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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT -1 19 "An Introduction to " }
{TEXT 256 5 "Maple" }{TEXT -1 22 " - Calculus Operations" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT -1 28 "Bob Bradshaw, Ohlone College" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;with(plots):\n" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 300 11 "Derivatives" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 316 "One of the uses of calculus is to determine the e
xtreme points of a curve. While a graph can be used to estimate the hi
ghest and lowest points, calculus can be used to determine the values \+
to any desired degree of accuracy. \n\nSuppose you want to determine t
he maximum and minimum values of the the following equation" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 14 "y:=x^3-15*x+8;" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 308 "\nThe first step is to graph the equation and get an e
stimate of the extrema. After the equation has been plotted, click on \+
the graph and move the mouse to the maximum point and click again. In \+
the upper left corner of the screen, you will see the coordinates of t
he point where you clicked. You should get a " }{TEXT 301 1 "y" }
{TEXT -1 27 " value of approximately 30." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "plot(y,x=-6..6);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
153 "\nTo use calculus to determine the extreme points, first take the
 derivative of the equation using the diff command. The structure of t
he command is diff(" }{TEXT 302 3 "eqn" }{TEXT -1 2 ", " }{TEXT 303 8 
"variable" }{TEXT -1 3 ").\n" }{MPLTEXT 1 0 14 "dy:=diff(y,x);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "\nWe can also take the second deri
vative using diff(" }{TEXT 304 3 "eqn" }{TEXT -1 1 "," }{TEXT 305 10 "
variable$2" }{TEXT -1 1 ")" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ddy:=
diff(y,x$2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "\nThe critical po
ints are found by setting the first derivative equal to zero." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "CritPt:=solve(dy=0,x);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 81 "\nTo get the decimal equivalent of these \+
critical points, use the \"evalf\" command." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "evalf(CritPt);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
18 "\nTo determine the " }{TEXT 306 1 "y" }{TEXT -1 22 " value for a v
alue of " }{TEXT 307 1 "x" }{TEXT -1 34 ", use the substitute command \+
subs(" }{TEXT 308 21 "x=value, y expression" }{TEXT -1 23 "). Since th
ere are two " }{TEXT 309 1 "x" }{TEXT -1 74 " values, we pick only one
 at a time by using CritPt[1] and then CritPt[2]." }}{PARA 0 "" 0 "" 
{TEXT -1 146 "Using the first value, we can substitute it into the exp
ression for the second derivative to determine if the point is a maxim
um or minimum value." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 
32 "concave1:=subs(x=CritPt[1],ddy);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 107 "\nSince the second derivative is positive, CritPt[1] must be a
 relative minimum.\n\nWe can also determine the " }{TEXT 310 1 "y" }
{TEXT -1 27 " value with a substitution." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "ycrit1:=subs(x=CritPt[1],dy);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 58 "\nIf you prefer a decimal value, include the evalf c
ommand." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ycrit1:=evalf(subs(x=Cri
tPt[1],y));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "\n\nWe can repeat \+
the process for the second critical value." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "concave2:=subs(x=CritPt[2],ddy);\nycrit2:=evalf(subs(
x=CritPt[2],y));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "\nIn summary,
 we have " }{TEXT 311 2 "\nx" }{TEXT -1 9 " = 2.23, " }{TEXT 312 1 "y
" }{TEXT -1 38 " = -14.36 is the relative minimum and\n" }{TEXT 313 1 
"x" }{TEXT -1 9 "= -2.23, " }{TEXT 314 1 "y" }{TEXT -1 131 " = 30.36 i
s the relative maximum.\n\nWe can redraw the graph and verify the info
rmation. We can also label those points on the graph." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 424 "toppoint:=pointplot([-2.23,30.36],
color=black,symbol=CIRCLE,symbolsize=18):\nbottompoint:=pointplot([2.2
3,-14.36],color=green,symbol=CIRCLE,symbolsize=18):\nt1:=textplot([2.2
3,-14.36,\"Relative Minimum at (2.23, -14.36)\"],align=BELOW,color=gre
en):\nt2:=textplot([-2.23,32,\"Relative Maximum at (-2.23, 30.36)\"],a
lign=ABOVE,color=black):\np1:=plot(y,x=-6..6):\ndisplay(p1,t1,t2,toppo
int,bottompoint,axes=boxed,view=[-7..7,-20..35]);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "We can add the tan
gent to the function at a point, say " }{TEXT 315 1 "x" }{TEXT -1 25 "
 = 1.5. First substitute " }{TEXT 316 1 "x" }{TEXT -1 12 " = 1.5 into \+
" }{TEXT 317 1 "y" }{TEXT -1 5 " and " }{TEXT 318 2 "y'" }{TEXT -1 49 
"  and then find the equation of the tangent line." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 66 "xpoint:=1.5;\nypoint:=subs(x=xpoint,y);\nmy
slope:=subs(x=xpoint,dy);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "myline:=myslope*(x-xpoint)+ypoint;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "mylinegraph:=plot(myline,x=-5..5,color=blue):" }}}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "display(p1,t1,t2,toppoint,bottompoi
nt,mylinegraph,axes=boxed,view=[-5..5,-20..35]);" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT 319 6 "Series" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 320 6 "Maple " }{TEXT -1 40 "can create Taylor se
ries around a point " }{TEXT 321 1 "c" }{TEXT -1 65 ". For example, to
 get the first 5 terms of the Taylor series for " }{XPPEDIT 18 0 "1/(1
+x);" "6#*&\"\"\"F$,&F$F$%\"xGF$!\"\"" }{TEXT -1 4 " at " }{TEXT 322 
1 "c" }{TEXT -1 10 " = 1, use " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "myseries:=series(1/(1+x),x=1,5);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 53 "To eliminate the error term, use the convert command." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "convert(myseries,polynom);
" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 15 "Sum of a Series" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 323 5 "Maple" }{TEXT -1 57 " can \+
determine the value of summations using the command " }{TEXT 324 32 "s
um(expression, x = start..end)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
g:=sum(1/x,x=1..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "\nIf you w
ant to see the summation sign, use an upper case \"S\" in the command.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "g:=Sum(1/x,x=1..100);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "\nTo create an equation, use the c
ommand twice, the first time using an upper case letter." }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "Sum(1/x,x=1..100)=sum(1/x,x=1..100);" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 325 11 "Integration" }{TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Maple can perform integration usi
ng the int command. Using the structure int(expression, variable) give
s an indefinite integral." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "int(x^2+5*x,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "\nUsing th
e structure " }{TEXT 329 39 "int(expression, variable = start..end) " 
}{TEXT -1 47 "gives a definite integral. As is true with the " }{TEXT 
330 3 "sum" }{TEXT -1 100 " command, using an upper case letter simply
 creates the symbols but does not perform a calculation.\n" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 43 "Int(exp(2*x),x=0..3):=int(exp(2*x),x=0..3);
" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 331 11 "You Try It!" }{TEXT -1 1 "
\n" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "(1)  Determine the maximum a
nd minimum points of the function " }{TEXT 334 2 " y" }{TEXT -1 3 " = \+
" }{XPPEDIT 18 0 "3*x^4+4*x^3-4*x^2-8*x+2;" "6#,,*&\"\"$\"\"\"*$)%\"xG
\"\"%F&F&F&*&F*F&*$)F)F%F&F&F&*&F*F&*$)F)\"\"#F&F&!\"\"*&\"\")F&F)F&F2
F1F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 78 "(2)  Create a gra
ph of the function that has the extrema labeled on the graph." }}
{PARA 0 "" 0 "" {TEXT -1 43 "(3)  Draw the tangent line to the graph a
t " }{TEXT 332 1 "x" }{TEXT -1 4 " = 2" }}{PARA 0 "" 0 "" {TEXT -1 28 
"(4)  Determine the value of " }{XPPEDIT 18 0 "sum(1/x,x = 1 .. 10);" 
"6#-%$sumG6$*&\"\"\"F'%\"xG!\"\"/F(;F'\"#5" }}{PARA 0 "" 0 "" {TEXT 
-1 52 "(5)  Determine the value of the following integrals\n" }
{XPPEDIT 18 0 "int(x^3,x = 1 .. 5);" "6#-%$intG6$*$%\"xG\"\"$/F';\"\"
\"\"\"&" }{TEXT -1 23 ",                      " }{XPPEDIT 18 0 "int(1/
sqrt(9-4*x^2),x = 0 .. 1)" "6#-%$intG6$*&\"\"\"F'-%%sqrtG6#,&\"\"*F'*&
\"\"%F'*$%\"xG\"\"#F'!\"\"F2/F0;\"\"!F'" }{TEXT -1 22 ",              \+
       " }{XPPEDIT 18 0 "int(sqrt(x^2+9),%? = 0 .. 1+sqrt(w));" "6#-%$
intG6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"\"\"*F-/%#%?G;\"\"!,&F-F--F'6#%\"w
GF-" }}}}}{MARK "5 0 0" 4 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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