{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 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 1 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 1 0 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 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 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 1 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 "" -1 268 "" 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 "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 0 256 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 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 54 "Use Maple to graph Cubic \+ and Higher Degree Polynomials" }}{PARA 0 "" 0 "" {TEXT 266 13 "Instruc tions:" }}{PARA 0 "" 0 "" {TEXT -1 127 "After experimenting the exampl es, you need to do the exercises 1 and 2. For exercises 1 and 2, you n eed to find the followings:" }}{PARA 0 "" 0 "" {TEXT -1 22 "[1] the x -intercepts;" }}{PARA 0 "" 0 "" {TEXT -1 64 "[2] the solutions to th e inequalities [work on these by hand]." }}{PARA 0 "" 0 "" {TEXT -1 33 "[3] the graphs of the functions." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 258 7 "Type I:" }{TEXT -1 26 " No repeated x -intercepts:" }}{PARA 0 "" 0 "" {TEXT -1 44 "Example 1: If f(x) = (x- 1)(x-3)(x+5)(x-4). " }}{PARA 0 "" 0 "" {TEXT -1 26 "(1) Find the x-int ercepts " }}{PARA 0 "" 0 "" {TEXT -1 122 "(2) Find the intervals where f(x)>0 and f(x)<0; [**Solve inequalities by hand first and use graph \+ to verify your answer]. " }}{PARA 0 "" 0 "" {TEXT -1 27 "(3) Sketch th e graph of f." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=x->(x- 1)*(x-3)*(x+5)*(x-4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "pl ot(f,-6..5,-500..100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s olve(f(x)=0,x);" }{TEXT -1 59 "#Notice that this command is to find th e x-intercepts of f." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sol ve(f(x)>0,x);" }{TEXT -1 65 " #Notice that this command is to find the intervals where f(x)>0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(f(x)<0,x);" }{TEXT -1 64 "#Notice that this command is to find \+ the intervals where f(x)<0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 259 6 "Remark" }{TEXT -1 179 ": Notice th at we see from the graph above that when x->infinity, f(x)->infinity. \+ Also, when x->-infinity, f(x)->infinity. We can verify this by using t he following Maple commands:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f(10000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f(-10000) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 14 "** Exercise 1:" }{TEXT -1 5 " For " }{TEXT 256 42 "f(x)= - (3x-10)(x-1)(2x+10)(x-3)(x-4)(x-5)" } {TEXT -1 50 ". Answer the questions from the preceding example." }} {PARA 256 "" 0 "" {TEXT 260 0 "" }}{PARA 0 "" 0 "" {TEXT 261 7 "Type I I" }{TEXT -1 42 ". Polynomials with repeated x-intercepts: " }}{PARA 0 "" 0 "" {TEXT 262 6 "Note1:" }{TEXT -1 84 " If a polynomial has an \+ even-repeated x-intercepts, then such x-intercepts must be " }{TEXT 268 8 "touching" }{TEXT -1 12 " the x-axis." }}{PARA 0 "" 0 "" {TEXT 263 6 "Note2:" }{TEXT -1 83 " If a polynomial has an odd-repeated x-i ntercepts, then such x-intercepts must be " }{TEXT 267 9 "crossing " } {TEXT -1 11 "the x-axis." }}{PARA 0 "" 0 "" {TEXT -1 14 "Example 2: If " }{TEXT 265 4 "f(x)" }{TEXT -1 2 "= " }{XPPEDIT 18 0 "(2*x-1)^2*(x-2 );" "6#*&,&*&\"\"#\"\"\"%\"xGF'F'F'!\"\"F&,&F(F'F&F)F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "(1) Graph the function f, and " }} {PARA 0 "" 0 "" {TEXT -1 57 "(2) make the table (by hand) to determin e the signs of " }{TEXT 264 1 "f" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 24 "f:=x->((2*x-1)^2)*(x-2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(f,-1..3,-10..10);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "solve(f(x)=0,x);" }{TEXT -1 34 "#this is to \+ find the x-intercepts." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s olve(f(x)>0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(f( x)<0,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Example 3: If g(x)= \+ " }{XPPEDIT 18 0 "-(2*x-1)*(x-2)^2;" "6#,$*&,&*&\"\"#\"\"\"%\"xGF(F(F( !\"\"F(*$,&F)F(F'F*F'F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "(1) Graph the function f , and " }}{PARA 0 "" 0 "" {TEXT -1 58 "(2) make the table (by hand) to determine the signs of g." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g:=x->-(2*x-1)*(x-2)^2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(g,-1..3,-10..10);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(g(x)=0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(g(x)>0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(g(x)<0,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 12 "**Exercise 2" }{TEXT -1 7 ": For " }{XPPEDIT 18 0 "h (x) = -(2*x-1)^2*(x-4)^2;" "6#/-%\"hG6#%\"xG,$*&,&*&\"\"#\"\"\"F'F-F-F -!\"\"F,,&F'F-\"\"%F.F,F." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "h:=x->-(2*x-1)^2*(x-4)^2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(h,-1..5,c..d); " } {TEXT -1 101 "#You need to determine appropriate values for c and d (t ry and error) so the picture of h looks fine." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Investigate and Experiment a general degree 4 polynomi al:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "with(plots):animate(a*x^4+x^3+x^2+x,x=-2..2,a=-3..3,t hickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "?animate;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "9 4 3" 11 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }