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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 77 "Lab # 4:  Analysis of Fun
ctions using first and second derivative information" }}{PARA 256 "" 
0 "" {TEXT -1 278 "In this lab we analyze functions with regards to in
tervals where the function is  increasing or decreasing, concave up or
 down, and the location of inflection points and relative extrema.  So
me of the Maple commands that aid in this task are demonstrated with a
 sample problem. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 60 "Sample Prob
lem: Plotting Derivatives and Solving for Roots. " }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 25 "Consider the function    " }{TEXT -1 4 "    " }
{XPPEDIT 18 0 "f(x) = sqrt(5)*x+3*x^(2/3);" "6#/-%\"fG6#%\"xG,&*&-%%sq
rtG6#\"\"&\"\"\"F'F.F.*&\"\"$F.)F'*&\"\"#F.F0!\"\"F.F." }{TEXT -1 6 " \+
     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 4 "(1) " }{TEXT -1 
105 "Plot the function over an appropriate interval that illustrates t
he important features of this function. " }}{PARA 0 "" 0 "" {TEXT 259 
4 "(2) " }{TEXT -1 95 "On what intervals is the function increasing an
d on what intervals is the function decreasing? " }}{PARA 0 "" 0 "" 
{TEXT 260 3 "(3)" }{TEXT -1 69 " List all critical points and of these
,  which are stationary points." }{TEXT 270 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 271 4 "(4) " }{TEXT -1 97 "Find the open interval
s on which f is concave down and the open intervals where f is concave
 up. " }}{PARA 0 "" 0 "" {TEXT 261 3 "(5)" }{TEXT -1 36 " Where are th
e inflection point(s)? " }}{PARA 0 "" 0 "" {TEXT 262 4 "(6) " }{TEXT 
-1 27 "Label all relative extrema." }}{PARA 0 "" 0 "" {TEXT -1 3 "   \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 4 "(1) " }{TEXT -1 25 "Try to pl
ot the function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot(sq
rt(5) * x + 3*x^(2/3),x=-3..3,labels=[\"x\",\"f(x)\"]);" }}{PARA 13 "
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\"3%)*****p$y^kYF-$\"3G.#p;n;/$\\F37$$\"3O******\\]2=jF-$\"3m&\\V?;9<<
'F37$$\"3s*****\\Z*=D'*F-$\"3'HS$*H&[.`%)F37$$\"3++++!RIKH\"F3$\"3]$Q9
74pj0\"!#<7$$\"3))******\\4+p=F3$\"3;gKnf>g)R\"Ffn7$$\"3/+++5:xWCF3$\"
3pq]`k\">'><Ffn7$$\"3-+++![n%)o$F3$\"3%['3#za;xO#Ffn7$$\"37++++rKt\\F3
$\"3#H6QA=C_*HFfn7$$\"3u******z:JIiF3$\"3]0z[g`a\"e$Ffn7$$\"3%********
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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 315 "Notice that Maple does not evaluate negative values of x
.  This is a problem with the algorithm Maple uses to find n'th roots.
  Since we are interested in the 3rd root of x^2, we use a command spe
cifically designed to do this. \"surd(g(x),n)\" evaluates the n'th roo
t of g(x). So we define our function as follows:   " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "f:=x-> sqrt(5) * x + 3*surd(x^2,3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(,&*&-%%sqrtG6#\"\"&\"\"\"9$F2F2*&\"\"$F2-%%surdG6$*$)F3\"\"#F2F5F2F
2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(f(x),x=-3..
3,labels=[\"x\",\"f(x)\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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tsF17$$\"33++vB4JB;F1$\"3IaX\"\\zSNx(F17$$\"3u*****\\KCnu\"F1$\"3[.'Qt
WXpD)F17$$\"3s***\\(=n#f(=F1$\"3]8O*45ayv)F17$$\"3P+++!)RO+?F1$\"3SWyj
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[X!Rn>5!#;7$$\"3G+++5jypBF1$\"39//\")[g9j5Ffz7$$\"3<++]Ujp-DF1$\"3za;s
/>i76Ffz7$$\"3++++gEd@EF1$\"3_\\-dX\"zl:\"Ffz7$$\"39++v3'>$[FF1$\"3M@.
3s)fJ?\"Ffz7$$\"37++D6EjpGF1$\"3-7rWOE[Z7Ffz7$$\"\"$F*$\"3=3b;Sb%[H\"F
fz-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F\\]l-%+AXESLABELSG6$Q\"x6\"Q%f(x)F
a]l-%%VIEWG6$;F(Fa\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 267 13 "Answer to (1)" }{TEXT -1 209 " is this graph.  In \+
the labs you hand in, you should label critical points, inflection poi
nts and relative extrema on this graph. You may do this by hand unless
 you want to figure out how to make Maple do it. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 3 "(2)" }{TEXT -1 137 "  We \+
will rely on the theorem that f is increasing where f' is positive and
 f is decreasing where f' is negative.  So lets define f': by " }
{TEXT 265 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "fprime := \+
x -> D(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'fprimeGf*6#%\"xG6
\"6$%)operatorG%&arrowGF(,&-%%sqrtG6#\"\"&\"\"\"*(\"\"#F1-%%surdG6$*$)
9$F3F1\"\"$F1F9!\"\"F1F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 232 "
This may look funny but a little algebra reveals this to be the deriva
tive you would derive using the power rule. Furthermore, the command \+
\"D(f)(x)\" works well for plotting and evaluating the derivative.  Le
ts plot the derivative.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
70 "plot(fprime(x),x=-3..3,y=-10..10,labels = [\"x\",\"f'(x)\"],discon
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nE]djm[<F1$\"3Iu#za&G9'*QF17$$\"3wop7\"e::\"=F1$\"3'\\eMS3=n(QF17$$\"3
+\\.jOaKs=F1$\"3Y8ODJ/weQF17$$\"3*4JqLTW)R>F1$\"3Ef_q*)4rRQF17$$\"3At*
)*p@80+#F1$\"3r84\\sKLBQF17$$\"3y>%pV6!Hl?F1$\"3`q1Dp?c1QF17$$\"3hCq76
w)R7#F1$\"33JMvw&f>z$F17$$\"3O\"oB\"z%f\")=#F1$\"3IIdS5')fwPF17$$\"3O>
z(e?S&[AF1$\"3u5MYgQoiPF17$$\"3.*o^KYb;J#F1$\"3_:<4+<m[PF17$$\"3evKvj@
OtBF1$\"3Nz1eEuVNPF17$$\"39t!*\\gL'zV#F1$\"3_WJ4f_2APF17$$\"37O'**4*>=
+DF1$\"3:qQz?[k4PF17$$\"3'3QAr_4Qc#F1$\"3KK.yhBN(p$F17$$\"3uz67&>5pi#F
1$\"3qDMdlpb&o$F17$$\"3!Q@Ic:$*[o#F1$\"3I+D+[l/vOF17$$\"3i.tur\"[8v#F1
$\"3%Rjs\\[mLm$F17$$\"32F%y.L'y5GF1$\"3[Iy(o&RB`OF17$$\"3_!oEY!)fT(GF1
$\"38)og!o0uUOF17$$\"3Mn`v0j\"[$HF1$\"3*p\\6`.#)Hj$F17$$\"\"$F*$\"3!*e
5?E0zAOF1-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F_am-%+AXESLABELSG6$Q\"x
6\"Q&f'(x)Fdam-%%VIEWG6$;F(Fd`m;$!#5F*$\"#5F*" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 40 "It looks like f' is positive from  on (-" }{XPPEDIT 18 0 
"infinity;" "6#%)infinityG" }{TEXT -1 15 ", -0.7) and (0," }{XPPEDIT 
18 0 "infinity;" "6#%)infinityG" }{TEXT -1 234 ") and negative on (-0.
7,0).  However, this -0.7 is just a guess found by clicking on the gra
ph at the location near the x-intercept.  You can have Maple find this
 value exactly with the \"solve\" command.  I will call this value \"b
\":   " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "b := solve(fprime
(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG,$*$-%%sqrtG6#\"\"&
\"\"\"#!\")\"#D" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Let's see if t
his is close to the original approximation of -0.7.  " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(b);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$!+IvTbr!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Not
 bad. As a check, let's see if the derivative is trully zero at this v
alue. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "fprime" }{TEXT -1 
0 "" }{MPLTEXT 1 0 4 "(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Does our original guess evaluate
 to zero?   " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(fprim
e(-.7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!)#yFk\"!\"*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 118 "No.  Lesson: You can get a good approxim
ation to values by clicking on the graph but these are not exact solut
ions.   " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 14 "Answer to (2):" }
{TEXT -1 34 "  The function is increasing on (-" }{XPPEDIT 18 0 "infin
ity;" "6#%)infinityG" }{TEXT -1 12 ", b] and [0," }{XPPEDIT 18 0 "infi
nity;" "6#%)infinityG" }{TEXT -1 51 ").  The function is decreasing on
 [b,0], where b = " }{XPPEDIT 18 0 "(-8)*sqrt(5)/25;" "6#*(,$\"\")!\"
\"\"\"\"-%%sqrtG6#\"\"&F'\"#DF&" }{TEXT -1 86 ".   Note, recall that y
ou get the endpoints for free, provided f is continuous there. " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 256 4 "(3) " }{TEXT -1 250 "The critical \+
points occur where f' is either zero or undefined.  From the previous \+
analysis we see that f' is zero at b and undefined at zero. But the cr
itical points are (b,f(b)) and (0,f(0)).   Furthermore stationary poin
ts are where f' is zero.   " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "y1 = f(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#y1G#\"\"%\"\"&" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "y2 = f(0);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%#y2G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }{TEXT 258 15 "Answer to (3): " }{TEXT -1 25 "The critical points a
re (" }{XPPEDIT 18 0 "(-8)*sqrt(5)/25;" "6#*(,$\"\")!\"\"\"\"\"-%%sqrt
G6#\"\"&F'\"#DF&" }{TEXT -1 31 ", 4/5) and (0, 0). Of these,  (" }
{XPPEDIT 18 0 "(-8)*sqrt(5)/25;" "6#*(,$\"\")!\"\"\"\"\"-%%sqrtG6#\"\"
&F'\"#DF&" }{TEXT -1 100 ", 4/5) is the only stationary point.  Note: \+
The critical numbers are the x-values of these points.  " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 3 "(4)" }{TEXT -1 374 "  Reca
ll that a function is concave up where the second derivative is positi
ve (Ie. the first derivative is increasing) and concave down where the
 second derivative is negative (Ie. the first derivative is decreasing
).  Judging from the graphs of f and f' it appears as though the funct
ion is always concave down.  We can verify this by investigating the s
econd derivative. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fdoub
leprime := x -> D(fprime)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-fd
oubleprimeGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&-%%surdG6$*$)9$\"\"#
\"\"\"\"\"$F5F3!\"##F7F6F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
111 "Due to the \"surd\" function it is hard to tell that this is alwa
ys negative but if you express this as f''(x) = " }{XPPEDIT 18 0 "(-2)
/(3*x^2)*x^(2/3);" "6#*(,$\"\"#!\"\"\"\"\"*&\"\"$F'*$%\"xGF%F'F&)F+*&F
%F'F)F&F'" }{TEXT -1 98 "  you see this is always negative.  As an inf
ormal display of this, we plot the second derivative:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot(fdoubleprime(x),x=-3..3,y = -2
..1,labels=[\"x\",\"f''(x)\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
300 {PLOTDATA 2 "6%-%'CURVESG6$7eo7$$!\"$\"\"!$!3:*H!*=$G!3a\"!#=7$$!3
!******\\2<#pG!#<$!3k5v!*[(\\^j\"F-7$$!3#)***\\7bBav#F1$!3->bO2=!es\"F
-7$$!36++]K3XFEF1$!3deZf2zxQ=F-7$$!3%)****\\F)H')\\#F1$!3')fN\"[_`i'>F
-7$$!3#****\\i3@/P#F1$!3k;A#f#[J4@F-7$$!3;++Dr^b^AF1$!3QuT%[w\"3fAF-7$
$!3$****\\7Sw%G@F1$!3v$=!erR\"\\V#F-7$$!3*****\\7;)=,?F1$!3o&4ww@uNk#F
-7$$!3/++DO\"3V(=F1$!3;ch42N$[)GF-7$$!3#******\\V'zV<F1$!3k(*HEwwCwJF-
7$$!3******\\d;%)G;F1$!3I.Bn?FfyMF-7$$!3!******\\!)H%*\\\"F1$!3s$[>Jc[
X)QF-7$$!3/+++vl[p8F1$!3E:`E5zg$Q%F-7$$!3\"******\\>iUC\"F1$!3w))e#*)4
$\\\")\\F-7$$!3-++DhkaI6F1$!31t/8Mb`gcF-7$$!3s******\\XF`**F-$!3AihR%*
zU3nF-7$$!3u*******>#z2))F-$!33D!zvf@i*yF-7$$!3S++]7RKvuF-$!3J^3)fGsl#
)*F-7$$!3s,+++P'eH'F-$!3MS>B4zZN7F17$$!3q)***\\7*3=+&F-$!3-/G1LZ3z;F17
$$!3[)***\\PFcpPF-$!3e)pd9mS#[CF17$$!3;)****\\7VQ[#F-$!3K)H%fZG#)pUF17
$$!35)**\\P9(\\$*=F-$!3@q()3>mMJhF17$$!32)***\\i6:.8F-$!3Z6hSap045!#;7
$$!3R)\\i!*=fR9\"F-$!3J!*e0$z(\\+7Fas7$$!3%p)*\\i:sw%)*!#>$!3W+]lI())f
Y\"Fas7$$!3/!*\\(=U_dD)Fjs$!3e\"Gl.9FX&=Fas7$$!39$***\\(oKQm'Fjs$!3)e$
*Qmg5wY#Fas7$$!3C'*\\7`H\">2&Fjs$!3A@pGzY'4b$Fas7$$!3O***\\(=K**zMFjs$
!3!**ex#3-uneFas7$$!3!4]i:NLSo#Fjs$!3eSS\"4a)z&H)Fas7$$!3W-]P%[t!))=Fj
s$!3$*>0QM/-E8!#:7$$!3A`7y]N4!\\\"Fjs$!38Lie4i7==F[v7$$!3+/v=<O6#4\"Fj
s$!3G-47qnQ^FF[v7$$!3qZv$f$oLTp!#?$!3mb43Pt#[.&F[v7$$!3Wb+++v`hHFiv$!3
yhTPj;an:!#97$$\"3![W7y]bB<\"Fiv$!3sg__q-*HR&Faw7$$\"3/X\\i:&[iI&Fiv$!
3(Rt;h+(>.sF[v7$$\"3GXuVB:9S%*Fiv$!3zPy#fso9M$F[v7$$\"3c%*\\7`MSd8Fjs$
!3K?!fTwv)e?F[v7$$\"3g%\\(oa?=%=#Fjs$!3?xoix'=>4\"F[v7$$\"3l%**\\ilg4,
$Fjs$!3G$Qx?&>3<rFas7$$\"3w%*\\Pfy^kYFjs$!3E>`@:!*QqRFas7$$\"3%[***\\i
]2=jFjs$!3'Gcst9(G\\EFas7$$\"3%\\*\\ilAjrzFjs$!3\\ie1pr<V>Fas7$$\"3/&*
*\\(o%*=D'*Fjs$!3Vw%*e]HM6:Fas7$$\"3^*\\(=nY(y7\"F-$!3fk?\"G!)yLA\"Fas
7$$\"3]****\\(QIKH\"F-$!3n6ca$=\"R>5Fas7$$\"3K******\\4+p=F-$!3oi=%[!z
sQiF17$$\"38****\\7:xWCF-$!33xD'4t\\5O%F17$$\"3E,++vuY)o$F-$!3w<Di$Gs-
_#F17$$\"3!z******4FL(\\F-$!3S11\">D=>p\"F17$$\"3A)****\\d6.B'F-$!3un@
Z8.%GD\"F17$$\"3s****\\(o3lW(F-$!3/C>&o?0t()*F-7$$\"35*****\\A))oz)F-$
!3wx0x#Ru#4zF-7$$\"3e******Hk-,5F1$!3Q#)**ydPbdmF-7$$\"36+++D-eI6F1$!3
SsFfR,JgcF-7$$\"3u***\\(=_(zC\"F1$!3q0Tw)RT<'\\F-7$$\"3M+++b*=jP\"F1$!
3EG\"H1*ohaVF-7$$\"3g***\\(3/3(\\\"F1$!3!\\uwW\")yE*QF-7$$\"33++vB4JB;
F1$!31Xr])4/W\\$F-7$$\"3u*****\\KCnu\"F1$!3!H[xR)3:pJF-7$$\"3s***\\(=n
#f(=F1$!3]spX/_^\")GF-7$$\"3P+++!)RO+?F1$!34lZ'fdE]k#F-7$$\"30++]_!>w7
#F1$!3P9[;&HAiV#F-7$$\"3O++v)Q?QD#F1$!3y#HB9$\\0cAF-7$$\"3G+++5jypBF1$
!3[s!)e+#o+6#F-7$$\"3<++]Ujp-DF1$!3MiE%o%[*>'>F-7$$\"3++++gEd@EF1$!3i^
)>$[sFW=F-7$$\"39++v3'>$[FF1$!3o!*QgVAvJ<F-7$$\"37++D6EjpGF1$!3(4V)R`S
$[j\"F-7$$\"\"$F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F[cl-%+AXESLABELSG
6$Q\"x6\"Q'f''(x)F`cl-%%VIEWG6$;F(Fbbl;$!\"#F*$\"\"\"F*" 1 2 0 1 10 0 
2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 14 "Answer to (4):" }{TEXT -1 35 " \+
The function is concave down on (-" }{XPPEDIT 18 0 "infinity;" "6#%)in
finityG" }{TEXT -1 13 ", 0) and (0, " }{XPPEDIT 18 0 "infinity;" "6#%)
infinityG" }{TEXT -1 3 "). " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 272 4 "(5) " }{TEXT -1 109 "Since inflection points are where a \+
function changes concavity and this function is always concave down, t
he " }{TEXT 273 13 "Answer to (5)" }{TEXT -1 40 " is: \"There are no i
nflection points\".  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
274 3 "(6)" }{TEXT -1 292 " We check the critical numbers to find if a
ny lead to relative extrema. At the critical number x = b, where b is \+
as previously defined, you can either use the first or second derivati
ve test to verify that this leads to a  relative maximum.  For example
, plugging b into the second derivative" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "fdoubleprime(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##
!#D\"#C" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Since this value is n
egative the function is concave down there and this leads to a relativ
e maximum at x = b. " }}{PARA 0 "" 0 "" {TEXT -1 279 "At the critical \+
number x = 0, you must rely on the first derivative test to verify tha
t this leads to a relative minimum.  This is done by noting that f' is
 negative just to the left of zero and f' is positive just to the righ
t.  This implies a relative minimum occurs at x = 0. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 275 10 "Answer
 (6)" }{TEXT -1 84 " There is a relative maximum of 4/5 at x = b and a
 relative minimum of zero at x=0. " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 10 "Assignment" }}{PARA 0 "" 0 "" {TEXT -1 397 "You may do the anal
ysis for these problems by hand or with Maple. Furthermore, you may la
bel any inflection points or relative extrema by hand as well.  Make s
ure your graphs are big enough to easily label these features.  The ab
ove example illustrates the techniques you may use to answer the follo
wing questions, however you are not being asked to repeat the above pr
ocess on these problems.    " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 277 11 "Problem #1 " }}{PARA 0 "" 0 "" {TEXT 278 1 " " }{TEXT 
279 145 "Generate a graph of the following function which illustrates \+
all of the relevant features.  Identify all inflection points and rela
tive extrema. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 280 1 " " }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x) =
 x^2*exp(2*x)" "6#/-%\"fG6#%\"xG*&F'\"\"#-%$expG6#*&F)\"\"\"F'F.F." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 281 11 "Problem #2 " }}
{PARA 0 "" 0 "" {TEXT 282 1 " " }{TEXT 283 145 "Generate a graph of th
e following function which illustrates all of the relevant features.  \+
Identify all inflection points and relative extrema. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }{TEXT 284 1 " " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x) = (x^2-1)/(x^2+1);" "6#/-%\"fG6#%
\"xG*&,&*$F'\"\"#\"\"\"F,!\"\"F,,&*$F'F+F,F,F,F-" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 276 11 "Pr
oblem # 3" }}{PARA 0 "" 0 "" {TEXT -1 59 "Consider the family of logis
tic growth curves defined by   " }{XPPEDIT 18 0 "y = 10/(1+10*exp(-kt)
);" "6#/%\"yG*&\"#5\"\"\",&F'F'*&F&F'-%$expG6#,$%#ktG!\"\"F'F'F/" }
{TEXT -1 5 " .   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 181 "Generate a graph of y for k = 1, 2, and 3 on the same ax
es.  Discuss the effect of varying k on the shape of the curve includi
ng the effect on the location of the inflection point. " }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }