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{SECT 0 {EXCHG {PARA 205 "" 0 "" {TEXT 231 42 "Maximum and Minimum Va
lues and Riemann Sum" }{TEXT 231 0 "" }}{PARA 214 "" 0 "" {TEXT 216 0 
"" }}}{SECT 1 {PARA 202 "" 0 "" {TEXT 200 85 "Derivatives can be used \+
to find local maximum and local minimum values of functions. " }{TEXT 
200 0 "" }}{PARA 219 "" 0 "" {TEXT 201 70 "Example 1:  Find the absolu
te and local maximum and minimum values of " }{XPPEDIT 212 0 "Typesett
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i(\"\", bold = \"false\", italic = \"true\", executable = \"true\", fo
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{TEXT 202 5 "     " }{TEXT 214 23 "on the interval [0, 3]." }{TEXT 
232 0 "" }}{PARA 214 "" 0 "" {TEXT 216 0 "" }}{EXCHG {PARA 216 "> " 0 
"" {MPLTEXT 1 228 8 "restart;" }{MPLTEXT 1 228 0 "" }}}{EXCHG {PARA 
216 "> " 0 "" {MPLTEXT 1 228 30 "f:=x->x^(2/5)*(2*x^2-3*x)^2+3;" }
{MPLTEXT 1 228 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 228 38 "p
lot(f(x),x=0..3,y=0..10,thickness=2);" }{MPLTEXT 1 228 0 "" }}}{EXCHG 
{PARA 216 "> " 0 "" {MPLTEXT 1 228 9 "fx:=D(f);" }{MPLTEXT 1 228 0 "" 
}}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 228 16 "s:=solve(fx(x));" }
{MPLTEXT 1 228 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 228 13 "s
s:=evalf(s);" }{MPLTEXT 1 228 0 "" }}}{EXCHG {PARA 216 "> " 0 "" 
{MPLTEXT 1 228 75 "p1:=[0,f(0)];p2:=[ss[2],f(ss[2])];p3:=[ss[1],f(ss[1
])];p4:=evalf([3,f(3)]);" }{MPLTEXT 1 228 0 "" }}}{PARA 214 "" 0 "" 
{TEXT 216 0 "" }}{PARA 219 "" 0 "" {TEXT 208 7 "       " }{TEXT 201 
131 "p1 and p3 are the absolute minimum points, p2 is a local maximum \+
point, p3 is a local minimum and p4 is the absolute maximum point." }
{TEXT 232 0 "" }}{PARA 219 "" 0 "" {TEXT 201 0 "" }{TEXT 201 1 "\n" }
{TEXT 201 76 "Example 2:  Find the local maximum and local minimum val
ues of the function " }{XPPEDIT 213 0 "Typesetting:-mrow(Typesetting:-
mi(\"\", bold = \"false\", italic = \"true\", executable = \"true\", f
ont_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:
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tion;\", bold = \"false\", mathvariant = \"normal\", fence = \"false\"
, separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+
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g:-mrow(Typesetting:-mi(\"x\", bold = \"false\", italic = \"true\", ma
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ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr
ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent
 = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typ
esetting:-mrow(Typesetting:-mi(\"\", bold = \"false\", italic = \"true
\", executable = \"true\", font_style_name = \"2D Input\", mathvariant
 = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"x\", bold = \"fals
e\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&
InvisibleTimes;\", bold = \"false\", mathvariant = \"normal\", fence =
 \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+
\"false\", largeop = \"false\", movablelimits = \"false\", accent = \"
false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Ty
pesetting:-mo(\"ⅇ\", bold = \"false\", mathvariant = \"no
rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\"
, symmetric = \"false\", largeop = \"false\", movablelimits = \"false
\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.1111111em\"),
 Typesetting:-mi(\"x\", bold = \"false\", italic = \"true\", mathvaria
nt = \"italic\"), superscriptshift = \"0\")), Typesetting:-mo(\"&minus
;\", bold = \"false\", mathvariant = \"normal\", fence = \"false\", se
parator = \"false\", stretchy = \"false\", symmetric = \"false\", larg
eop = \"false\", movablelimits = \"false\", accent = \"false\", lspace
 = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Types
etting:-msup(Typesetting:-mi(\"x\", bold = \"false\", italic = \"true
\", mathvariant = \"italic\"), Typesetting:-mn(\"3\", bold = \"false\"
, mathvariant = \"normal\"), superscriptshift = \"0\"), Typesetting:-m
o(\"⁢\", bold = \"false\", mathvariant = \"normal\", fe
nce = \"false\", separator = \"false\", stretchy = \"false\", symmetri
c = \"false\", largeop = \"false\", movablelimits = \"false\", accent \+
= \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msu
p(Typesetting:-mo(\"ⅇ\", bold = \"false\", mathvariant = \+
\"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal
se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa
lse\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.1111111em
\"), Typesetting:-mi(\"x\", bold = \"false\", italic = \"true\", mathv
ariant = \"italic\"), superscriptshift = \"0\")), Typesetting:-mi(\"\"
, bold = \"false\", italic = \"true\", executable = \"true\", font_sty
le_name = \"2D Input\", mathvariant = \"italic\")), Typesetting:-mi(\"
\", bold = \"false\", italic = \"true\", executable = \"true\", font_s
tyle_name = \"2D Input\", mathvariant = \"italic\")), Typesetting:-mi(
\"\", bold = \"false\", italic = \"true\", executable = \"true\", font
_style_name = \"2D Input\", mathvariant = \"italic\"));" "-I%mrowG6#/I
+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6(Q!F'/%%boldGQ&fal
seF'/%'italicGQ%trueF'/%+executableGF4/%0font_style_nameGQ)2D~InputF'/
%,mathvariantGQ'italicF'-F#6'F+-F#6%-F,6&Q\"fF'F/F2F:-I#moGF$6.Q0&Appl
yFunction;F'F//F;Q'normalF'/%&fenceGF1/%*separatorGF1/%)stretchyGF1/%*
symmetricGF1/%(largeopGF1/%.movablelimitsGF1/%'accentGF1/%'lspaceGQ&0.
0emF'/%'rspaceGFZ-I(mfencedGF$6%-F#6#-F,6&Q\"xF'F/F2F:F/FH-FE6.Q\"=F'F
/FHFJFLFNFPFRFTFV/FYQ,0.2777778emF'/FfnFco-F#6'F+-F#6%F\\o-FE6.Q1&Invi
sibleTimes;F'F/FHFJFLFNFPFRFTFVFXFen-I%msupGF$6%-FE6.Q/ⅇF
'F/FHFJFLFNFPFRFTFVFX/FfnQ,0.1111111emF'F\\o/%1superscriptshiftGQ\"0F'
-FE6.Q(−F'F/FHFJFLFNFPFRFTFV/FYQ,0.2222222emF'/FfnF[q-F#6%-F]p6%
F\\o-I#mnGF$6%Q\"3F'F/FHFdpFioF\\pF+F+F+" }{TEXT 202 5 "    ." }{TEXT 
232 0 "" }}{PARA 214 "" 0 "" {TEXT 216 0 "" }}{EXCHG {PARA 216 "> " 0 
"" {MPLTEXT 1 228 26 "f:=x->x*exp(x)-x^3*exp(x);" }{MPLTEXT 1 228 0 ""
 }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 228 54 "plot(f(x),x=-5..5,y=
-2..2,thickness=2,numpoints=5000);" }{MPLTEXT 1 228 0 "" }}}{EXCHG 
{PARA 216 "> " 0 "" {MPLTEXT 1 228 9 "fx:=D(f);" }{MPLTEXT 1 228 0 "" 
}}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 228 27 "s1:=fsolve(fx(x),x=-4
..-2);" }{MPLTEXT 1 228 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 
228 26 "s2:=fsolve(fx(x),x=-1..0);" }{MPLTEXT 1 228 0 "" }}}{EXCHG 
{PARA 216 "> " 0 "" {MPLTEXT 1 228 25 "s3:=fsolve(fx(x),x=0..1);" }
{MPLTEXT 1 228 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 228 68 "l
ocal_max1:=[s1,f(s1)];local_min:=[s2,f(s2)];local_max2:=[s3,f(s3)];" }
{MPLTEXT 1 228 0 "" }}}{PARA 219 "" 0 "" {TEXT 201 0 "" }{TEXT 201 1 "
\n" }{TEXT 201 12 "Example 3:  " }{TEXT 208 30 "Find a polynomial of t
he form " }{XPPEDIT 209 0 "Typesetting:-mrow(Typesetting:-mi(\"\", bol
d = \"false\", italic = \"true\", executable = \"true\", font_style_na
me = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mrow(Typese
tting:-mi(\"\", bold = \"false\", italic = \"true\", executable = \"tr
ue\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Types
etting:-mrow(Typesetting:-mi(\"P\", italic = \"true\", mathvariant = \+
\"italic\"), Typesetting:-mo(\"⁡\", bold = \"false\", ma
thvariant = \"normal\", fence = \"false\", separator = \"false\", stre
tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel
imits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"
0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\"
, italic = \"true\", mathvariant = \"italic\")), mathvariant = \"norma
l\")), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"fals
e\", separator = \"false\", stretchy = \"false\", symmetric = \"false
\", largeop = \"false\", movablelimits = \"false\", accent = \"false\"
, lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mr
ow(Typesetting:-mi(\"\", bold = \"false\", italic = \"true\", executab
le = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic
\"), Typesetting:-mrow(Typesetting:-mi(\"A\", italic = \"true\", mathv
ariant = \"italic\"), Typesetting:-mo(\"⁢\", bold = \"f
alse\", mathvariant = \"normal\", fence = \"false\", separator = \"fal
se\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\"
, movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", r
space = \"0.0em\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-m
i(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mn
(\"2\", mathvariant = \"normal\"), superscriptshift = \"0\")), Typeset
ting:-mi(\"\", bold = \"false\", italic = \"true\", executable = \"tru
e\", font_style_name = \"2D Input\", mathvariant = \"italic\")), Types
etting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separat
or = \"false\", stretchy = \"false\", symmetric = \"false\", largeop =
 \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"
0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesettin
g:-mi(\"B\", italic = \"true\", mathvariant = \"italic\"), Typesetting
:-mo(\"⁢\", bold = \"false\", mathvariant = \"normal\",
 fence = \"false\", separator = \"false\", stretchy = \"false\", symme
tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce
nt = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-
mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-
mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"f
alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false
\", movablelimits = \"false\", accent = \"false\", lspace = \"0.222222
2em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"C\", italic = \"tr
ue\", mathvariant = \"italic\")), Typesetting:-mi(\"\", bold = \"false
\", italic = \"true\", executable = \"true\", font_style_name = \"2D I
nput\", mathvariant = \"italic\")), Typesetting:-mi(\"\", bold = \"fal
se\", italic = \"true\", executable = \"true\", font_style_name = \"2D
 Input\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,T
ypesettingGI(_syslibGF'6%-I#miGF$6(Q!F'/%%boldGQ&falseF'/%'italicGQ%tr
ueF'/%+executableGF4/%0font_style_nameGQ)2D~InputF'/%,mathvariantGQ'it
alicF'-F#6'F+-F#6%-F,6%Q\"PF'F2F:-I#moGF$6.Q0⁡F'F//F;Q'n
ormalF'/%&fenceGF1/%*separatorGF1/%)stretchyGF1/%*symmetricGF1/%(large
opGF1/%.movablelimitsGF1/%'accentGF1/%'lspaceGQ&0.0emF'/%'rspaceGFZ-I(
mfencedGF$6$-F#6#-F,6%Q\"xF'F2F:FH-FE6-Q\"=F'FHFJFLFNFPFRFTFV/FYQ,0.27
77778emF'/FfnFco-F#6(F+-F#6&-F,6%Q\"AF'F2F:-FE6.Q1⁢F'F/
FHFJFLFNFPFRFTFVFXFen-F#6#-I%msupGF$6%F\\o-I#mnGF$6$Q\"2F'FH/%1supersc
riptshiftGQ\"0F'F+-FE6-Q\"+F'FHFJFLFNFPFRFTFV/FYQ,0.2222222emF'/FfnF_q
-F#6%-F,6%Q\"BF'F2F:F\\pF\\oF[q-F,6%Q\"CF'F2F:F+F+" }{TEXT 208 13 "   \+
such that " }{XPPEDIT 209 0 "Typesetting:-mrow(Typesetting:-mi(\"\", b
old = \"false\", italic = \"true\", executable = \"true\", font_style_
name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mrow(Type
setting:-mi(\"\", bold = \"false\", italic = \"true\", executable = \"
true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typ
esetting:-mrow(Typesetting:-mi(\"P\", italic = \"true\", mathvariant =
 \"italic\"), Typesetting:-mo(\"⁡\", bold = \"false\", m
athvariant = \"normal\", fence = \"false\", separator = \"false\", str
etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable
limits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \+
\"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn(\"2
\", mathvariant = \"normal\")), mathvariant = \"normal\")), Typesettin
g:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \+
\"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa
lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.277
7778em\", rspace = \"0.2777778em\"), Typesetting:-mn(\"5\", mathvarian
t = \"normal\")), Typesetting:-mi(\"\", bold = \"false\", italic = \"t
rue\", executable = \"true\", font_style_name = \"2D Input\", mathvari
ant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysl
ibGF'6%-I#miGF$6(Q!F'/%%boldGQ&falseF'/%'italicGQ%trueF'/%+executableG
F4/%0font_style_nameGQ)2D~InputF'/%,mathvariantGQ'italicF'-F#6&F+-F#6%
-F,6%Q\"PF'F2F:-I#moGF$6.Q0⁡F'F//F;Q'normalF'/%&fenceGF1
/%*separatorGF1/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movablelim
itsGF1/%'accentGF1/%'lspaceGQ&0.0emF'/%'rspaceGFZ-I(mfencedGF$6$-F#6#-
I#mnGF$6$Q\"2F'FHFH-FE6-Q\"=F'FHFJFLFNFPFRFTFV/FYQ,0.2777778emF'/FfnFd
o-F]o6$Q\"5F'FHF+" }{TEXT 208 31 "    , P'(2) = 3, and P\"(2) = 2." }
{TEXT 232 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 206 8 "restart;
" }{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 206 
18 "f:=x->A*x^2+B*x+C;" }{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 215 "> " 
0 "" {MPLTEXT 1 206 9 "fx:=D(f);" }{MPLTEXT 1 206 0 "" }}}{EXCHG 
{PARA 215 "> " 0 "" {MPLTEXT 1 206 11 "fxx:=D(fx);" }{MPLTEXT 1 206 0 
"" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 206 39 "eq1:=f(2)=5;eq2:=f
x(2)=3;eq3:=fxx(2)=2;" }{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 215 "> " 0
 "" {MPLTEXT 1 206 21 "solve(\{eq1,eq2,eq3\});" }{MPLTEXT 1 206 0 "" }
}}{EXCHG {PARA 210 "" 0 "" {TEXT 208 27 "So here is that polynomial;" 
}{TEXT 208 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 206 14 "P(x):
=x^2-x+3;" }{MPLTEXT 1 206 0 "" }}}{PARA 214 "" 0 "" {TEXT 222 0 "" }
{TEXT 222 1 "\n" }{TEXT 222 12 "Example 4:  " }{TEXT 225 17 "For the f
unction " }{XPPEDIT 237 0 "Typesetting:-mrow(Typesetting:-mi(\"\", bol
d = \"false\", italic = \"true\", executable = \"true\", font_style_na
me = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mrow(Typese
tting:-mi(\"\", bold = \"false\", italic = \"true\", executable = \"tr
ue\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Types
etting:-mrow(Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \+
\"italic\"), Typesetting:-mo(\"⁡\", bold = \"false\", ma
thvariant = \"normal\", fence = \"false\", separator = \"false\", stre
tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel
imits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"
0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\"
, italic = \"true\", mathvariant = \"italic\")), mathvariant = \"norma
l\")), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"fals
e\", separator = \"false\", stretchy = \"false\", symmetric = \"false
\", largeop = \"false\", movablelimits = \"false\", accent = \"false\"
, lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mr
ow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mn(\"3\", mathvar
iant = \"normal\"), Typesetting:-mo(\"⁢\", bold = \"fal
se\", mathvariant = \"normal\", fence = \"false\", separator = \"false
\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+
movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp
ace = \"0.0em\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(
\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mn(
\"4\", mathvariant = \"normal\"), superscriptshift = \"0\")), Typesett
ing:-mi(\"\", bold = \"false\", italic = \"true\", executable = \"true
\", font_style_name = \"2D Input\", mathvariant = \"italic\")), Typese
tting:-mrow(Typesetting:-mn(\"20\", mathvariant = \"normal\")), lineth
ickness = \"1\", denomalign = \"center\", numalign = \"center\", bevel
led = \"false\"), Typesetting:-mo(\"−\", mathvariant = \"normal
\", fence = \"false\", separator = \"false\", stretchy = \"false\", sy
mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a
ccent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\")
, Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \+
\"true\", mathvariant = \"italic\"), Typesetting:-mn(\"3\", mathvarian
t = \"normal\"), superscriptshift = \"0\")), Typesetting:-mo(\"+\", ma
thvariant = \"normal\", fence = \"false\", separator = \"false\", stre
tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel
imits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspac
e = \"0.2222222em\"), Typesetting:-mfrac(Typesetting:-mrow(Typesetting
:-mn(\"3\", mathvariant = \"normal\"), Typesetting:-mo(\"&InvisibleTim
es;\", bold = \"false\", mathvariant = \"normal\", fence = \"false\", \+
separator = \"false\", stretchy = \"false\", symmetric = \"false\", la
rgeop = \"false\", movablelimits = \"false\", accent = \"false\", lspa
ce = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-ms
up(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")
, Typesetting:-mn(\"2\", mathvariant = \"normal\"), superscriptshift =
 \"0\")), Typesetting:-mi(\"\", bold = \"false\", italic = \"true\", e
xecutable = \"true\", font_style_name = \"2D Input\", mathvariant = \"
italic\")), Typesetting:-mrow(Typesetting:-mn(\"5\", mathvariant = \"n
ormal\")), linethickness = \"1\", denomalign = \"center\", numalign = \+
\"center\", bevelled = \"false\"), Typesetting:-mo(\"+\", mathvariant \+
= \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f
alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \"
false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.22
22222em\"), Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mn(\"24
\", mathvariant = \"normal\"), Typesetting:-mo(\"⁢\", b
old = \"false\", mathvariant = \"normal\", fence = \"false\", separato
r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+
\"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0
.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\",
 mathvariant = \"italic\")), Typesetting:-mrow(Typesetting:-mn(\"5\", \+
mathvariant = \"normal\")), linethickness = \"1\", denomalign = \"cent
er\", numalign = \"center\", bevelled = \"false\"), Typesetting:-mo(\"
−\", mathvariant = \"normal\", fence = \"false\", separator = \"
false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals
e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.22222
22em\", rspace = \"0.2222222em\"), Typesetting:-mn(\"2\", mathvariant \+
= \"normal\")), Typesetting:-mi(\"\", bold = \"false\", italic = \"tru
e\", executable = \"true\", font_style_name = \"2D Input\", mathvarian
t = \"italic\")), Typesetting:-mi(\"\", bold = \"false\", italic = \"t
rue\", executable = \"true\", font_style_name = \"2D Input\", mathvari
ant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysl
ibGF'6%-I#miGF$6(Q!F'/%%boldGQ&falseF'/%'italicGQ%trueF'/%+executableG
F4/%0font_style_nameGQ)2D~InputF'/%,mathvariantGQ'italicF'-F#6'F+-F#6%
-F,6%Q\"fF'F2F:-I#moGF$6.Q0⁡F'F//F;Q'normalF'/%&fenceGF1
/%*separatorGF1/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movablelim
itsGF1/%'accentGF1/%'lspaceGQ&0.0emF'/%'rspaceGFZ-I(mfencedGF$6$-F#6#-
F,6%Q\"xF'F2F:FH-FE6-Q\"=F'FHFJFLFNFPFRFTFV/FYQ,0.2777778emF'/FfnFco-F
#6+-I&mfracGF$6(-F#6&-I#mnGF$6$Q\"3F'FH-FE6.Q1⁢F'F/FHFJ
FLFNFPFRFTFVFXFen-F#6#-I%msupGF$6%F\\o-F]p6$Q\"4F'FH/%1superscriptshif
tGQ\"0F'F+-F#6#-F]p6$Q#20F'FH/%.linethicknessGQ\"1F'/%+denomalignGQ'ce
nterF'/%)numalignGFhq/%)bevelledGF1-FE6-Q(−F'FHFJFLFNFPFRFTFV/FY
Q,0.2222222emF'/FfnFar-F#6#-Ffp6%F\\oF\\pF[q-FE6-Q\"+F'FHFJFLFNFPFRFTF
VF`rFbr-Fho6(-F#6&F\\pF`p-F#6#-Ffp6%F\\o-F]p6$Q\"2F'FHF[qF+-F#6#-F]p6$
Q\"5F'FHFcqFfqFiqF[rFgr-Fho6(-F#6%-F]p6$Q#24F'FHF`pF\\oFesFcqFfqFiqF[r
F]rFbsF+F+" }{TEXT 216 3 "   " }{TEXT 220 3 " , " }{TEXT 225 47 "plot \+
it and its first and second derivatives.  " }{TEXT 216 0 "" }}{PARA 
208 "" 0 "" {TEXT 211 82 "a)  Find the coordinates of the points where
 the function has horizontal tangents." }{TEXT 211 0 "" }}{PARA 208 ""
 0 "" {TEXT 211 180 "b)  Find the coordinates of the points in the int
erval -2 < x < 4 where the function is increasing and decreasing most \+
rapidly and the rate of increase or decrease at those points." }{TEXT 
211 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 206 8 "restart;" }
{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 206 36 "f
:=x->3/20*x^4-x^3+3/5*x^2+24/5*x-2;" }{MPLTEXT 1 206 0 "" }}}{EXCHG 
{PARA 215 "> " 0 "" {MPLTEXT 1 206 9 "fx:=D(f);" }{MPLTEXT 1 206 0 "" 
}}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 206 11 "fxx:=D(fx);" }
{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 206 54 "p
lot(\{f(x),fx(x),fxx(x)\},x=-5..6,y=-8..8,thickness=2);" }{MPLTEXT 1 
206 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 211 61 "This command will s
et the derivative equal to zero and solve." }{TEXT 211 0 "" }}{PARA 
214 "" 0 "" {TEXT 216 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 206 16 "s:
=solve(fx(x));" }{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT
 204 72 "The following are the points where the function has a horizon
tal tangent" }{TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 227 0 "" }}
{PARA 215 "> " 0 "" {MPLTEXT 1 206 81 "ht1:=[s[1],evalf(f(s[1]))];ht2:
=[s[2],evalf(f(s[2]))];ht3:=[s[3],evalf(f(s[3]))];" }{MPLTEXT 1 206 0 
"" }}}{EXCHG {PARA 214 "" 0 "" {TEXT 216 0 "" }}{PARA 208 "" 0 "" 
{TEXT 211 68 "The next command sets the second derivative equal to zer
o and solves" }{TEXT 211 0 "" }}{PARA 214 "" 0 "" {TEXT 216 0 "" }}
{PARA 215 "> " 0 "" {MPLTEXT 1 206 18 "h:=fsolve(fxx(x));" }{MPLTEXT 
1 206 0 "" }}}{EXCHG {PARA 214 "" 0 "" {TEXT 216 0 "" }}{PARA 208 "" 0
 "" {TEXT 211 53 "These are the points of maximum increase and decreas
e" }{TEXT 211 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 206 52 "r1:=[h[1],
evalf(f(h[1]))];r2:=[h[2],evalf(f(h[2]))];" }{MPLTEXT 1 206 0 "" }}}
{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 206 62 "max_rate_of_increase:=fx
(h[1]);max_rate_of_decrease:=fx(h[2]);" }{MPLTEXT 1 206 0 "" }}}}
{PARA 0 "" 0 "" {TEXT 219 0 "" }}{PARA 0 "" 0 "" {TEXT 238 0 "" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT 239 11 "Riemann sum" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "with(student);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 238 26 "Here are the sums commands" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "leftsum(x^2,x=0..30,10);" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "rightsum(x^2,x=0..30,10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 238 12 "The commands" 
}{MPLTEXT 1 19 1 " " }{TEXT 234 7 "leftbox" }{TEXT 238 3 ",  " }{TEXT 
235 8 "rightbox" }{TEXT 238 6 ", and " }{TEXT 236 9 "middlebox" }{TEXT
 238 63 " will draw the curve with any number of rectangles, as follow
s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "leftbox(x^2,x=0..30,1
0,shading=cyan);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "rightbo
x(x^2,x=0..30,10,shading=aquamarine);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "middlebox(x^2,x=0..30,10,color=blue,shading=yellow);"
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "limit(evalf(leftsum(x^2,
x=0..30,n)),n=infinity);" }}}}{PARA 0 "" 0 "" {TEXT 219 0 "" }}{SECT 1
 {PARA 202 "" 0 "" {TEXT 200 9 "Exercises" }{TEXT 200 0 "" }}{PARA 219
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italic\")), Typesetting:-mi(\"\", bold = \"false\", italic = \"true\",
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{PARA 214 "" 0 "" {TEXT 216 0 "" }}{PARA 204 "" 0 "" {TEXT 233 54 "Exe
rcise 2:  Find a third degree polynomial such that " }{TEXT 226 45 "P(
1) = 1, P'(1) = 3, P\"(1) = 6, P'''(1) = 12." }}{PARA 204 "" 0 "" 
{TEXT 226 0 "" }}{PARA 204 "" 0 "" {TEXT 226 0 "" }}{PARA 204 "" 0 "" 
{TEXT 226 178 "Exercise 3: Use Maple's sum function to create and eval
uate the left-hand and right hand sums for n=10 subintervals. Use the \+
leftsum and rightsum command to confirm your answers." }}{PARA 204 "" 
0 "" {TEXT 226 0 "" }}{PARA 204 "" 0 "" {TEXT 226 0 "" }{XPPEDIT 226 0
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