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e 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 207 "We see that th
e closer we look at (1,0), the closer the tangent line is to the graph
. Moreover, we notice the following. At every step we halved the size \+
of the x-interval. The distance between the graph of " }{XPPEDIT 18 0 
"f(x) = ln(x);" "6#/-%\"fG6#%\"xG-%#lnG6#F'" }{TEXT -1 271 " and the t
angent line is not only halved but gets smaller more rapidly than that
. This is the meaning of the tangent line approximation. We can see th
e numerics as well. The closer we are to 1, the closer the value of th
e tangent line to the actual value of the function." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 215 "for k from 1 to 10 \ndo \n    printf (\"
x: %f; f(x): %f; Lf(x): %f; error %f\\n\", \n           1+2^(-k),\n   \+
        evalf(f(1+2^(-k))), \n           evalf(Lf(1+2^(-k))), \n      \+
     evalf(f(1+2^(-k))-Lf(1+2^(-k))));  \nod;" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 61 "x: 1.500000; f(x): 0.4054
65; Lf(x): 0.500000; error -0.094535" }}{PARA 6 "" 1 "" {TEXT -1 61 "x
: 1.250000; f(x): 0.223144; Lf(x): 0.250000; error -0.026856" }}{PARA 
6 "" 1 "" {TEXT -1 61 "x: 1.125000; f(x): 0.117783; Lf(x): 0.125000; e
rror -0.007217" }}{PARA 6 "" 1 "" {TEXT -1 61 "x: 1.062500; f(x): 0.06
0625; Lf(x): 0.062500; error -0.001875" }}{PARA 6 "" 1 "" {TEXT -1 61 
"x: 1.031250; f(x): 0.030772; Lf(x): 0.031250; error -0.000478" }}
{PARA 6 "" 1 "" {TEXT -1 61 "x: 1.015625; f(x): 0.015504; Lf(x): 0.015
625; error -0.000121" }}{PARA 6 "" 1 "" {TEXT -1 61 "x: 1.007812; f(x)
: 0.007782; Lf(x): 0.007812; error -0.000030" }}{PARA 6 "" 1 "" {TEXT 
-1 61 "x: 1.003906; f(x): 0.003899; Lf(x): 0.003906; error -0.000008" 
}}{PARA 6 "" 1 "" {TEXT -1 61 "x: 1.001953; f(x): 0.001951; Lf(x): 0.0
01953; error -0.000002" }}{PARA 6 "" 1 "" {TEXT -1 61 "x: 1.000977; f(
x): 0.000976; Lf(x): 0.000977; error -0.000000" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 44 "The errors get smaller the closer we are to " }
{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 177 ". The errors ar
e negative, because the tangent line overestimates the function. The t
angent line lies above the graph of the function, since the function i
s concave downwards.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "
" 0 "" {TEXT -1 14 "Assignment 1  " }{TEXT 257 34 "Make similar tables
 for values of " }{XPPEDIT 257 0 "x < 1;" "6#2%\"xG\"\"\"" }{TEXT -1 
1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 98 "On the other hand, if we are far away from (1,0), the tangent l
ine approximation is not very good:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "evalf(f(2)); Lf(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#$\"+1=ZJp!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(f(3)); Lf(3);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+*G7')4\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The reason is that the
 function " }{XPPEDIT 18 0 "ln(x);" "6#-%#lnG6#%\"xG" }{TEXT -1 125 " \+
bends, as a concave downwards function, while the tangent line does no
t. Somehow we have to take into account the concavity." }}{PARA 0 "" 
0 "" {TEXT -1 77 "In Calculus 1 we saw that the concavity is measured \+
by the second derivative." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "Qf(
x) = x-1-(x-1)^2/2;" "6#/-%#QfG6#%\"xG,(F'\"\"\"F)!\"\"*&,&F'F)F)F*\"
\"#F-F*F*" }{TEXT -1 31 " has the following properties: " }}{PARA 0 "
" 0 "" {XPPEDIT 18 0 "Qf(1) = f(1);" "6#/-%#QfG6#\"\"\"-%\"fG6#F'" }
{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 12 "Qf'(1)=f'(1)" }}{PARA 0 
"" 0 "" {TEXT -1 15 "Qf''(1)=f''(1):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "Qf:= x -> x-1-(x-1)^2/2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#QfGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(9$\"\"\"F.!
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"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Let's graph  f(x),   the linear ap
proximation Lf(x)  and the quadratic approximation  Qf(x);" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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BELSG6$Q!6\"Fg^m-%%VIEWG6$;$\"$v)!\"$$\"%D6F__m%(DEFAULTG" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv
e 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 206 "At this moment we can ha
rdly distinguish the quadratic function, which is a parabola, from the
 logarithm. We make also a table showing the error of appoximation for
 both the linear and quadratic  functions." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "for k from
 1 to 10 \ndo \n    printf (\"x: %f; Linear Error: %f; Quadratic Error
: %f\\n\", \n           1+2^(-k), \n           evalf(f(1+2^(-k))-Lf(1+
2^(-k))), \n           evalf(f(1+2^(-k))-Qf(1+2^(-k))));  \nod;" }}
{PARA 6 "" 1 "" {TEXT -1 63 "x: 1.500000; Linear Error: -0.094535; Qua
dratic Error: 0.030465" }}{PARA 6 "" 1 "" {TEXT -1 63 "x: 1.250000; Li
near Error: -0.026856; Quadratic Error: 0.004394" }}{PARA 6 "" 1 "" 
{TEXT -1 63 "x: 1.125000; Linear Error: -0.007217; Quadratic Error: 0.
000596" }}{PARA 6 "" 1 "" {TEXT -1 63 "x: 1.062500; Linear Error: -0.0
01875; Quadratic Error: 0.000078" }}{PARA 6 "" 1 "" {TEXT -1 63 "x: 1.
031250; Linear Error: -0.000478; Quadratic Error: 0.000010" }}{PARA 6 
"" 1 "" {TEXT -1 63 "x: 1.015625; Linear Error: -0.000121; Quadratic E
rror: 0.000001" }}{PARA 6 "" 1 "" {TEXT -1 63 "x: 1.007812; Linear Err
or: -0.000030; Quadratic Error: 0.000000" }}{PARA 6 "" 1 "" {TEXT -1 
63 "x: 1.003906; Linear Error: -0.000008; Quadratic Error: 0.000000" }
}{PARA 6 "" 1 "" {TEXT -1 63 "x: 1.001953; Linear Error: -0.000002; Qu
adratic Error: 0.000000" }}{PARA 6 "" 1 "" {TEXT -1 64 "x: 1.000977; L
inear Error: -0.000000; Quadratic Error: -0.000000" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 93 "We see that the errors are smaller for the quadra
tic polynomial. The errors are positive for " }{XPPEDIT 18 0 "h(x);" "
6#-%\"hG6#%\"xG" }{TEXT -1 13 " because for " }{XPPEDIT 18 0 "1 < x;" 
"6#2\"\"\"%\"xG" }{TEXT -1 51 " the parabola was below the graph of th
e logarithm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 258 13 "Assignment 2 " }{TEXT -1 44 " :  Make the correspondin
g table for x < 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 220 "The errors are negative because the tangent li
ne and the parabola are above the logarithm. The errors with the parab
ola are smaller than the tangent line approximation. The approximation
 is not very good far away from 1." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "plot([f,Lf,Qf], 0..3, color=[red, blue, green]);" }}
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a starts decreasing at " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }
{TEXT -1 206 ", while the logarithm keeps increasing. So there is no h
ope further away from 2. If one wants a better approximation, one has \+
to use higher degree polynomials. We would like to find a 3rd degree p
olynomial " }{XPPEDIT 18 0 "P[3](x);" "6#-&%\"PG6#\"\"$6#%\"xG" }
{TEXT -1 13 ", such that: " }{XPPEDIT 18 0 "P[3](1) = f(1);" "6#/-&%\"
PG6#\"\"$6#\"\"\"-%\"fG6#F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "P[3];" "
6#&%\"PG6#\"\"$" }{TEXT -1 11 "'(1)=f'(1)," }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "P[3];" "6#&%\"PG6#\"\"$" }{TEXT -1 17 "''(1)=f''(1) and
 " }{XPPEDIT 18 0 "P[3];" "6#&%\"PG6#\"\"$" }{TEXT -1 106 "'''(1)=f'''
(1). Luckily Maple can gives us this polynomial, called the third degr
ee Taylor polynomial for " }{XPPEDIT 18 0 "ln(x);" "6#-%#lnG6#%\"xG" }
{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 
17 ". The command is:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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XESLABELSG6$Q!6\"Fhbn-%%VIEWG6$;Fh`lF\\`l%(DEFAULTG" 1 2 0 1 10 0 2 9 
1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "
Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "We see that tay3 (br
own, orange curve) is closer to the logarithm (red curve) than the tan
gent line (blue) and the second degree polynomial Qf(x) (green)." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" 
{TEXT -1 14 "Assignment #3 " }{TEXT 259 53 " Find the 4th, 5th, 6th de
gree Taylor polynomials of " }{XPPEDIT 260 0 "ln(x);" "6#-%#lnG6#%\"xG
" }{TEXT 261 50 " at x=1. Plot them and make a table of values for " }
{XPPEDIT 262 0 "x = 1.1;" "6#/%\"xG-%&FloatG6$\"#6!\"\"" }{TEXT 263 
17 " and another for " }{XPPEDIT 264 0 "x = 1.2;" "6#/%\"xG-%&FloatG6$
\"#7!\"\"" }{TEXT 265 17 " and another for " }{XPPEDIT 266 0 "x = .9;
" "6#/%\"xG-%&FloatG6$\"\"*!\"\"" }{TEXT 267 1 "." }}}}{MARK "31 0 0" 
195 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }