,(x-1)/(x^2-1)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(subs(x=0.02,sin(x)/x))
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(sin(x)/x,x=-1..1)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(sin(1/x),x=-1..1)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(x*sin(1/x),x=-1..
1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(x*sin(1/x),x=-0
.2..0.2,\nnumpoints=1500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "plot([x*sin(1/x),x,-x],x=-0.2..0.2,color=[red,blue,blue]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([x^2*sin(1/x),x^2,-x^2]
,x=-0.2..0.2,color=[red,blue,blue]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "limit(1/x,x=0):" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 79 "The derivative: \nCalling sequences:\n1.  diff(f(x),x):\n2. Dif
f(f(x),x):\n3. D(f):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "diff
((x^3-1)/(x^2+x+1),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "d
iff(1/(1+x^2),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Diff(1
/(1+x^2),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Diff(1/(1+x
^2),x) = diff(1/(1+x^2),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "f:=(x)->1/(1+x^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(
f):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(f(x)*g(x),x):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(f(x)/g(x),x):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(f*g):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 7 "D(1/f):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "D(sin):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(cos(
x),x):" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "The student package:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}{PARA 
7 "" 1 "" {TEXT -1 29 "Warning, new definition for D" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 46 "showtangent(1/(x^2+1),x=3,x=-5..5,color=b
lue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "movetan:=a->showtangent(1/(x
^2+1),x=a,x=0..5,\ncolor=[blue,red]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "a:=display(seq(movetan(t/20),t=0..100),view=[0..5,-1.
.1.5],\ninsequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
48 "b:=plot(diff(1/(1+x^2),x),x=0..5,color=magenta):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 110 "display(\{a,b\},\ntitlefont=[TIMES, BOLD
,18],\ntitle=`Graph of 1/(1+x^2) shown in red its derivative in magent
a`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Trigonometry:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "a:=plot(\{[[0,0],[cos(Pi/3)
,0]],\n[[cos(Pi/3),0],[cos(Pi/3),sin(Pi/3)]],\n[[cos(Pi/3),sin(Pi/3)],
[0,0]]\},\ncolor=aquamarine, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "b:=plot([cos(t),sin(t),t=0..2*Pi], color=red, thick
ness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "c:=textplot(\{
[0.1513, 0.08068,`x`],\n[0.2477, -0.09914,`cos(x)`],\n[0.6715, 0.3376,
`sin(x)`],\n[0.2016, 0.478,`1`]\},\nfont=[TIMES,BOLD,18]):" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 53 "display(\{a,b,c\},tickmarks=[0,0],scaling=c
onstrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "d:=animate(
[cos(x*t),sin(x*t),x=0..1],t=0..2*Pi,\ncolor=red,frames=40):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "e:=animate([x*cos(t),x*sin(t
),x=0..1],t=0..2*Pi,\ncolor=blue,frames=40):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 69 "f:=animate([x*t,sin(x*t),x=0..1],t=0..2*Pi,\ncol
or=magenta,frames=40):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "d
isplay(\{d,e,f\},\nscaling=constrained, thickness=2, view=[-1..2*Pi,-1
..1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "g:=animate([t-cos
(t)+cos(x*t),sin(x*t),x=0..1],t=0..2*Pi,\ncolor=red,frames=40):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "h:=animate([t-cos(t)+x*cos(t
),x*sin(t),x=0..1],t=0..2*Pi,\ncolor=blue,frames=40):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "display(\{f,g,h\},\nscaling=constra
ined, thickness=2, view=[-1..2*Pi+1,-1..1]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Maximu
m and minimum values:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "wit
h(student):with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "
f:=(x)->3*x^4-16*x^3+18*x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG
:6#%\"xG6\"6$%)operatorG%&arrowGF(,(*$9$\"\"%\"\"$*$F.F0!#;*$F.\"\"#\"
#=F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "minimize(f(x),x,
\{x=-1..1\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(f(x)=0,x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6&\"\"!F#,&#\"\")\"\"$\"\"\"*$\"#5#F(\"\"##F(F',&F%F(F)
#!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "minimize(f(x),x
,\{x=-1..4\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#F" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(f(x)=-27,x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6&,&#!\"\"\"\"$\"\"\"*&%\"IGF'\"\"##F'F*#F*F&,&F$F'F
(#!\"#F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "maximize(f(
x),x,\{x=-0.2..1\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(f(x)=5,x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6&,&#\"\"&\"\"$\"\"\"*$\"#5#F'\"\"##F+F&,&F$F'
F(#!\"#F&F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "maximize(f
(x),x,\{x=-1..4\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(diff(f(x),x)=0,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!\"\"\"\"\"$" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 38 "M:=(a)->showtangent(f(x),x=a,x=-1..4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG:6#%\"aG6\"6$%)operatorG%&arrowG
F(-%,showtangentG6%-%\"fG6#%\"xG/F29$/F2;!\"\"\"\"%F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display([M(0),M(1),M(3)],\nthicknes
s=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f(0):f(1):f(3):" }
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "The mean value theorem:" }}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 81 "`mv' denotes the set of points at
 wich the derivative is the slope of the secant:" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 55 "mv:=(g,a,b)-> fsolve(diff(g(x),x)=(g(b)-g(a))/
(b-a),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mvG:6%%\"gG%\"aG%\"bG6
\"6$%)operatorG%&arrowGF*-%'fsolveG6$/-%%diffG6$-9$6#%\"xGF8*&,&-F66#9
&\"\"\"-F66#9%!\"\"F>,&F=F>FAFBFBF8F*F*" }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 72 "This routine plots the graph, the secant and all parallel
 tangent lines:" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 27 "The first choi
ce for f,c,d:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=(x)->3*x
^4-16*x^3+18*x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"
6$%)operatorG%&arrowGF(,(*$9$\"\"%\"\"$*$F.F0!#;*$F.\"\"#\"#=F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "c:=-1:d:=4:" }}}}{SECT 0 
{PARA 5 "" 0 "" {TEXT -1 12 "The routine:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "e:=[mv(f,c,d)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"eG7#$\"+Du(f&H!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(
e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "m:=D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"m
G:6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&,&\"\"\"F/*$9$\"\"#F/!\"#F1F/F3
F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "plot(\{f(x),f(c)+(
f(d)-f(c))/(d-c)*(x-c),\nseq(f(op(i,e))+m(op(i,e))*(x-op(i,e)),i=1..no
ps(e))\},\nx=c..d, \nthickness=2);" }}}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 41 "Clear f,c,d,e and reassign new variables:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:=`f`:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "f:=(x)->1/(1+x^2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "c:=`c`:d:=`d`:e:=`e`:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "c:=0:d:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}}}{MARK "0 0" 20 }{VIEWOPTS 1 1 0 3 2 1804 }