{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple O utput" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Heading 2" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 269 33 "Continuity and Limits at Infinity " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 9 "Example 1" }{TEXT -1 48 "\nFor the following function, find th e value of " }{TEXT 256 1 "a" }{TEXT -1 37 " that makes the function c ontinuous. " }}{PARA 0 "" 0 "" {TEXT -1 79 "Plot the continuous functi on. Then take different values of the variable a and " }}{PARA 0 "" 0 "" {TEXT -1 45 "plot the associated discontinuous functions. " }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "f(x) = a*x" "6#/-%\"fG6#%\"xG*&%\"aG\" \"\"F'F*" }{TEXT -1 21 " for x <= 2 and " }{XPPEDIT 18 0 "f(x) = \+ a*x^2 + x + 1" "6#/-%\"fG6#%\"xG,(*&%\"aG\"\"\"*$F'\"\"#F+F+F'F+F+F+" }{TEXT -1 12 " for x > 2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Define the two pie ces" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f1 := x->a*x; f2 := x->a*x^2 +x+1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Substitute the break poi nt x=2 into both functions, set them equal, and then solve for a:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "a := solve(f1(2)=f2(2), a);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f1(x); f2(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Plot the function." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f:= x -> piecewise( x <= 2, f1(x), f2(x));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(f(x), x = -5 ..5);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "What if w e'd gotten " }{TEXT 268 1 "a" }{TEXT -1 35 " wrong? Suppose we though t a = -4." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a := -4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f1(x); f2(x);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 34 "Plot the (discontinuous) function." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "f:= x -> piecewise( x <= 2, f1(x), f2(x));" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(f(x), \+ x = -5..5, discont=true);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 42 "Plot the graph of the following function. " }}{PARA 0 "" 0 "" {TEXT -1 37 "Where is the fun ction discontinuous? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "q(t) = 0 for t < -4 " }}{PARA 0 "" 0 "" {TEXT -1 11 " = " }{XPPEDIT 18 0 "4*t + 16" "6#,&*&\"\"%\"\"\"%\"tGF&F& \"#;F&" }{TEXT -1 19 " for -4 <= t <= -2 " }}{PARA 0 "" 0 "" {TEXT -1 11 " = " }{XPPEDIT 18 0 "2t^2" "6#*&\"\"#\"\"\"*$%\"tGF$F%" } {TEXT -1 18 " for -2 < t <= 1 " }}{PARA 0 "" 0 "" {TEXT -1 11 " \+ = " }{XPPEDIT 18 0 "5*sqrt(t+3)-8;" "6#,&*&\"\"&\"\"\"-%%sqrtG6#,&% \"tGF&\"\"$F&F&F&\"\")!\"\"" }{TEXT -1 18 " for 1 < t <= 6 " }} {PARA 0 "" 0 "" {TEXT -1 23 " = 7 for t > 6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "c:= x -> piecewise( x < -4, 0, -4 <= x a nd x <= -2, 4*x + 16, -2 < x and x <= 1, 2*x^2, 1 < x and x <= 6, 5 * \+ sqrt(x + 3) - 8, 7);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(c(x),x= -5..7, discont=true, color=magenta);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 61 "answer: Function is continuous everywhere; no discontinuity. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 9 "Example 3" }{TEXT -1 38 "\nFor each of the following func tions: " }}{PARA 0 "" 0 "" {TEXT -1 48 " (i) Plot each function for l arge vlaues of x. " }}{PARA 0 "" 0 "" {TEXT -1 70 "(ii) Use the plot t o conjecture the limitof the function as x goes to " }{XPPEDIT 18 0 "i nfinity" "6#%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 65 "(iii) Determine precisely the limit of the function as x goes to \+ " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "e(t) = (1+t) *( sin(t) ) / t" "6#/-%\"eG6#%\"tG*(,&\"\"\"F*F'F*F *-%$sinG6#F'F*F'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "b ) " }{XPPEDIT 18 0 "f(x) = sin(t) / sqrt(t)" "6#/-%\"fG6#%\"xG*&-%$ sinG6#%\"tG\"\"\"-%%sqrtG6#F,!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 3 "c) \+ " }{XPPEDIT 18 0 "g(x) = (2x^3 + 7*x) / x^2" "6#/-%\"gG6#%\"xG*&,&*& \"\"#\"\"\"*$F'\"\"$F,F,*&\"\"(F,F'F,F,F,*$F'F+!\"\"" }{TEXT -1 2 " \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e:= ( ( 1+x)/x )* sin(x);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(e(x), x = 100..200);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "limit((( 1+ x)/x)* sin(x), x = infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 " limit of e(x) as x goes to " }{XPPEDIT 18 0 "infinity" "6#%)infinityG " }{TEXT -1 54 " DNE, since the function oscillates between 1 and -1. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:= ( 1/(sqrt(x)) * sin (x));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plo t(f(x), x = 1000..2000, numpoints=1000);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "limit(( 1/(sqrt(x)) * sin(x)), x = \+ infinity);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g:= (2*x^3 + 7*x)/(x^2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(g(x), x= 1000..2000);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "limit((2*x^3 + 7*x)/(x^2), x = infinity);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 9 "Example 4" }{TEXT -1 102 "\nA hot piece of steel which is 94 degrees C is pla ced in a room that is kept at a constant temperature" }}{PARA 0 "" 0 " " {TEXT -1 102 "of 20 degrees C. The steel begins to cool, and has it s temperature t minutes after being placed in " }}{PARA 0 "" 0 "" {TEXT -1 21 "the room given by: " }{XPPEDIT 18 0 "T(t) = 20+74*exp(l n(.4)*t/40);" "6#/-%\"TG6#%\"tG,&\"#?\"\"\"*&\"#uF*-%$expG6#*(-%#lnG6# -%&FloatG6$\"\"%!\"\"F*F'F*\"#SF8F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "a) Graph T(t) " }}{PARA 0 "" 0 "" {TEXT -1 48 "b) \+ How long till the steel reaches 22 degrees? " }}{PARA 0 "" 0 "" {TEXT -1 48 "c) How long till the steel reaches 21 degrees? " }}{PARA 0 "" 0 "" {TEXT -1 49 "d) What will be the ultimate temp of the steel? " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "T:= 20 + 74 * exp( ln(.4) *x/40 );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(T(x), x =0..200 );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "limit(20 + 74 * exp( ln(.4)*x)/40 ), x = infinity); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eq:= 20 \+ + 74 * exp(( (ln(.4)*x)/40 ) )= 22;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(eq,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eq:= 20 + 74 * exp(( (ln(.4)*x)/40 \+ ) )= 21;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " solve(eq);" }{TEXT -1 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "It takes approximately 157 minutes to rea ch 22 F and 187 minutes to reach 21F. The limiting " }}{PARA 0 "" 0 " " {TEXT -1 13 "temp is 20F. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 262 10 "Example 5\n" }{TEXT -1 12 "Use Maple's \+ " }{TEXT 263 5 "limit" }{TEXT -1 47 " command to find each of the foll owing limits. " }}{PARA 0 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "Li mit((3^x - 1) / x, x=0)" "6#-%&LimitG6$*&,&)\"\"$%\"xG\"\"\"F+!\"\"F+F *F,/F*\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "b) " } {XPPEDIT 18 0 "Limit( (2 - 2^x) / (4^x - 4),x=1)" "6#-%&LimitG6$*&,&\" \"#\"\"\")F(%\"xG!\"\"F),&)\"\"%F+F)F/F,F,/F+F)" }{TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 4 "c) " }{XPPEDIT 18 0 "Limit( (1 - cos(x) ) / (x^2), x=0) " "6#-%&LimitG6$*&,&\"\"\"F(-%$cosG6#%\"xG!\"\"F(*$F, \"\"#F-/F,\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "d) " } {XPPEDIT 18 0 "Limit((x-sin(x))/(x^3),x = 0);" "6#-%&LimitG6$*&,&%\"xG \"\"\"-%$sinG6#F(!\"\"F)*$F(\"\"$F-/F(\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 4 "e) " }{XPPEDIT 18 0 "Limit( (1 + x)^(1/x),x=0) " "6#-%&LimitG6$ ),&\"\"\"F(%\"xGF(*&F(F(F)!\"\"/F)\"\"!" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "limit( (3 ^x -1)/x, x = 0);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "limit( ( 2 \+ - 2^x)/(4^x - 4), x = 1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "limit((1 - cos(x))/(x^2),x = 0);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "limit( (x - sin(x))/(x^3), x = 0);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "li mit( ( 1 + x)^(1/x), x = 0);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 9 "Example 6" }{TEXT -1 29 " \nPlot each of the following: " }}{PARA 0 "" 0 "" {TEXT -1 4 "a) " } {XPPEDIT 18 0 "f(x) = ( x^3 - 3) / x" "6#/-%\"fG6#%\"xG*&,&*$F'\" \"$\"\"\"F+!\"\"F,F'F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 " b) " }{XPPEDIT 18 0 "f(x) = ( 1 + 2*x - x^4) / x^2 " "6#/-%\"fG6#% \"xG*&,(\"\"\"F**&\"\"#F*F'F*F**$F'\"\"%!\"\"F**$F'F,F/" }}{PARA 0 "" 0 "" {TEXT -1 82 " In each case, plot the asymptotic curves on the sa me axes with different color. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot( [(x^3 - 3)/x, x^2], x= 1..10, color=[green,magenta]);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Asymptotic curve \+ is " }{XPPEDIT 18 0 "y = x^2." "6#/%\"yG)%\"xG-%&FloatG6$\"\"#\"\"!" } {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot([ (1 \+ + 2*x - x^4)/(x^2), -x^2], x = 1..10, color=[green, magenta]);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 9 "Example 7" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " f(x) = x -3 for x > 0 " }}{PARA 0 "" 0 "" {TEXT -1 25 " = \+ 5 for x = 0 " }}{PARA 0 "" 0 "" {TEXT -1 13 " = " } {XPPEDIT 18 0 "x^2 + 4*x -1" "6#,(*$%\"xG\"\"#\"\"\"*&\"\"%F'F%F'F'F '!\"\"" }{TEXT -1 13 " for x < 0 " }}{PARA 0 "" 0 "" {TEXT -1 57 "Pl ot f(f(x)) and find limit f(f(x)) as x goes to 0 with `" }{TEXT 265 5 "limit" }{TEXT -1 11 "' command. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "z:= x -> piecewise( x > 0, x-3, x = 0, 5, x^2 + 4*x - 1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot (z(x), x = -5..5, discont=true, color=brown);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(z(z(x)), x = -5..5, dis cont=true, color=brown);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(z(z(x)), x = 0);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 9 "Example 8" } {TEXT -1 25 "\nFind all asymptotes for " }{XPPEDIT 18 0 "f(x) = (3*x^3 - 4*x^2 - 2*x + 5)*(x^2 - 1)" "6#/-%\"fG6#%\"xG*&,**&\"\"$\"\"\"*$F'F +F,F,*&\"\"%F,*$F'\"\"#F,!\"\"*&F1F,F'F,F2\"\"&F,F,,&*$F'F1F,F,F2F," } {TEXT -1 18 ". Graph f(x) and " }}{PARA 0 "" 0 "" {TEXT -1 53 "all as ymptotes on the same axes with different color." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "plot([(3*x^ 3 - 4*x^2 - 2*x + 5)/(x^2 - 1), 3*x - 4], x = -4..4, color=[red, green ], thickness=2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "plot([(3*x^3 - 4*x^2 - 2*x + 5)/(x^2 - 1), 3*x - 4], x = 1.5.. 10, color=[red, green], thickness=2);" }{TEXT -1 0 "" }}}}{MARK "2 0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }