{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Outpu t" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 66 "MAT 155 Project III \+ Fall 2002" }}{PARA 257 "" 0 "" {TEXT -1 44 " \+ Examining Graphs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 256 25 "Do not work on this file." }{TEXT -1 38 " This is just the list of problems. " }{TEXT 257 18 "Open your own \+ file" }{TEXT -1 82 " by selecting \"File\" on the Maple V bar and choo sing \"New\". Save the new file as " }{TEXT 258 14 "yourname3.mws " } {TEXT -1 7 "and as " }{TEXT 259 14 "yourname3b.mws" }{TEXT -1 41 " reg ularly using the \"save as\" command. " }{TEXT 260 81 "Be careful not to save this file with that name or you will overwrite your work! " } {TEXT -1 247 " This window should always have 155.00.03.mws printed on the top. If you click on the button on this window next to the X ( the one marked with two boxes) then you will be able to see this file \+ and your file at the same time. You can adjust the" }}{PARA 0 "" 0 " " {TEXT -1 52 "sizes of the windows displaying each file as needed." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 30 "Do not \+ hit return on this file" }{TEXT -1 122 " or the computer will remember it when working on your file and may overwrite variables and function s that you've defined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 89 "Sign your name as a comment at the top of your file by backspacing in front of the prompt" }{TEXT -1 104 " and typing it \+ in. Also write Project III and the names of any other students who ar e working with you." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 262 28 "At the start of each problem" }{TEXT -1 160 " in your \+ file, backspace in front of the prompt and type the Problem Number 1 o r Problem Number 2 etc. Then hit return and on the first line with a \+ prompt type" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "This will clear all previous work. " }{TEXT 264 48 "Also remember to save your file with both names." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 86 "Do all the problems in order. Ask a student or the teach er if you are having trouble." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 41 "Problem 1: Solving a \+ quadratic equation." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Define a f unction f(x)= x^2+5x+6 using the command:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:= x -> x^2 +5*x+6;" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 72 "To find out where f(x)=0, there are several methods. Y ou can factor it:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(f(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "So that you see that it is zero at the two roots: x = 3 and x = 2 . Note that if f(x) has no rational solutions then Maple cannot facto r it. Try g(x)=x^2-2 and h(x)= x^2+4x+6." }}{PARA 0 "" 0 "" {TEXT -1 31 "You could also use the command " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(f(x)=0, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "The computer essentially solves the quadratic \+ equation to get these answers. It will give you irrational answers wi th square root signs and imaginary answers with " }{TEXT 265 7 "I . I " }{TEXT -1 222 "is just the square root of negative one. It is not \+ a real number. Try the solve command on g(x) and h(x) to see why Mapl e couldn't factor them. Write a comment about their solutions by backs pacing in front of the prompt." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Now try graphing f(x), g(x) and h(x) using the command:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(f(x), x=-10.. 10);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "You can focus the graph near where it crosses the x axis \+ by focussing on x in [-6,0]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot (f(x), x=-6..0);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 55 "Does it cross at the correct places? Write a com ment." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Now p lot g(x) and h(x) and write comments about the places where they cross the x axis." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 1 " " }}{PARA 261 "" 0 "" {TEXT -1 45 "Problem 2: Solving more compli cated problems" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 44 "Remember to save your file and type restart;" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 204 "Define a function f(x)=cos(x )-x and try to solve f(x)=0. It cannot be factored because it is not \+ a polynomial and the solve command doesn't work. However, you can gra ph the function, so try to graph it." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(cos(x)-x, x=-10..10);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 365 "This graph appears to cross the x axis only once and it appears to cross near x=1. It definitely cross es the axis somewhere between 0 and 2. So we can try the fsolve comma nd. This command will try alot of numbers between 0 and 2 until it fi nds one that solves f(x)=0. However, it will only look for one answer and then quit even if there are two answers. Type:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "fsolve(f(x)=0, x=0..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 433 "Now graph the picture again very close to the answer \+ say x in .5 to 1 and see how it crosses the axis. Can you be sure tha t the function only crosses the axis once? Maybe it crosses again nea r x=100. One way to be sure it only crosses once would be to show tha t the function is decreasing (that its graph goes down from left to r ight). Later in the semester, we will be able to prove that f(x)=cos( x)-x is a decreasing function." }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 238 "Try to find a so lution to g(x)=(x^2-7)^(1/3) by graphing and using fsolve. You will h ave to find more than one solution by focusing your graph on two diffe rent locations and using fsolve twice. Then check your answer by gra phing again." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 36 "Problem 3: The irrat ional number e:" }}{PARA 0 "" 0 "" {TEXT -1 543 "Let f(x)=(1+x)^(1/x). Notice that this function is undefined at two numbers, x=-1, and x=0 . When x=-1 we get f(-1)=(0)^(-1)=1/0 and at x=0 we get f(0)=(1)^(1/0 ). Any number divided by 0 is undefined. f(x) is also undefined at x =-2 because f(-2)=(-1)^(-1/2)= 1/ (-1)^(1/2) and the square root of ne gative one is not a real number. If x>-1 we don't have to worry about having roots of negative numbers. So we can say that the domain of f is (-1,0) u (0, infinity). What does the function look like near x =0 and x=-1? Try the command: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot((1+x)^(1/x), x=-1...1);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 209 "f(x) is going to infinity near x= -1. This is called a vertical asmptote. However, near x=0, the funct ion's graph appears to be perfectly nice. You might even think that \+ f(x)=(1+x)^(1/x) is defined at zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 410 "Suppose you say that 1/0=infinity. Th en f(0)=(1+0)^(infinity)=1^(infinity)=infinity. This doesn't make se nse. It doesn't help you guess where the graph appears to cross the x axis. Instead you must focus on x values near 0 and see where it app ears to cross. Focus the graph very close to x=0. Try plotting it fr om x=-0.1 to 0.1. Write a comment estimating the number where the gra ph appears to cross. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Now plot the graph from x=-0.01 to 0.01. " }{TEXT 270 92 "Remember you can cut and paste your plot command onto a new li ne and just edit the x values." }{TEXT -1 126 " Then write a comment telling where it appears to cross. What is the largest y value on th e graph and what is the smallest?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 144 "Now plot it from x=-0.001 to 0.001 and w rite down where it appears to cross. What is the largest y value on t he graph and what is the smallest?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "Now plot it from x=-0.0001 to 0.0001 and write down where it appears to cross. What is the largest y value on the graph and what is the smallest?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 148 "Now plot it from x=-0.00001 to 0.00001 a nd write down where it appears to cross. What is the largest y value \+ on the graph and what is the smallest?" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 123 "Even though f(0) is not defined, we \+ know that as x gets closer and closer to 0, f(x) gets closer and close r to some number." }}{PARA 0 "" 0 "" {TEXT -1 123 "You already know an approximate value for this number. It turns out that it is an irrati onal number. The number is called" }{TEXT 266 5 " e. " }{TEXT -1 149 "It turrns out that e is as important a number as Pi and sqrt(2) a nd is used in many applications of mathematics. We will see it a lot \+ in Calculus. " }{TEXT 267 53 "We say that the limit of f(x) as x appr oaches 0 is e." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 " " 0 "" {TEXT -1 12 "Exploration:" }{MPLTEXT 1 0 3 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 27 "A: Solving Inequalities. " }}{PARA 0 " " 0 "" {TEXT -1 156 "Let f(x)=x^2 -3x+2 and solve for f(x) >0. To do \+ this solve(f(x)=0) and then graph the function. Find the x values wh ere the graph of f is above the axis." }}{PARA 0 "" 0 "" {TEXT -1 71 " Let g(x)= sin(x)-x^2+3 and solve for f(x) less than or equal to zero. \+ " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 18 "B: Ano ther limit:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Let f(x)=(x^5-32)/(x-2). Discuss where it is defined and check o ut the limit where it is not defined." }{MPLTEXT 1 0 0 "" }}}}{MARK "3 3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }