{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 "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 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 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{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 "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 52 "Asymptotes: Vertical, Horizontal, Oblique, and mo re!" }}{PARA 257 "" 0 "" {TEXT 256 27 "Calculus I Lab -- Fall 2002" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 19 "" 0 "" {TEXT -1 11 "prepared by" }}{PARA 19 "" 0 "" {TEXT -1 16 "Douglas B. Meade" }}{PARA 257 "" 0 "" {TEXT -1 25 "Department of Mathematics" }}{PARA 257 "" 0 "" {TEXT -1 28 "University of South Carolina" }}{PARA 257 "" 0 "" {TEXT -1 18 "Columbia, SC 29208" }}{PARA 257 "" 0 "" {TEXT -1 25 "E-mail: me ade@math.sc.edu" }}{PARA 19 "" 0 "" {TEXT -1 17 "10 September 2002" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Purp ose" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 340 "Asymptotes of a function have a variety of uses. \+ The primary use in this course is as one step in the creation of a gra ph of the function over a large interval without evaluating the functi on at too many points. You are likely to encounter the same ideas in c ourses dealing with complexity analysis of algorithms and differential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 287 "Remember that it should not be necessary for you to do too muc h typing to complete this project. Except for entering specific functi ons, all of the Maple code needed should be available somewhere within this worksheet. Please see Jay, Wally, or Professor Meade if you have any questions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 497 "The laboratory project begins with an investigation of h orizontal and vertical asymptotes using Maple. The first example is on e of your homework problems from Section 2.8. The first problem which \+ you will be solving is an example worked in class last Friday (6 Septe mber). Maple will be used to evaluate limits and to produce graphs. Th e second part of the lab introduces the oblique asymptote. This is a s ituation in which the symbolic capabilities within Maple can be a cons iderable time saver.." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Before submitting your report check that you have answ ered each part of the two exercises." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The project is due in the dropbox at " }{TEXT 257 37 "midnight, Thursday, 12 September 2002" }{TEXT -1 59 ". \+ Do not put off work on this project until Thursday night!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 "Exercise 1 : Horizontal and Vertical Asymptotes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "To illustrate the use of Maple to create a plot of a function \+ and its horizontal and vertical asymptotes, consider the function" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "F := 3/(9-x^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "If we ask Maple to plot this function on the interval [-10, 10 \+ ], we will see" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot( F, \+ x=-10..10 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 314 "This plot looks somewhat like the output from an echocardiogram (EKG). We have seen similar plots in earlier l abs. It is produced because Maple simply plots points and then connect s the dots. To see only the points in the plot, position the cursor on top of the plot and click the right mouse button. Then, select " } {TEXT 258 5 "Point" }{TEXT -1 11 " under the " }{TEXT 259 5 "Style" } {TEXT -1 151 " menu --- or click on the blue sine-like icon in the con text bar. The horizontal and vertical asymptotes of this example fairl y apparent in this graph:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 22 "horizontal asymptote: " }{XPPEDIT 18 0 "y = 0 " "6#/%\"yG\"\"!" }{TEXT -1 10 " " }}{PARA 259 "" 0 "" {TEXT -1 21 "vertical asymptotes: " }{XPPEDIT 18 0 "x = -3" "6#/%\"xG,$\"\"$ !\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Sometimes it will be helpful to know the asymptotes before plottin g the function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "The search for horizontal asympto tes is simpler in that it depends only on the evaluation of two limits ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "asymp_H := \{ y=limit( F, x=infinity )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " y =limit( F, x=-infinity ) \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_H := implicitplot( asymp_H," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " x=-10..10, y=-20..20," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " style=point, color=green ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot_H; " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "For \+ this example the vertical asymptotes occur at the zeros of the denomin ator:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q1 := solve( denom (F)=0, \{x\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "asymp_V \+ := map( op, \{q1\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "pl ot_V := implicitplot( asymp_V," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " \+ x=-10..10, y=-20..20," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " style=point, color=blue ):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot_V;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 57 "plot_fn := plot( F, x=-10..10, y=-20..20, discont=t rue ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot_fn;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "W hen all three plots are assembled as one plot,the result is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display( [ plot_H, plot_V, p lot_fn ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Notes:" }}{PARA 15 "" 0 "" {TEXT -1 189 "I n general, the search for vertical asymptotes must include any points \+ where the function is not defined. For trigonometric functions this in cludes, but is not limited to, odd multiples of " }{XPPEDIT 18 0 "Pi/2 " "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 38 " for the tangent and secan t functions." }}{PARA 15 "" 0 "" {TEXT -1 172 "The extra step of conve rting the list of sets of solutions into a single set of equations of \+ vertical lines will make it easier to plot all vertical asymptotes at \+ one time." }}{PARA 15 "" 0 "" {TEXT -1 26 "The third argument in the \+ " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 328 " command is an optio nal argument that specifies the vertical window for the plot. The vert ical window is normally selected automatically by Maple to ensure that all points fit in the display. However, for functions with vertical a symptotes this results in a window that is so large that important det ails are rendered invisible." }}{PARA 15 "" 0 "" {TEXT -1 26 "The fina l argument in the " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 10 " co mmand, " }{HYPERLNK 17 "discont=true" 2 "plot,options" "" }{TEXT -1 160 ", is an optional argument that instructs Maple to look for discon tinuities in the function and to not connect the points on either side of this discontinuities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 246 "Determine the horizontal an d vertical asymptotes for the graphs of the functions listed below. Yo ur report should include a summary of the asymptotes and a plot of the function with an appropriately selected window (both horizontal and v ertical)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 18 0 "F(x) = 2*x/sqrt(x^2-5 )" "6#/-%\"FG6#%\"xG*(\"\"#\"\"\"F'F*-%%sqrtG6#,&*$F'F)F*\"\"&!\"\"F1 " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F := 2*x/sqrt(x^2-5);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "asymp_H := \{ y=limit( F, x= infinity )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " y=limit ( F, x=-infinity ) \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "p lot_H := implicitplot( asymp_H," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " x=%?..%?, y=%?..%?," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " style=point, color=green ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot_H;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q1 := solve( denom(F)=0, \{x\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "asymp_V := map( op, \{q1\} );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_V := implicitplot( asym p_V," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " x=% ?..%?, y=%?..%?," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " \+ style=point, color=blue ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot_V;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot_fn := pl ot( F, x=%?..%?, y=%?..%? ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot_fn;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display( [ plot_H, plot_V, plot_fn ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "b) " }{XPPEDIT 18 0 "F( x) = 2*x/sqrt(x^2+5)" "6#/-%\"FG6#%\"xG*(\"\"#\"\"\"F'F*-%%sqrtG6#,&*$ F'F)F*\"\"&F*!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "F := %? ;\nasymp_H := \{ y=limit( F, x=infinity )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " y=limit( F, x=-infinity ) \};" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "plot_H := implicitplot( asymp_H," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " x=%?..%?, y=%?..%?," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " style=point , color=green ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot_H; " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_V := implicitplot( asym p_V," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " x=% ?..%?, y=%?..%?," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " \+ style=point, color=blue ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot_V;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot_fn := pl ot( F, x=%?..%?, y=%?..%?, discont=true ):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "plot_fn;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display( [ plot_H, plot_V, plot_fn ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "c) " } {XPPEDIT 18 0 "F(x) = sin(abs(x-3))/(x-3) + (x+1)/x" "6#/-%\"FG6#%\"xG ,&*&-%$sinG6#-%$absG6#,&F'\"\"\"\"\"$!\"\"F1,&F'F1F2F3F3F1*&,&F'F1F1F1 F1F'F3F1" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "F := %?;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "asymp_H := \{ y=limit( F, x=infinit y )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " y=limit( F, x= -infinity ) \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_H : = implicitplot( asymp_H," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " \+ x=%?..%?, y=%?..%?," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " style=point, color=green ):" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "plot_H; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_V := implicitplot( asymp_V," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 43 " x=%?..%?, y=%?..%?," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " style=point , color=blue ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot_V;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot_fn := plot( F, x=%?..%? , y=%?..%?, discont=true ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot_fn;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display( [ p lot_H, plot_V, plot_fn ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "d) F(x) = tan(x/2) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "F := %?;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "asymp_H := \{ y=limit( F, x=infinity )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " y=limit( F, x=-infinity ) \};" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_H := implicitplot( asym p_H," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " x=% ?..%?, y=%?..%?," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " \+ style=point, color=green ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot_H; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_V := \+ implicitplot( asymp_V," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " \+ x=%?..%?, y=%?..%?," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " style=point, color=blue ):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "plot_V;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot_fn := plot( F, x=%?..%?, y=%?..%?, discont=true ):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot_fn;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 39 "display( [ plot_H, plot_V, plot_fn ] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Exercise \+ 2 : Oblique Asymptotes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "An " }{TEXT 260 17 "oblique asymp tote" }{TEXT -1 17 " to the graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\" yG-%\"fG6#%\"xG" }{TEXT -1 20 " is a straight line " }{XPPEDIT 18 0 "y = a*x+b" "6#/%\"yG,&*&%\"aG\"\"\"%\"xGF(F(%\"bGF(" }{TEXT -1 14 " whe re either " }{XPPEDIT 18 0 "limit( f(x)-(a*x+b), x=infinity ) = 0" "6# /-%&limitG6$,&-%\"fG6#%\"xG\"\"\",&*&%\"aGF,F+F,F,%\"bGF,!\"\"/F+%)inf inityG\"\"!" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "limit( f(x)-(a*x+b), x=-infinity ) = 0" "6#/-%&limitG6$,&-%\"fG6#%\"xG\"\"\",&*&%\"aGF,F+F ,F,%\"bGF,!\"\"/F+,$%)infinityGF1\"\"!" }{TEXT -1 203 ". In addition t o aiding in the graphing of a function, oblique asymptotes can be used to obtain a simple approximation to the function for very large (or v ery small) values of its argument. Note that if " }{XPPEDIT 18 0 "a=0 " "6#/%\"aG\"\"!" }{TEXT -1 160 " then the oblique asymptote is actual ly a horizontal asymptote. This means that oblique asymptotes generali ze the horizontal asymptotes discussed in Exercise 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "To illustrate the con cept of oblique asymptote, the oblique asymptote for the following fun ction will be found and plotted together with the vertical asymptotes " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "F := ( 2*x^4 + 3*x^3 - \+ 2*x - 4 ) / ( x^3 - 1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "Begin, as usual, by asking M aple for a plot of the function. After a little trial-and-error, the f ollowing options are found to provide a reasonable picture." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot( F, x=-10..10, y=-20..2 0, discont=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Notice the vertical asymptote near " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 75 " and how the gr aph resembles a straight line for large and small values of " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 63 ". This straight line is the \+ oblique asymptote for this problem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "To find an o blique asymptote one general begins by performing long division on the rational function. This is a simple but tedious step that is ideally \+ performed by Maple. The " }{HYPERLNK 17 "quo" 2 "quo" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "rem" 2 "rem" "" }{TEXT -1 213 " commands can be \+ used to find the quotient and remainder when two polynomials are divid ed. The quotient of the numerator and denominator of the original rati onal function is the oblique asymptote for the function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q := quo( numer(F), denom(F), x ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r := rem( numer(F), den om(F), x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "This means that it should be possible to \+ re-write the original function as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "FF := q + r/denom(F);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "To verify Map le's calculuation," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "norma l(FF) = F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "To confirm that the quotient of the nume rator and denominator of this rational function is the oblique asympto te, check against the definition:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "limit( F-q, x=infinity );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "limit( F-q, x=-infinity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Observe \+ that these results are consistent with a plot of " }{XPPEDIT 18 0 "y = f(x) - (a*x+b)" "6#/%\"yG,&-%\"fG6#%\"xG\"\"\",&*&%\"aGF*F)F*F*%\"bGF *!\"\"" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " plot( F-q, x=-10..10, y=-10..10, discont=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Remem ber that we are not interested in the behavior for \"small\" values of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 10 ", only as " }{XPPEDIT 18 0 "abs(x) -> infinity" "6#f*6#-%$absG6#%\"xG7\"6$%)operatorG%&arrow G6\"%)infinityGF-F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 76 "We conclude by storing this result an d the corresponding plot for later use." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "asymp_O := \{ y = q \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_O := implicitplot( asymp_O," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 45 " x=-10..10, y=-20..20," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " style=point , color=green ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot_O;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Vertical asympotes can be found as before." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q1 := solve( denom(F)=0, \{x\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 47 "Note that Maple returns all three solutions to " } {XPPEDIT 18 0 "x^3-1=0" "6#/,&*$%\"xG\"\"$\"\"\"F(!\"\"\"\"!" }{TEXT -1 154 ", not just the real-valued solution. To extract all complex-va lued solutions one might use the following command to remove all solut ions that contain the " }{XPPEDIT 18 0 "I = sqrt(-1)" "6#/%\"IG-%%sqrt G6#,$\"\"\"!\"\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "q2 := op(remove( has, \{q1\}, I ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "asymp_V := map( op, \{q2\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_V := implicitplot( asymp_V," } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " x=-10..10, y=-20..20," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " \+ style=point, color=blue ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "pl ot_V;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot_fn := plot( F, x=-10..10, y=-20..20, discont=true ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display( [ plot _O, plot_V, plot_fn ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 243 "Determine the oblique and ve rtical asymptotes for the graphs of the functions listed below. Your r eport should include a summary of the asymptotes and a plot of the fun ction with an appropriately selected window (both horizontal and verti cal)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 18 0 "F(x) = ( 3*x^3 + 4*x^2 - x \+ + 1 ) / (x^2 + 1)" "6#/-%\"FG6#%\"xG*&,**&\"\"$\"\"\"*$F'F+F,F,*&\"\"% F,*$F'\"\"#F,F,F'!\"\"F,F,F,,&*$F'F1F,F,F,F2" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "F := %?;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "q := quo( numer(F), denom(F), x );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r := rem( numer(F), denom(F) , x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "FF := q + r/denom (F);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "normal(FF) = F;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "limit( F-q, x=infinity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit( F-q, x=-infinity ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot( F-q, x=%?..%?, y =%?..%?, discont=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "asymp_O := \{ y = q \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_O := implicitplot( asymp_O," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " x=%?..%?, y=%?..%?," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " style=point, color=green ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot_O;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q1 := solve( denom(F)=0, \{x\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "q2 := op(remove( has, \{q1\}, I )); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "asymp_V := map( op, \{q 2\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_V := implici tplot( asymp_V," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " \+ x=%?..%?, y=%?..%?," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " \+ style=point, color=blue ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot_V;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " plot_fn := plot( F, x=%?..%?, y=%?..%?, discont=true ):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "display( [ plot_O, plot_V, plot_fn ] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "b)" }{XPPEDIT 18 0 "F(x) = ( 3*x^3 + 4*x^2 - x + 1 ) / (x^2 - 1)" "6#/-%\"FG6#%\"xG*&,**&\"\"$\"\"\"*$F' F+F,F,*&\"\"%F,*$F'\"\"#F,F,F'!\"\"F,F,F,,&*$F'F1F,F,F2F2" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "F := %?;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q := quo( numer(F), denom(F), x ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r := rem( numer(F), den om(F), x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "FF := q + r/ denom(F);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "normal(FF) = F ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "limit( F-q, x=infinity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit( F-q, x=-infin ity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot( F-q, x=%?.. %?, y=%?..%?, discont=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "asymp_O := \{ y = q \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_O := implicitplot( asymp_O," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " x=%?..%?, y=%?..%?," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " style=point, color=green ) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot_O;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "q1 := solve( denom(F)=0, \{x\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "q2 := op(remove( has, \{q1\}, I )); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "asymp_V := map( op, \{q 2\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot_V := implici tplot( asymp_V," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " \+ x=%?..%?, y=%?..%?," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " \+ style=point, color=blue ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot_V;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " plot_fn := plot( F, x=%?..%?, y=%?..%?, discont=true ):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "display( [ plot_O, plot_V, plot_fn ] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "Exercise \+ 3 : Nonlinear asymptotics" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Consi der the function" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "F := sq rt( 3*x^3+1 ) / (2*x + 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "a)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "This function is " }{TEXT 261 3 "not" }{TEXT -1 90 " a rational function and so it does not have a quotient and remainder. H owever, note that " }{XPPEDIT 18 0 "F(x)^2" "6#*$-%\"FG6#%\"xG\"\"#" } {TEXT -1 44 " is a rational function. Find the quotient, " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 17 ", and remainder, " } {XPPEDIT 18 0 "r(x)" "6#-%\"rG6#%\"xG" }{TEXT -1 39 ", for the numerat or and denominator of " }{XPPEDIT 18 0 "F(x)^2" "6#*$-%\"FG6#%\"xG\"\" #" }{TEXT -1 8 ", i.e., " }{XPPEDIT 18 0 "F(x)^2 = q(x) + r(x)/(2*x+1) ^2" "6#/*$-%\"FG6#%\"xG\"\"#,&-%\"qG6#F(\"\"\"*&-%\"rG6#F(F.*$,&*&F)F. F(F.F.F.F.F)!\"\"F." }{TEXT -1 7 " where " }{XPPEDIT 18 0 "limit( r(x) /(2*x+1)^2, x=infinity ) = 0" "6#/-%&limitG6$*&-%\"rG6#%\"xG\"\"\"*$,& *&\"\"#F,F+F,F,F,F,F0!\"\"/F+%)infinityG\"\"!" }{TEXT -1 2 " ." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "b)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Note that " } {XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(q(x)+r(x)/(2*x+1)^2) <> sqrt(q(x)) + sqrt(r(x))/(2*x+1)" "6#0- %%sqrtG6#,&-%\"qG6#%\"xG\"\"\"*&-%\"rG6#F+F,*$,&*&\"\"#F,F+F,F,F,F,F4! \"\"F,,&-F%6#-F)6#F+F,*&-F%6#-F/6#F+F,,&*&F4F,F+F,F,F,F,F5F," }{TEXT -1 50 ". However, if the remainder term tends to zero as " }{XPPEDIT 18 0 "x -> infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinity GF*F*F*" }{TEXT -1 18 ", it is true that " }{XPPEDIT 18 0 "limit( F(x) - sqrt(q(x)), x=infinity ) = 0" "6#/-%&limitG6$,&-%\"FG6#%\"xG\"\"\"- %%sqrtG6#-%\"qG6#F+!\"\"/F+%)infinityG\"\"!" }{TEXT -1 14 ". Verify th at " }{XPPEDIT 18 0 "limit( sqrt(r(x))/(2*x+1), x=infinity ) = 0" "6#/ -%&limitG6$*&-%%sqrtG6#-%\"rG6#%\"xG\"\"\",&*&\"\"#F/F.F/F/F/F/!\"\"/F .%)infinityG\"\"!" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "c)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Because " }{XPPEDIT 18 0 "y=sqrt(q(x))" "6#/%\"yG-%%sqrtG6 #-%\"qG6#%\"xG" }{TEXT -1 79 " is not a straight line it is not called an oblique asymptote for the graph of " }{XPPEDIT 18 0 "y=F(x)" "6#/% \"yG-%\"FG6#%\"xG" }{TEXT -1 29 ". Instead, we would say that " } {XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sqrt(q(x))" "6#-%%sqrtG6#-%\"qG6#%\"xG" }{TEXT -1 5 " are " } {TEXT 262 10 "asymptotic" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x -> infi nity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" } {TEXT -1 49 ". Create a graph that illustrates this statement." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "d)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Find a functio n that is asymptotic to " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" } {TEXT -1 4 " as " }{XPPEDIT 18 0 "x -> -infinity" "6#f*6#%\"xG7\"6$%)o peratorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 113 "Hint: Think carefully. This requires no additi onal computation is you have a good picture from the previous step." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "Create a comp osite plot containing a good plot of the function, all horizontal and \+ oblique asymptotes, and the asymptotic functions found in parts c) and d). Conclude your lab report with this plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 10 "Exercise 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 269 "Mapl e contains a built-in command to find horizontal, vertical, and obliqu e asymptotes. Find the name of this command. Use this command to verif y the results of one of the examples in Exercises 1 and 2. Include the full Maple command(s) and their output in your report." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }