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{SECT 0 {PARA 258 "" 0 "" {TEXT -1 43 "CONSTRUCTING A FUNCTION FROM IT
S DERIVATIVE" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 
-1 18 "Calculus I Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 256 11 "Objectives:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 " To construct a fu
nction numerically and graphically from its derivative." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT 258 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 83 " To use a computer algebra system to construct a table fo
r a function and its graph" }}{PARA 0 "" 0 "" {TEXT -1 30 " when the d
erivative is given." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 
259 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 8 "For f '(" }
{TEXT 266 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "1/(x^2+1);" "6#*&\"
\"\"F$,&*$%\"xG\"\"#F$F$F$!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 9 "f(0) = \+
2," }}{PARA 0 "" 0 "" {TEXT -1 18 "do the following :" }}{PARA 0 "" 0 
"" {TEXT -1 44 "a)  Construct a table of ten values for  f (" }{TEXT 
267 1 "x" }{TEXT -1 16 ") on [-5, 5].   " }}{PARA 0 "" 0 "" {TEXT -1 
28 "b)  Plot the graphs of  f '(" }{TEXT 268 1 "x" }{TEXT -1 10 ") and
  f (" }{TEXT 269 1 "x" }{TEXT -1 13 ") on [-5, 5]." }}{PARA 0 "" 0 "
" {TEXT -1 103 "c) Verify that the graph of  f '  is indeed the graph \+
of the derivative of the constructed finction  f." }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT -1 1 " " }{TEXT 260 9 "Solution:" }{TEXT 261 1 " " }}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 263 42 "a) Construct a t
able of ten values for  f(" }{TEXT 271 1 "x" }{TEXT 272 37 ")  from it
s derivative function  f '(" }{TEXT 264 1 "x" }{TEXT 270 2 ")." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "   To con
struct a table we first define the derivative function.  We make a tab
le using" }}{PARA 0 "" 0 "" {TEXT -1 5 "  a  " }{TEXT 275 9 "For .. do
" }{TEXT -1 26 " loop.  Note the syntax.  " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "fprime := x -> 1/(x^2 + 1);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'fprimeGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-
,&F-F-*$)9$\"\"#F-F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "fRight:= n -> fR
ight(n-1)+fprime(n-1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'fRightG
f*6#%\"nG6\"6$%)operatorG%&arrowGF(,&-F$6#,&9$\"\"\"F1!\"\"F1-%'fprime
GF.F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fRight(0):= \+
2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%'fRightG6#\"\"!\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "L:= k -> [k, fRight(k)];  \n
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGf*6#%\"kG6\"6$%)operatorG%&a
rrowGF(7$9$-%'fRightG6#F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 43 "for k from 1 by 1 to 5 do print (L(k)) od;\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#7$\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$
\"\"##\"\"(F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"$#\"#P\"#5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"%#\"#>\"\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$\"\"&#\"$G$\"#&)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "fLeft:=
 k -> fLeft(k+1) - fprime(k);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
fLeftGf*6#%\"kG6\"6$%)operatorG%&arrowGF(,&-F$6#,&9$\"\"\"F1F1F1-%'fpr
imeG6#F0!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fLef
t(0):= 2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&fLeftG6#\"\"!\"\"#
" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "L:= k -> [k,fLeft(k)];  \n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"LGf*6#%\"kG6\"6$%)operatorG%&arrowGF(7$9$-%&fLeftG6
#F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "for k from -1 \+
by -1 to -5 do print (L(k)) od;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7
$!\"\"#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$!\"##\"#8\"#5" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$!\"$#\"\"'\"\"&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$!\"%#\"#(*\"#&)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7$!\"&#\"%PC\"%5A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 273 31 "b)  Graphs of f \+
(x) and f '(x) " }}{PARA 0 "" 0 "" {TEXT -1 19 "   We first define " }
{TEXT 276 6 "fgraph" }{TEXT -1 56 ", the graph of  f as a set of line \+
segments obtained by " }}{PARA 0 "" 0 "" {TEXT -1 30 "the two parts of
 the function " }{TEXT 277 15 "graph of fprime" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 73 "Then we display both  fgraph  and  graph \+
of  fprime in the same window.  " }}{PARA 0 "" 0 "" {TEXT -1 59 "Note \+
the use of a colon (:) to postpone the actual display." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 42 "fprimegraph:= plot(fprime(y), y = -5..5): " }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 171 "li:= [[-5,fLeft(-5)],[-4,fLeft(-4)],[-3,fLeft(-3)]
,[-2,fLeft(-2)],[-1,fLeft(-1)],[0,fLeft(0)],[1, fRight(1)],[2, fRight(
2)],[3, fRight(3)],[4, fRight(4)],[5, fRight(5)]];\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#liG7-7$!\"&#\"%PC\"%5A7$!\"%#\"#(*\"#&)7$!\"$#\"
\"'\"\"&7$!\"##\"#8\"#57$!\"\"#\"\"$\"\"#7$\"\"!F>7$\"\"\"F=7$F>#\"\"(
F>7$F=#\"#PF97$\"\"%#\"#>F47$F4#\"$G$F/" }}}{EXCHG {PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fgraph:= pl
ot(li,style = line): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "di
splay(fprimegraph, fgraph); \n" }}{PARA 13 "" 1 "" {GLPLOT2D 235 206 
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44.000000 47.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 265 86 "c) Veri
fy that the graph of  f '  is indeed the graph of the constructed func
tion  f. " }}{PARA 0 "" 0 "" {TEXT -1 82 " The function  f  is increas
ing on [-5, 5].  It is concave up, meaning increasing " }}{PARA 0 "" 
0 "" {TEXT -1 87 "at an increasing rate on [-5, 0], and concave down, \+
meaning increasing at a decreasing " }}{PARA 0 "" 0 "" {TEXT -1 55 "ra
te on [0, 5], with a point of inflection at  x = 0.  " }}{PARA 0 "" 0 
"" {TEXT -1 81 " The graph of the derivative of  fprime  is positive i
ncreasing on [-5,0], has a " }}{PARA 0 "" 0 "" {TEXT -1 55 "maximum at
 0 and then is decreasing and still positive." }}{PARA 0 "" 0 "" 
{TEXT -1 94 " In this manner the graph of  f '  is indeed the graph of
 the derivative of the constructed f." }}}}{PARA 5 "" 0 "" {TEXT -1 
64 "________________________________________________________________" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 274 10 "ASSI
GNMENT" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 262 8 "Problem
:" }}{PARA 0 "" 0 "" {TEXT -1 15 "  For f '(x) = " }{XPPEDIT 18 0 "ln(
1+x^2);" "6#-%#lnG6#,&\"\"\"F'*$%\"xG\"\"#F'" }}{PARA 0 "" 0 "" {TEXT 
-1 12 "  f(0) = -2," }}{PARA 0 "" 0 "" {TEXT -1 29 "a) Construct f(x) \+
on [-4, 4]." }}{PARA 0 "" 0 "" {TEXT -1 37 "b) Plot f(x) and f '(x) on
  [-4, 4]. " }}{PARA 0 "" 0 "" {TEXT -1 107 "c) Verify that the graph \+
of  f '  is indeed the graph of the derivative of the constructed  fun
ction  f  on" }}{PARA 0 "" 0 "" {TEXT -1 13 "     [-4, 4]." }}}{PARA 
0 "" 0 "" {TEXT -1 67 " ______________________________________________
____________________" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{PARA 0 "" 
0 "" {TEXT -1 99 "MSIP Grant #P120A80089-98:  \"Three Urban Calculus R
eform Programs: Adopting the Best\" 1998-2001    " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 3 "   " }}}{MARK "9 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }