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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT -1 64 "MAT 155                 Project VI                     Fa
ll 2002" }}{PARA 259 "" 0 "" {TEXT -1 70 "                       Findi
ng Tangent Lines                          " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 258 25 "Do not work on this file." }{TEXT -1 108 "  This is ju
st the list of problems.  Open your own file and save it as yourname6.
mws and as yourname6b.mws." }{TEXT 259 1 " " }{TEXT -1 275 " Sign your
 name as a comment at the top of your file by backspacing in front of \+
the prompt and typing.  Also write Project VI and the names of any oth
er students you are working with.  Don't forget to number your problem
s and to type restart at the beginning of each problem." }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 89 "If yo
u cannot complete a problem, go on to the next one and return to the p
roblem later. " }{TEXT -1 9 " You can " }{TEXT 257 16 "get a new promp
t" }{TEXT -1 236 " by selecting the prompt button right below the word
 \"spreadsheet\".  It has the symbol \"[>\" on it.  You must hit enter
 on every line of the problem in order, including the restart line, to
 review what you've done for the Maple program. " }{MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 260 
31 "If you cannot recall a command " }{TEXT -1 160 "from a previous la
b you may consult the command index which can be opened up as a second
 window.  The name of the file with the command index is 155.00.00.htm
l." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
261 10 "Problem 1:" }{TEXT -1 2 "  " }{TEXT 275 6 "Loops:" }{TEXT -1 
153 "  Before beginning to study tangent lines we will learn one of th
e most fundamental techniques used in computer programming.  This tech
nique is called a " }{TEXT 262 4 "loop" }{TEXT -1 49 ". In later probl
ems we will use a loop program to" }}{PARA 0 "" 0 "" {TEXT -1 158 "gra
ph many secant lines to a curve without having to repeatedly type the \+
plot command.  To make life a little easier we won't do any calculus i
n this problem." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 83 "Suppose we want the computer to write a list 1^2, 2^
2, 3^2...10^2.  We tell it that" }{TEXT 269 7 " for j " }{TEXT -1 8 "r
unning " }{TEXT 264 13 "from 1 to 10 " }{TEXT -1 9 "it should" }{TEXT 
270 4 " do " }{TEXT -1 11 "the command" }{TEXT 265 4 " j^2" }{TEXT -1 
44 ".  The maple program understands words like " }{TEXT 263 4 "for," 
}{TEXT -1 1 " " }{TEXT 266 8 "from, to" }{TEXT -1 5 " and " }{TEXT 
267 2 "do" }{TEXT -1 131 ".  Sometimes there is a long list of command
s for the computer to complete so we end the list of things it should \+
do with the word " }{TEXT 268 3 "od " }{TEXT -1 11 "as follows:" }}
{PARA 0 "" 0 "" {TEXT 272 32 ">  for j from 1 to 10 do j^2 od;" }}
{PARA 0 "" 0 "" {TEXT -1 92 "This is a one line program called a loop \+
because it repeats the same command over and over. " }{TEXT 273 1 " " 
}{TEXT 271 109 "Before hitting enter on a loop command like the one be
low, always save your work.  Your computer might crash." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "for j from 1 to 10 do j^2 od;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 197 "Copy this loop program using copy
 and paste and try it out on your file.  Then  write a program which f
inds j^2 for j running from 1 to 25.  Then write a program which finds
 sin(Pi*j) for j=1 to 5." }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 274 34 "Problem
 2: Plotting Secant Lines: " }{TEXT -1 22 " Define f(x)=x^3 using" }}
{PARA 0 "" 0 "" {TEXT -1 17 "[>  f:= x -> x^3." }}{PARA 0 "" 0 "" 
{TEXT -1 288 "Now we want to find the secant line which meets the grap
h of f(x) above x=2 and x=3.  To do this we need to use the point slop
e formula:  y= m( x- a) + b where m is the slope and (a,b) is a point \+
on the line.  Since (2, f(2)) is on the line a=2 and b=f(2) will do fi
ne.  Use the commands " }}{PARA 0 "" 0 "" {TEXT -1 17 "[>      a:=2;  \+
  " }}{PARA 0 "" 0 "" {TEXT -1 18 "[>      b:=f(a);  " }}{PARA 0 "" 0 
"" {TEXT -1 145 "to set these values on the computer.  To find the slo
pe, m, we use the fact that the secant line is passing through (2, f(2
)) and (3, f(3)).  So " }}{PARA 0 "" 0 "" {TEXT -1 30 "[>      m:= (f(
3)-f(2))/(3-2);" }}{PARA 0 "" 0 "" {TEXT -1 58 "Now you can plot the s
ecant line with f using the command:" }}{PARA 0 "" 0 "" {TEXT -1 43 "[
>     plot( \{f(x), m*(x-a)+f(a)\}, x=0..3);" }}{PARA 0 "" 0 "" {TEXT 
-1 80 "Verify that the secant line does in fact cross the graph of f a
bove x=2 and x=3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 82 "Now find the secant line which meets the graph of  f(x)=x
^3 above  x=2 and x=1.9 ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}
{PARA 257 "" 0 "" {TEXT -1 61 "Problem 3:  Using Loops to Find and Und
erstand Tangent Lines:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 276 3 "a) " }{TEXT -1 557 " To find the tangent line to f(x)=x^3
 through the point (2, f(2)), we need to look at many secant lines eac
h crossing the graph of f(x) a a point above 2 and a point near 2.  Th
ese secant lines will look more and more like the tangent line at 2 wh
en the second point gets very close to 2.  For example, we could find \+
the secant lines which cross f(x) above 2 and above (2+1/j)  where j i
s getting larger and larger.   This would require a lot of work on our
 part to plug in different values of j, compute the slope and then plo
t the curve, so we use a loop." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "To compute secant lines between t
he point 2 and points closer and closer to 2, we tell the computer tha
t" }}{PARA 0 "" 0 "" {TEXT -1 127 "for  j= 1 to 10 it should do the co
mputation m:=(f(2+1/j)-f(2))/(2+1/j-2) to find the slope and then to p
lot the secant line.  " }{TEXT 285 15 "Save your work " }{TEXT 284 88 
"and then type the following program without hitting return until afte
r the od is typed. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "for \+
j from 1 to 10 do m:= (f(2+1/j)-f(2))/(1/j); plot(\{f(x), m*(x-2)+f(2)
\}, x=1..3); od; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT -1 144 "You should get a lot of pictures and definitions
 of slopes, m, above each picture.  If you don't, it is probably becau
se you forgot to define f." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 270 "Now scroll back up to see all the
 graphs you've made.  You will see that these secant lines appear to r
epeat.  In fact they don't quite repeat but get very close to a line c
alled a tangent line.  The last slope m is a good approximation of the
 slope of the tangent line." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 3 "b) " }{TEXT 
-1 281 " Now define a new function f(x) = sin(x) and use a loop like t
he one above to estimate the tangent line to sin(x) which passes throu
gh (0, sin(0)).  Remember to look at secant lines which cross the grap
h of f above x=0 and above x=0+1/j and plot the curve over [-2,2] not \+
[1,3].   " }{TEXT 283 73 "Be very careful to save your work before hit
ting enter on a loop command." }{TEXT -1 95 "  Then write a comment wh
ich gives the value of the slope of the tangent line and explains why.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 13 "Exploration: " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 42 "A: Tange
nt Lines to Exponential Functions:" }}{PARA 0 "" 0 "" {TEXT -1 319 "Us
ing the same loop that worked for f(x)=sin(x), find the slopes for 2^x
, 3^x and e^x at x=0.  Note that in maple the command exp(x) is the co
rrect way to define e^x because maple does not understand the letter e
.  Write down your conclusions about the slopes and compare with ln(2)
 and ln(3) using the evalf command.  " }{TEXT 282 73 "Be very careful \+
to save your work before hitting enter on a loop command." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 14 "B: More Loops:" }
}{PARA 0 "" 0 "" {TEXT -1 211 "Use a loop to estimate the limit as x a
pproaches 0 of g(x) and then test if your loop works by substituting v
arious functions for g(x) and comparing your estimates with the ones f
ound using maple's limit command" }}{PARA 0 "" 0 "" {TEXT -1 22 "[> li
mit( g(x), x=0 );" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 286 38 "C:  When the technique doesn't work:  " }{TEXT -1 157 "T
his method for finding tangent lines by taking secant lines through (a
, f(a)) and (a+1/j, f(a+1/j)) does not work if the tangent line doesn'
t exist at a .  " }{TEXT 288 82 "If the tangent line to f(x) at a does
 not exist then two things might happen:  1) " }{TEXT -1 136 " The var
ious plots of the secant lines will not look like each other even if y
ou take j very large  ( try a=0, f(x)=sin(Pi/4x) )   or   " }{TEXT 
287 2 "2)" }{TEXT -1 113 "  the secant lines will get closer and close
r to a line but it won't be a tangent line  (try a=0 f(x)=abs(x) ).  \+
" }{TEXT 289 73 "Be very careful to save your work before hitting ente
r on a loop command." }}{PARA 0 "" 0 "" {TEXT 279 2 "  " }{TEXT -1 1 "
 " }{MPLTEXT 1 0 0 "" }}}}{MARK "23" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }