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"Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 212 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 213 "Times" 1 14 255 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 214 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 215 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 216 "T imes" 1 14 255 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 217 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 218 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 219 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 220 "Times" 1 14 255 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 221 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {PSTYLE "" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 222 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 223 "Times" 1 14 255 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 224 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 225 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 226 "Times" 1 14 255 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 227 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {PARA 205 "" 0 "" {TEXT 201 38 "Estimating Distance Traveled a nd Areas" }}{PARA 0 "" 0 "" {TEXT 228 7 "Name: " }}{SECT 1 {PARA 3 "" 0 "" {TEXT 229 38 "Estimating Distance from Velocity Data" }}{PARA 0 "" 0 "" {TEXT 228 297 "A car is traveling with increasing velocity and the velocity is measured every second for 5 seconds as it accelerates from rest to 88 feet/second (60 miles per hour). In the table belo w, t is the number of seconds since the beginning of the measurement a nd v is the velocity in feet per second." }}{PARA 0 "" 0 "" {TEXT 228 0 "" }}{PARA 0 "" 0 "" {TEXT 228 79 "t | 0 | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0" }}{PARA 0 "" 0 "" {TEXT 228 67 "-------------------------------------------------------- -----------" }}{PARA 0 "" 0 "" {TEXT 228 84 "v | 0 | 30 | 42 | 54 | 60 | 70 | 78 | 80 | 83 | 85 | 88 \n" }}{PARA 0 "" 0 "" {TEXT 228 32 "First, the data will be plotted." }}{PARA 0 "" 0 "" {TEXT 228 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "t:=[0, 0.5,1,1.5,2,2.5,3,3.5,4,4.5,5];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "v:=[0,30,42,54,60,70,78,80,83,85,88];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "points:=[seq([t[i],v[i]],i=1..11)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(points,thickness=2,xtickmarks= 10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a:=plot(points,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "b:=polygonplot([[t[5],0],[t[6],0],[t[6],v[5]],[t [5],v[5]]],color=cyan):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " display(\{a,b\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "c:=polygonplot([[t[5],0],[t[ 6],0],[t[6],v[6]],[t[5],v[6]]],color=maroon):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "d:=plot(points,thickness=2,color=green):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{c,d\});" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "lbplot:=seq(polygonplot([[t[i],0],[t[i+1],0],[t[i+1],v[i]],[t[i], v[i]]],color=COLOR(RGB,i/5,i/10,i/15)),i=1..10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "d:=plot(points,thickness=2,color=green):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "display(\{d,lbplot\});" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "lower_bound:=sum(v[i]*(t[i+1]-t[i]),i=1..10);" }{TEXT 228 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 228 49 "The following shows the sum using sigms notation." }}}{EXCHG {PARA 206 "> " 0 "" {XPPEDIT 201 1 "l_boun d := sum(v[i]*(t[i+1]-t[i]), i = 1 .. 10);" "6#>%(l_boundG-%$sumG6$*&& %\"vG6#%\"iG\"\"\",&&%\"tG6#,&F,F-F-F-F-&F0F+!\"\"F-/F,;F-\"#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "ubplot:=seq(polygonplot([[t[i],0],[t[i+1],0],[t[i+1],v[i+1]],[t[i ],v[i+1]]],color=COLOR(RGB,i/20,i/10,i/30)),i=1..10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "d:=plot(points,thickness=3,color=re d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "display(\{ubplot,d\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "upper_bound:=sum(v[i+1]*(t[i+1]-t[i]),i=1..10);" }{TEXT 228 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 228 49 "The following shows the sum using sigms notation." }}}{EXCHG {PARA 207 "> " 0 "" {XPPEDIT 201 1 " u_bound := sum(v[i+1]*(t[i+1]-t[i]), i = 1 .. 10);" "6#>%(u_boundG-%$s umG6$*&&%\"vG6#,&%\"iG\"\"\"F.F.F.,&&%\"tGF+F.&F16#F-!\"\"F./F-;F.\"#5 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 228 155 "The total distance traveled is therefore between 291 feet and 335 feet. Better estimates could b e found by measuring the velocity every 0.25 seconds, etc." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 229 44 "Estimating Distance Given a Velocity f ormula" }}{PARA 208 "" 0 "" {TEXT 201 9 "Example 1" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := t -> 50*(1-exp(-0.196*t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(f(t),t=0..30,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "lSum:=Sum(f(3*n)*3,n=0..9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "lsum:=sum(f(3*n)*3,n=0..9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rSum:=S um(f(3*n)*3,n=1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rs um:=sum(f(3*n)*3,n=1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "leftsum(f(t),t=0..30,10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "rightsum(f(t),t=0..30,10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "leftbox(f(t),t=0..30,10,shad ing=cyan);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "rightbox(f(t) ,t=0..30,10,shading=aquamarine);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "middlebox(f(t),t=0..30,10,color=blue,shading=yellow);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 228 0 "" }}{PARA 209 "" 0 "" {TEXT 201 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT 228 8 " " }{TEXT 204 8 "Use the " }{TEXT 205 3 "sum" } {TEXT 206 140 " function to create and evaluate the left-hand and righ t-hand sums for n=10 subintervals for the following function over the \+ given interval." }}{PARA 0 "" 0 "" {TEXT 228 12 " " } {XPPEDIT 201 1 "f(x) = x^2+1;" "6#/-%\"fG6#%\"xG,&*$)F'\"\"#\"\"\"F,F, F," }{TEXT 228 2 ", " }{TEXT 207 9 "2 < x < 5" }}{PARA 0 "" 0 "" {TEXT 228 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=x->x^2+1:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(f(x),x=0..5,y=0..26,th ickness=2,color=blue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "d x:=(5-2)/10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "lSum:=Sum(f (2+n*dx)*dx,n=0..9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "lsu m:=evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "check1:=eva lf(leftsum(f(x),x=2..5,10));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rSum:=Sum(f(2+n*dx)*dx,n=1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rsum:=evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "check2:=evalf(rightsum(f(x),x=2..5,10));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 208 13 "Exercise 1: " }{TEXT 209 12 "Use Maple's " }{TEXT 210 3 "sum" }{TEXT 211 71 " function to c reate and evaluate the left-hand and right-hand sums for " }}{PARA 0 " " 0 "" {TEXT 212 30 "n = 10 subintervals. Use the " }{TEXT 213 7 "lef tsum" }{TEXT 214 4 " and" }{TEXT 215 1 " " }{TEXT 216 8 "rightsum" } {TEXT 217 33 " commands to confirm your answers" }{TEXT 201 2 ". " } {TEXT 228 1 " " }}{PARA 0 "" 0 "" {TEXT 228 0 "" }}{PARA 0 "" 0 "" {TEXT 228 8 " " }{XPPEDIT 201 0 "f(x) = 4-x^2;" "6#/-%\"fG6#%\" xG,&\"\"%\"\"\"*$)F'\"\"#F*!\"\"" }{TEXT 218 12 " , 0 < x < 2" }} {PARA 0 "" 0 "" {TEXT 228 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 "" }}}{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 "" }}}}{PARA 0 "" 0 "" {TEXT 20 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 229 13 "Exercise 2: " }{TEXT 219 12 "Use Maple's " }{TEXT 220 3 "sum" }{TEXT 221 71 " function to c reate and evaluate the left-hand and right-hand sums for " }}{PARA 210 "" 0 "" {TEXT 222 30 "n = 20 subintervals. Use the " }{TEXT 223 7 "l eftsum" }{TEXT 224 4 " and" }{TEXT 225 1 " " }{TEXT 226 8 "rightsum" } {TEXT 227 33 " commands to confirm your answers" }{TEXT 201 2 ". " }} {PARA 211 "" 0 "" {XPPEDIT 201 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$ expGF&" }{TEXT 201 12 " , 1 < x < 4" }}{PARA 0 "" 0 "" {TEXT 228 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 "" }}} {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 "" }}}}{PARA 0 "" 0 "" {TEXT 228 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 229 75 "Exercise 3: Find the \+ actual definite integrals using maple (Use help menu) " }}{PARA 0 "" 0 "" {TEXT 228 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 "" }}}{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 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }