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{SECT 0 {EXCHG {PARA 206 "" 0 "" {TEXT 200 50 "MATH 156   MIDTERM  - P
rof Keen   Due Oct 20, 2006" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 203 55 "I
n this take home exam you will work with the integral " }{XPPEDIT 19 1
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ng:-mi(\"\"), Typesetting:-msubsup(Typesetting:-mo(\"∫\", form = \+
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12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), Typ
esetting:-mverbatim(\"\"\"!\"), Typesetting:-mn(\"1\"), superscriptshi
ft = \"2\", subscriptshift = \"0\"), Typesetting:-mi(\"\"), Typesettin
g:-mfrac(Typesetting:-mrow(Typesetting:-mverbatim(\"\"\"\"\")), Typese
tting:-mverbatim(\"*$,&\"\"\"\"\"\"*$)%\"xG\"\"#\"\"\"\"\"\"\"\"\"\"),
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, bevelled = \"false\", foreground = \"[255,0,0]\", background = \"[25
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, Typesetting:-mo(\"ⅆ\", form = \"prefix\", fence = \"fa
lse\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stre
tchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsi
ze = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+
\"false\", font_style_name = \"2D Input\", size = \"12\", foreground =
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m(\"%\"xG\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,
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nameGQ)2D~InputF'/%%sizeGQ#12F'/%+foregroundGQ*[255,0,0]F'/%+backgroun
dGQ.[255,255,255]F'-I*mverbatimGF$6#Q$\"\"!F'-I#mnGF$6#FM/%1superscrip
tshiftGQ\"2F'/%/subscriptshiftGQ\"0F'F+-I&mfracGF$6*-F#6#-F[o6#Q$\"\"
\"F'-F[o6#Q>*$,&\"\"\"\"\"\"*$)%\"xG\"\"#\"\"\"\"\"\"\"\"\"F'/%.lineth
icknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFgp/%)bevelledGF<FZF
gnF+-I'mspaceGF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.3~emF'/%&depthGFaq/
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HFKFNFPFRFTFWFZFgn-F[o6#Q%%\"xGF'F+" }{TEXT 203 43 ".   Using Maple to
 find the integral we see" }}{PARA 0 "" 0 "" {TEXT 203 0 "" }}{PARA 0 
"" 0 "" {TEXT 203 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "E
xample := Int(1/(1+x^2), x = 0 .. 1); value(Example);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(ExampleG6\"-
I$IntG6$%*protectedGI(_syslibGF$6$*$,&\"\"\"F-*$)I\"xGF$\"\"#F-F-!\"\"
/F0;\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&#\"\"\"\"\"%F%I#PiG%*
protectedGF%F%" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "" "%#%?G" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
200 173 "We will use this example to approximate the value of Pi.   Al
l the questions below refer to the function inside the integral sign. \+
  That is, we will work with the function " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "f := x-> 4/(x^2+1);" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I\"fG6\"f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$,$*&\"\"%\"\"\",&*$
)F'\"\"#F.F.F.F.!\"\"F.F$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19
 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "" "%#%?G" }}}
{EXCHG {PARA 200 "" 0 "" {TEXT 204 266 "Answers all the questions. Whe
n an explanation is required, your answer should not just be: \"I saw \+
it on the graph\" or \"The numbers show that A<B\". Explain your answe
rs using the properties of the function involved. Do not forget the st
udent and plots packages. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 203 39 "1.
 Graph the function in the range [0, " }{XPPEDIT 18 0 "Typesetting:-mr
ow(Typesetting:-mi(\"\"), Typesetting:-mn(\"1\"), Typesetting:-mi(\"\"
));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6
#Q!F'-I#mnGF$6#Q\"1F'F+" }{TEXT 203 24 "]. Use as range for the " }
{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT 203 57 "-values [0,4]. Recall that
 in a plot you can specify the " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT 
203 19 "-range with y=0..4." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 203 123 "
2. Plot on the same graph the function and the left-hand sums and righ
t-hand sums with the same number of subintervals for " }{XPPEDIT 18 0 
"n;" "6#%\"nG" }{TEXT 203 39 "=2, 4, 8, 16,  e.g. LHS(8) and RHS(8). "
 }}{PARA 0 "" 0 "" {TEXT 203 117 "3. Plot on the same graph the functi
on and the left-hand sums and midpoint sums with the same number of su
bintervals " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 203 39 "=2, 4, 8, 16
,  e.g. LHS(8) and MID(8). " }}{PARA 0 "" 0 "" {TEXT 203 119 "4. Plot \+
on the same graph the function and the right-hand sums and midpoint su
ms with the same number of subintervals  " }{XPPEDIT 18 0 "n;" "6#%\"n
G" }{TEXT 203 39 "=2, 4, 8, 16,  e.g. RHS(8) and MID(8). " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 203 86 "5. Compute the left-hand sums, right-han
d sums, midpoint sums and trapezoid rule with " }{XPPEDIT 18 0 "n;" "6
#%\"nG" }{TEXT 203 123 "=2, 4, 8, 16, 32, 64, 126, 256, 512, 1024 subi
ntervals. Which ones are overestimates and which ones are underestimat
es for " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 203 59 "? Why?  Put the
se numbers in increasing order, e.g. LHS(8)<" }{XPPEDIT 18 0 "Pi;" "6#
%#PiG" }{TEXT 203 173 "<TRAP(8)<MID(8)<RHS(8). This order is not  corr
ect, your work should show the correct order.  Explain your ordering, \+
i.e. explain why LHS(8)<TRAP(8) or the other way around." }}{PARA 0 ""
 0 "" {TEXT 203 30 "6. How many decimal digits of " }{XPPEDIT 18 0 "Pi
;" "6#%#PiG" }{TEXT 203 74 " can you get using left-hand sums and righ
t-hand sums? How many digits of " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT 203 56 " can you get using the trapezoid rule and midpoint sums?
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 203 67 "7. Plot on the same graph th
e function and the trapezoid rule with " }{XPPEDIT 18 0 "n = 1;" "6#/%
\"nG\"\"\"" }{TEXT 203 69 " subintervals. Explain why it is an underes
timate or an overestimate." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 203 67 "8.
 Plot on the same graph the function and the trapezoid rule with " }
{XPPEDIT 18 0 "n = 2;" "6#/%\"nG\"\"#" }{TEXT 203 56 " subintervals. W
hich is larger? TRAP(1) or TRAP(2)? Why?" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 203 134 "9. Explain why the midpoint sums are underestimates or \+
overestimates by showing on a graph the midpoint sums using tangent li
nes. Use " }{XPPEDIT 18 0 "n = 1;" "6#/%\"nG\"\"\"" }{TEXT 203 5 " and
 " }{XPPEDIT 18 0 "n = 2;" "6#/%\"nG\"\"#" }{TEXT 203 14 " subinterval
s." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 203 59 "10. EXTRA CREDIT: Use Mapl
e to decide whether the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%
\"xG" }{TEXT 203 42 " is concave upwards or concave downwards. " }}}}
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