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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 67 "MAT 155                 Project IX                       \+
 Fall 2002" }}{PARA 0 "" 0 "" {TEXT 256 69 "                         I
ncreasing and Decreasing                   " }{TEXT -1 5 "     " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
129 "Your work should be done neatly in the same format as previous la
bs and saved regularly as yourname9.mws and as yourname9b.mws.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 38 "You shou
ld write comments as you work." }{TEXT -1 199 "  To do so just backspa
ce in front of the prompt > and type in the comments.  They should app
ear in black not in red.   When you finish the lab be sure to go over \+
your comments and check the grammar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 258 10 "Problem 1:" }{TEXT -1 2 "  " }{TEXT 268 32 "Local Maxi
ma and Critical Points" }}{PARA 0 "" 0 "" {TEXT 263 3 "a) " }{TEXT -1 
45 "Define a function f which takes x to x^3-x.  " }}{PARA 0 "" 0 "" 
{TEXT 264 3 "b) " }{TEXT -1 398 "Graph it from -1.5 to 1.5.  Write a c
omment explaining where the function increases and where it decreases.
 Try to estimate where it switches from increasing to decreasing.  Suc
h points are called local maxima.  What is the slope at the points whe
re it switches?  Now check where it switches from decreasing to increa
sing.  These are the local minima.  What is the slope of f at the loca
l minima?  " }}{PARA 0 "" 0 "" {TEXT 265 2 "c)" }{TEXT -1 88 " Find it
s derivative using the D command and graph the derivative from -1.5 to
 1.5 using" }}{PARA 0 "" 0 "" {TEXT -1 29 "> plot(D(f)(x), x=-1.5..1.5
);" }}{PARA 0 "" 0 "" {TEXT 266 2 "d)" }{TEXT -1 326 " Write a comment
 discussing where the derivative is positive and where it is negative.
 Use solve to find out where the derivative is 0.  A points where the \+
derivative is zero is called a critical point.  How are the critical p
oints you've found related to the graph of f ?  How are they related t
o the local minima and maxima?" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 
0 "" }}{PARA 258 "" 0 "" {TEXT -1 11 "Problem 2: " }{TEXT 267 26 "Don'
t forget to restart.  " }{TEXT -1 2 "  " }}{PARA 259 "" 0 "" {TEXT 
259 78 "Repeat problem 1 with f(x)= 3x-exp(3x).  You may need to use f
solve in step d." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 262 36 "Problem 3: A special critical point:" }}
{PARA 0 "" 0 "" {TEXT -1 331 "Repeat problem 1 with f(x)=tan(x)-x.  No
tice that this function is always increasing but the slope is zero at \+
one point between -1.5 and 1.5.  Explain.      Can a point be a critic
al point even though it isn't a local minimum nor a local maximum?  Ex
amine the graph near that point.  What is its linear approximation at \+
that point?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
260 52 "Problem 4:  Locating the interesting part of a graph" }}{PARA 
0 "" 0 "" {TEXT -1 405 "a)  The graph of f(x)=x^2-200x+10101   is a pa
rabola but it is difficult to choose good x and y bounds to make it lo
ok like one. The best way to graph it well is to find the location of \+
its local minimum.  A graph switches from decreasing to increasing at \+
a local minimum, so it has a critical point there.  You can find it by
 solving  D(f)(x)=0.   Then graph f,  making sure to include the local
 minimum." }}{PARA 0 "" 0 "" {TEXT -1 53 "b)  repeat part a) with f(x)
= 202x-x^2   and comment." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 261 11 "Exploration" }{TEXT -1 181 ": Examine other graph
s of functions and their derivatives to study where the function incre
ases and decreases.  Some functions to look at are sqrt(6-x), 1/x, x-2
/x and/or 1/(1+x^2)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }