{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "Highlight" -1 256 "" 0 0 0 255 0 1 0 1 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }3 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 6 "Limits" }{MPLTEXT 1 0 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 53 "In t his worksheet we will illustrate limits in Maple." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "An Illustration of the Limit of a Function" }} {PARA 0 "" 0 "" {TEXT -1 76 "Below we define a procedure which illustr ates the limit of a given function " }{XPPEDIT 18 0 "f" "6#%\"fG" } {TEXT -1 15 " at some point " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 40 ". The plot is created on some interval [" }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT -1 59 "The exact contents of this pr ocedure need not concern you. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 304 "animate_limit := proc(f, x0, a, b)\n local h, d, t, left, rig ht;\n h := (b-a)/1000.;\n left := plots[animate]([a+d*t, f(a+d*t), t=0..x0-h-a], d=0..1, color=green, frames=50):\n right := plots[ani mate]([b+d*t, f(b+d*t), t=0..x0+h-b], d=0..1, color=red, frames=50):\n plots[display](left, right);\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Now we can define some function, as well as " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a" "6#% \"aG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 41 " , and then use our newly created command:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "f := x -> 3*x^4 + sqrt(abs(tan(x))) + sin(x)/x;\nx0 : = 0: a:= -1: b:=1:\nanimate_limit(f, x0, a, b);" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 115 "Click on the the plot and then use the V CR controls that will appear at the top of the screen to run the anima tion." }}}{PARA 0 "" 0 "" {TEXT -1 59 "From the graph we can guess tha t in this case the limit of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 44 " at 0 seems to be 1.\nWe can use the command " }{TEXT 256 5 "limit " }{TEXT -1 33 " to let Maple verify this for us." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "limit(f(x), x=0);" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "An Animation of Secant Lines" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Now we define a procedure which illustrates how the tan gent line is obtained as the limit of secant lines. " }}{PARA 0 "" 0 " " {TEXT -1 55 "The parameters used by this procedure are the function \+ " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 12 ", the point " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 57 " at which we want to displ ay the tangent, some interval [" }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 53 "] and the number o f frames we want for the animation." }}{PARA 0 "" 0 "" {TEXT -1 59 "Th e exact contents of this procedure need not concern you. " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 343 "animate_secants := proc(f, x0, a, b, n)\n \+ local i, h, h0, slope, secant, x;\n h0 := 1;\n h := h0;\n for i from 1 to n do\n h := h/1.2;\n slope := (f(x0+h) - f(x0))/h ;\n secant := unapply( slope*(x-x0) + f(x0), x);\n p.i := pl ot([f(x), secant(x)], x=a..b):\n od:\n plots[display]( seq(p.i, i= 1..n), insequence=true );\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "We can now illustrate the above computations graphically. (We are \+ still using the function " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 21 " defined previously):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "animate_se cants(f, .1, -1, 1, 20); \n" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Assignment 1" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.1:" }}{PARA 0 "" 0 "" {TEXT -1 21 "a) Define a function " }{XPPEDIT 18 0 "f(x) = (x^ 2-2) / ( x - sqrt(2))" "6#/-%\"fG6#%\"xG*&,&*$F'\"\"#\"\"\"\"\"#!\"\"F ,,&F'F,-%%sqrtG6#\"\"#F.F." }{TEXT -1 27 " and explore its limit at \+ " }{XPPEDIT 18 0 "x[0] = sqrt(2)" "6#/&%\"xG6#\"\"!-%%sqrtG6#\"\"#" } {TEXT -1 35 " as done above.\nb) Do the same for " }{XPPEDIT 18 0 "f(x ) = (1-cos(x))/x^2" "6#/-%\"fG6#%\"xG*&,&\"\"\"\"\"\"-%$cosG6#F'!\"\"F +*$F'\"\"#F/" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x[0] = 0" "6#/&%\"xG6 #\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "c) Repeat for " }{XPPEDIT 18 0 "f(x) = (x^2-2*x-3)/(x^2-4*x+3)" "6#/-%\"fG6#%\"xG*& ,(*$F'\"\"#\"\"\"*&\"\"#F,F'F,!\"\"\"\"$F/F,,(*$F'\"\"#F,*&\"\"%F,F'F, F/\"\"$F,F/" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x[0] = 3" "6#/&%\"xG6# \"\"!\"\"$" }{TEXT -1 1 "." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex. 2:" }}{PARA 0 "" 0 "" {TEXT -1 21 "a) Plot the function " }{XPPEDIT 18 0 "f(x) = x^2*cos(1/x^3)" "6#/-%\"fG6#%\"xG*&F'\"\"#-%$cosG6#*&\"\" \"\"\"\"*$F'\"\"$!\"\"F/" }{TEXT -1 24 " on the interval [-1,1]." }} {PARA 0 "" 0 "" {TEXT -1 21 "b) Use the procedure " }{TEXT 256 15 "ani mate_secants" }{TEXT -1 66 " to create an animation of the secants app roaching the tangent of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 4 " a t " }{XPPEDIT 18 0 "x=.5" "6#/%\"xG$\"\"&!\"\"" }{TEXT -1 1 "." }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.3:" }}{PARA 0 "" 0 "" {TEXT -1 74 "A mythical roller coaster is one kilometer long, and takes three m inutes. " }}{PARA 0 "" 0 "" {TEXT -1 48 "Your location as a function o f time is given by " }{XPPEDIT 18 0 "x(t) = 3-sqrt(9-t^2)" "6#/-%\"xG6 #%\"tG,&\"\"$\"\"\"-%%sqrtG6#,&\"\"*F**$F'\"\"#!\"\"F2" }{TEXT -1 1 ". " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Here is a picture of this func tion:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "x:=t->3-sqrt(9-t^2);\nplot (x(t),t=0..3);" }}}{PARA 0 "" 0 "" {TEXT -1 33 "a) What is the average velocity " }}{PARA 0 "" 0 "" {TEXT -1 34 "1. From 2 minutes to 2.5 m inutes? " }}{PARA 0 "" 0 "" {TEXT -1 34 "2. From 2 minutes to 2.1 minu tes? " }}{PARA 0 "" 0 "" {TEXT -1 34 "3. From 2 minutes to 2.01 minute s?" }}{PARA 0 "" 0 "" {TEXT -1 41 "b) What is the instantaneous veloci ty at " }{XPPEDIT 18 0 "t=2" "6#/%\"tG\"\"#" }{TEXT -1 70 "? In order \+ to compute this, first find the average velocity from 2 to " } {XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 48 " will be close to 2, and then take the limit a s " }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 14 " approaches 2." }} {PARA 0 "" 0 "" {TEXT -1 60 "c) What is the instantaneous velocity at \+ an arbitrary point " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 1 "?" }} {PARA 0 "" 0 "" {TEXT -1 68 "d) Would you be willing to ride this roll er coaster? Why or why not?" }}{PARA 0 "" 0 "" {TEXT -1 95 "Hint: What is the velocity as you get close to the end of your ride? What is the acceleration? " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.4:" }} {PARA 0 "" 0 "" {TEXT -1 21 "a) Plot the function " }{XPPEDIT 18 0 "f( x) = (3*x^2 - (7*x+1)*sqrt(x)+5)/(x-1)" "6#/-%\"fG6#%\"xG*&,(*&\"\"$\" \"\"*$F'\"\"#F,F,*&,&*&\"\"(F,F'F,F,\"\"\"F,F,-%%sqrtG6#F'F,!\"\"\"\"& F,F,,&F'F,\"\"\"F7F7" }{TEXT -1 16 " near the point " }{XPPEDIT 18 0 " x[0]=1" "6#/&%\"xG6#\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "b) Guess the value of " }{XPPEDIT 18 0 "L = limit(f(x),x \+ = x[0]);" "6#/%\"LG-%&limitG6$-%\"fG6#%\"xG/F+&F+6#\"\"!" }{TEXT -1 43 " and then evaluate the limit using Maple's " }{TEXT 256 5 "limit" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 19 "c) Using the va lue " }{XPPEDIT 18 0 "epsilon = 0.2" "6#/%(epsilonG$\"\"#!\"\"" } {TEXT -1 26 ", graph the banding lines " }{XPPEDIT 18 0 "y[1] = L-epsi lon;" "6#/&%\"yG6#\"\"\",&%\"LG\"\"\"%(epsilonG!\"\"" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "y[2]=L+epsilon" "6#/&%\"yG6#\"\"#,&%\"LG\"\"\"%(ep silonGF*" }{TEXT -1 33 " together with the function near " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 61 "d) From your graph in part c), estimate the largest possible " }{XPPEDIT 18 0 "delta > 0" "6#2\"\"!%&deltaG" }{TEXT -1 19 " such that for all " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 18 " 0 < " }{XPPEDIT 18 0 "abs(x-x[0]) < de lta" "6#2-%$absG6#,&%\"xG\"\"\"&F(6#\"\"!!\"\"%&deltaG" }{TEXT -1 15 " implies " }{XPPEDIT 18 0 "abs(f(x)-L)