{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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 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 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 259 301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 259 302 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 307 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 310 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 311 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 312 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 316 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 317 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 318 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 319 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 320 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 321 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 322 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 323 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 324 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 325 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 326 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 327 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 328 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 329 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 330 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 331 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 332 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 333 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 334 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 335 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 336 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 337 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 338 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 339 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 340 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 341 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 342 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 343 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 344 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 345 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 346 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 347 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 348 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 349 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 350 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 351 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 352 "" 0 1 0 0 0 0 0 1 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -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 "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 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" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 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 }1 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 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 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" -1 261 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 {PARA 3 "" 0 "" {TEXT 352 52 "The meaning and uses of the deri vative of a function" }{TEXT -1 0 "" }{TEXT 298 1 " " }}{PARA 0 "" 0 " " {TEXT -1 38 "by Peter Stone, Nanaimo, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 22.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Th e derivative and small changes " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y=f(x )" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 17 ", the derivative " }{XPPEDIT 18 0 "dy/dx = `f '`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%$f~'G6#%\"xG" } {TEXT -1 43 " gives information about how the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 67 " change \"locally\", that \+ is, at or close to a particulalar value of " }{TEXT 309 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "Changing " }{TEXT 319 1 "x" } {TEXT -1 24 " by a small amount from " }{TEXT 310 1 "x" }{TEXT -1 4 " \+ to " }{XPPEDIT 18 0 "x+Delta;" "6#,&%\"xG\"\"\"%&DeltaGF%" }{TEXT 311 1 "x" }{TEXT -1 36 " leads to a corresponding change of " }{TEXT 312 1 "y" }{TEXT -1 6 " from " }{TEXT 314 1 "y" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "y+Delta;" "6#,&%\"yG\"\"\"%&DeltaGF%" }{TEXT 313 1 "y" }{TEXT -1 38 ", such that, if the changes are small," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*y/(Delta*x);" "6#*(%&DeltaG \"\"\"%\"yGF%*&%&DeltaGF%%\"xGF%!\"\"" }{TEXT -1 1 " " }{TEXT 318 1 "~ " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 315 1 "y" } {TEXT -1 1 " " }{TEXT 316 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(dy/ dx)*Delta;" "6#*&-%!G6#*&%#dyG\"\"\"%#dxG!\"\"F)%&DeltaGF)" }{TEXT 317 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The fact tha t " }{XPPEDIT 18 0 "dy/dx = Limit(Delta*y/(Delta*x),Delta*x = 0);" "6# /*&%#dyG\"\"\"%#dxG!\"\"-%&LimitG6$*(%&DeltaGF&%\"yGF&*&%&DeltaGF&%\"x GF&F(/*&%&DeltaGF&F1F&\"\"!" }{TEXT -1 101 " means that the approximat ions improve as we consider progressively smaller changes in the varia bles " }{TEXT 320 1 "x" }{TEXT -1 5 " and " }{TEXT 321 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 364 263 263 {PLOTDATA 2 "60-%'CURVESG6&7&7$$\"3+++ +++++]!#=$\"35+++r7s[;!#<7$$\"3a**************pF*$\"34+++2Fv8?F-7$F/F+ F'-%'COLOURG6&%$RGBG$\"\"!F9F8$\"*++++\"!\")-%*THICKNESSG6#\"\"#-%*LIN ESTYLEG6#\"\"\"-F$6$7S7$F8$FDF97$$\"3h*******\\ech#!#>$\"3mrQlq;]E5F-7 $$\"3-+++v*G:*[FM$\"3-b&z+RJ,0\"F-7$$\"3u******\\L)4X(FM$\"3+z(\\w$fNx 5F-7$$\"3)******\\MSF+\"F*$\"3XTd`9QZ06F-7$$\"3#)****\\Fy:f7F*$\"3%)[/ =Ym=M6F-7$$\"3')****\\d'*)o\\\"F*$\"3y1%z'GHZh6F-7$$\"3w****\\(>ZIu\"F *$\"3qi7^a#=/>\"F-7$$\"3u****\\xOi(*>F*$\"3E4w&za76A\"F-7$$\"3#)****\\ FPQ^AF*$\"3;n'>b,'\\_7F-7$$\"3/+++IrS7DF*$\"357xT7&>cG\"F-7$$\"3p***** \\o;Bu#F*$\"3*\\%ePJ&>bJ\"F-7$$\"3*********QS6+$F*$\"3'*yA!Hv7+N\"F-7$ $\"3[******\\o-hKF*$\"3+UUbPwb&Q\"F-7$$\"3(*******4cZ6NF*$\"3)oF3*\\pp ?9F-7$$\"3S****\\xq!*QPF*$\"3I1?z)HyLX\"F-7$$\"3&********3X$4SF*$\"3aI 6'GZ>K\\\"F-7$$\"3s******f:WQUF*$\"3kc$3JZBy_\"F-7$$\"3f****\\<_$\\]%F *$\"3u%*yosj3p:F-7$$\"3**)*****fs#3u%F*$\"3*p\")[T))Rlg\"F-7$$\"3!)*** *\\<#Q'**\\F*$\"3!3yGRih'[;F-7$$\"33++]_u3Y_F*$\"3'p1'pxvz*o\"F-7$$\"3 [*****\\PJK]&F*$\"3'eZ:u=8Qt\"F-7$$\"3%*****\\n(p$RdF*$\"3Oq]e*RU_x\"F -7$$\"3A*****\\#p2%*fF*$\"3(oN@`')R5#=F-7$$\"3o****\\xgkeiF*$\"3eSw'e& >')p=F-7$$\"3g****\\-V&*)['F*$\"3TY-o[hU8>F-7$$\"3.+++&\\$pPnF*$\"3c_1 YCuhh>F-7$$\"37******>am%*pF*$\"35.?YT(yE,#F-7$$\"3?*****\\JigC(F*$\"3 Yo$\\RB=R1#F-7$$\"3G****\\PDi,%e #F-7$$\"3y)***\\PM&=v*F*$\"3kL]zL'e;l#F-7$$\"3*******fzs++\"F-$\"3EdR2\"F-$\"39Gu$zFRp#HF-7$$\"3&*****\\o#R05\"F- $\"3i$)>&H^'y0IF-7$$\"3()*****>`9V7\"F-$\"3_))[J#=1\"yIF-7$$\"3)****\\ <#Rm\\6F-$\"3eb-3)oJr:$F-7$$\"3%****\\A_ER<\"F-$\"3(=E8XtoYB$F-7$$\"3% **************>\"F-$\"3CZltAp6?LF--F56&F7F:F8F8-F$6&7$F'F.-%'SYMBOLG6# %'CIRCLEG-F56&F7F9F9F9-%&STYLEG6#%&POINTG-F$6&Fi\\l-F[]l6#%(DIAMONDGF^ ]lF`]l-F$6&Fi\\l-F[]l6#%&CROSSGF^]lF`]l-%%TEXTG6&7$FI$\"#K!\"\"Q)y~=~f (x)6\"Fe\\l-%%FONTG6$%*HELVETICAG\"#5-F_^l6&7$$\"#f!\"#$\"$d\"Fa_lQ\"D Ff^lF^]l-Fh^l6$F[]lF[_l-F_^l6&7$$\"#uFa_l$\"$&=Fa_lFd_lF^]lFe_l-F_^l6& 7$$\"#UFa_l$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "y = sqrt(x);" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 26 ". Then the derivative is " }{XPPEDIT 18 0 "dy/dx = 1/(2*sqrt(x));" "6# /*&%#dyG\"\"\"%#dxG!\"\"*&F&F&*&\"\"#F&-%%sqrtG6#%\"xGF&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "We may use the value of the deri vative at " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" }{TEXT -1 9 ", namely " }{XPPEDIT 18 0 "1/(2*sqrt(4))=1/4" "6#/*&\"\"\"F%*&\"\"#F%-%%sqrtG6 #\"\"%F%!\"\"*&F%F%F+F," }{TEXT -1 13 " to estimate " }{XPPEDIT 18 0 " sqrt(4.2)" "6#-%%sqrtG6#-%&FloatG6$\"#U!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 12 "We think of " }{TEXT 268 1 "x" }{TEXT -1 15 " changing from " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x+Delta;" "6#,&%\"xG\"\"\"%&DeltaGF%" } {XPPEDIT 18 0 "x=4.2" "6#/%\"xG-%&FloatG6$\"#U!\"\"" }{TEXT -1 6 " wit h " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{XPPEDIT 18 0 "x=``" "6#/%\" xG%!G" }{TEXT -1 5 "0.2. " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y =sqrt(4)" "6#/%\"yG-%%sqrtG6#\"\"%" }{XPPEDIT 18 0 "``=2" "6#/%!G\"\"# " }{TEXT -1 5 " and " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 270 1 "y" }{TEXT -1 1 " " }{TEXT 271 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(dy/dx)*Delta;" "6#*&-%!G6#*&%#d yG\"\"\"%#dxG!\"\"F)%&DeltaGF)" }{XPPEDIT 18 0 "x = 1/4" "6#/%\"xG*&\" \"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }{TEXT 272 1 "." }{TEXT -1 15 " (0.2 ) = 0.05, " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(4.2)" "6#-%%sqrtG6#-%&FloatG6$\" #U!\"\"" }{TEXT -1 1 " " }{TEXT 269 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "y+Delta;" "6#,&%\"yG\"\"\"%&DeltaGF%" }{XPPEDIT 18 0 "y=2.05" "6 #/%\"yG-%&FloatG6$\"$0#!\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "A more accurate value for " }{XPPEDIT 18 0 "sqrt(4.2)" "6#-%%sq rtG6#-%&FloatG6$\"#U!\"\"" }{TEXT -1 17 " is 2.049390153. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "y = 1/x;" "6#/% \"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 26 ". Then the derivative is " } {XPPEDIT 18 0 "dy/dx = -1/(x^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&F&F&* $%\"xG\"\"#F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "We ma y use the value of the derivative at " }{XPPEDIT 18 0 "x = 5;" "6#/%\" xG\"\"&" }{TEXT -1 9 ", namely " }{XPPEDIT 18 0 "-1/25;" "6#,$*&\"\"\" F%\"#D!\"\"F'" }{XPPEDIT 18 0 "`` = -0" "6#/%!G,$\"\"!!\"\"" }{TEXT -1 17 ".04, to estimate " }{XPPEDIT 18 0 "1/5.1;" "6#*&\"\"\"F$-%&Floa tG6$\"#^!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "We th ink of " }{TEXT 263 1 "x" }{TEXT -1 15 " changing from " }{XPPEDIT 18 0 "x = 5;" "6#/%\"xG\"\"&" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x+Delta; " "6#,&%\"xG\"\"\"%&DeltaGF%" }{XPPEDIT 18 0 "x = 5.1;" "6#/%\"xG-%&Fl oatG6$\"#^!\"\"" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "Delta;" "6#%&Del taG" }{XPPEDIT 18 0 "x=``" "6#/%\"xG%!G" }{TEXT -1 5 "0.1. " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x = 5;" "6#/%\"xG\"\"&" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "y = 1/5;" "6#/%\"yG*&\"\"\"F&\"\"&!\" \"" }{TEXT -1 10 " = 0.2 and" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 265 1 "y" }{TEXT -1 1 " " } {TEXT 266 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(dy/dx)*Delta;" "6#* &-%!G6#*&%#dyG\"\"\"%#dxG!\"\"F)%&DeltaGF)" }{XPPEDIT 18 0 "x = -1/25; " "6#/%\"xG,$*&\"\"\"F'\"#D!\"\"F)" }{TEXT -1 1 " " }{TEXT 267 1 "." } {TEXT -1 6 " (0.1)" }{XPPEDIT 18 0 "`` = -0" "6#/%!G,$\"\"!!\"\"" } {TEXT -1 6 ".004, " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/5.1;" "6#*&\"\"\"F$-%&Fl oatG6$\"#^!\"\"F)" }{TEXT -1 1 " " }{TEXT 264 1 "~" }{TEXT -1 2 " " } {XPPEDIT 18 0 "y+Delta;" "6#,&%\"yG\"\"\"%&DeltaGF%" }{XPPEDIT 18 0 "y =2-0" "6#/%\"yG,&\"\"#\"\"\"\"\"!!\"\"" }{TEXT -1 14 ".004 = 0.196. " }}{PARA 0 "" 0 "" {TEXT -1 26 "A more accurate value for " }{XPPEDIT 18 0 "1/5.1;" "6#*&\"\"\"F$-%&FloatG6$\"#^!\"\"F)" }{TEXT -1 18 " is 0 .1960784314. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 273 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 118 "Suppose that a circula r metal plate with radius 10 cm. is heated so that its radius increase s by 0.04 cm. to 10.04 cm." }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Estimat e the change in the area by using the derivative " }{XPPEDIT 18 0 "dA/ dr" "6#*&%#dAG\"\"\"%#drG!\"\"" }{TEXT -1 5 " of " }{XPPEDIT 18 0 "A= Pi*r^2" "6#/%\"AG*&%#PiG\"\"\"*$%\"rG\"\"#F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 35 "(b) Calculate the change directly. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(a) Let " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 275 1 "A" }{TEXT -1 50 " be the change in the area produced by the change \+ " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{XPPEDIT 18 0 "r=0" "6#/%\"rG \"\"!" }{TEXT -1 18 ".04 in the radius " }{TEXT 276 1 "r" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "A=Pi*r^2" " 6#/%\"AG*&%#PiG\"\"\"*$%\"rG\"\"#F'" }{TEXT -1 10 ", we have " } {XPPEDIT 18 0 "dA/dr=2*Pi*r" "6#/*&%#dAG\"\"\"%#drG!\"\"*(\"\"#F&%#PiG F&%\"rGF&" }{TEXT -1 7 ". When " }{XPPEDIT 18 0 "r=10" "6#/%\"rG\"#5" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "dA/dr = 2*Pi*`.`*10;" "6#/*&%#dAG\"\" \"%#drG!\"\"**\"\"#F&%#PiGF&%\".GF&\"#5F&" }{XPPEDIT 18 0 "``=20*Pi" " 6#/%!G*&\"#?\"\"\"%#PiGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta; " "6#%&DeltaG" }{TEXT 277 1 "A" }{TEXT -1 1 " " }{TEXT 278 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``(dA/dr)*Delta;" "6#*&-%!G6#*&%#dAG\" \"\"%#drG!\"\"F)%&DeltaGF)" }{TEXT 287 1 "r" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 20*Pi;" "6#/%!G*&\"#? \"\"\"%#PiGF'" }{TEXT -1 1 " " }{TEXT 285 1 "." }{TEXT -1 12 " (0.04) \+ = 0." }{XPPEDIT 18 0 "8*Pi" "6#*&\"\")\"\"\"%#PiGF%" }{TEXT -1 1 " " } {TEXT 286 1 "~" }{TEXT -1 10 " 2.5133. " }}{PARA 0 "" 0 "" {TEXT -1 47 "The change in the area is approximately 2.5133 " }{XPPEDIT 18 0 "c m^2" "6#*$%#cmG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 " (b) The change in the area is " }{XPPEDIT 18 0 "Pi*(r+Delta*r)^2-Pi*r ^2 = Pi*((r+Delta*r)^2-r^2);" "6#/,&*&%#PiG\"\"\"*$,&%\"rGF'*&%&DeltaG F'F*F'F'\"\"#F'F'*&F&F'*$F*F-F'!\"\"*&F&F',&*$,&F*F'*&%&DeltaGF'F*F'F' F-F'*$F*F-F0F'" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "``=Pi*(10.04^2-10^2)" "6#/%!G*&%#PiG\"\"\",&*$-%&FloatG 6$\"%/5!\"#\"\"#F'*$\"#5F/!\"\"F'" }{TEXT -1 1 " " }{TEXT 279 1 "~" } {TEXT -1 9 " 2.5183. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 77 " There is a geometrical interpretation of the approximation used in par t (a). " }}{PARA 261 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 241 249 249 {PLOTDATA 2 "6.-%'CURVESG6$7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$ \"3Ikwb#=y_O\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#* F/$\"3gLj&[K5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$ \"3t_H-qceDhF/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$ \"3!\\@@&\\!>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF /$\"3aK+16bcT#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F /$\"33)fVPO<\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB [i8F/$\"3ap%>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3J x$F/$\"3IzF#e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[Rg F/$\"3!G;)RQ+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\" 3u3HvXQF/7$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V ?00]Ug8F/7$$!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4 *>c=8]8F/7$$!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$ F/7$$!3%\\F)4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7 $$!3XD84Pfc5rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@ UD=%)o!*\\F/$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30 yCH\"el'3EF/$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3Ezy OHMQdIFas$!3W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRq X8/bDF/$!3U$[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0 I.]F/$!3wHgb5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$ *4(F/$!3)4d)eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H)zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F /$!3%)f]nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)! #F-%'COLOURG6&%$RGBGF*F*F*-F$6$7S7$$\"3U+++++++!)F/F+7$$\"3Tt)o\"**)*3 DzF/$\"3aJh/YDA#4\"F/7$$\"3]\"\\e!z;/RxF/$\"3$*Rfnn6jE?F/7$$\"3oG))3$= p))R(F/$\"3@n]))f#)[UIF/7$$\"3#))[P]tcC#pF/$\"3/7Gc^n$*4SF/7$$\"3wW+IE ;SBjF/$\"3_i$=g`o/!\\F/7$$\"3iz9\"\\Lfgm&F/$\"3C!*QrgTjZcF/7$$\"3!QYbq fkJ*[F/$\"3#>(phR_0HjF/7$$\"3'G%f/Xsh3SF/$\"3Jb`,64ABpF/7$$\"3ifmt:<6c IF/$\"3qE![)3CD$R(F/7$$\"3Ztr(4-4.-#F/$\"3d3^e2XpSxF/7$$\"3y)4#4e%4h2 \"F/$\"3py[*4*QHFzF/7$$!3A'*QB>e&ox%Fao$\"3Y/H&Qd)****zF/7$$!3[)>$R()e )**3\"F/$\"3Iwb4IyRDzF/7$$!3vCdG?d%p6#F/$\"37QbMRg#[r(F/7$$!3F.:&[9([= IF/$\"36C#e'GKp3uF/7$$!30)*z*)=A&Q.%F/$\"3U9HYe![&3pF/7$$!3K][3\"e\"fJ [F/$\"3CI&=2.%=wjF/7$$!3#eR6u2`9n&F/$\"3\"y/%H[w@UcF/7$$!3:m?WkLOBjF/$ \"36n$\\u!z^+\\F/7$$!3$[A[=RXu#pF/$\"3^NYfFBJ,SF/7$$!3Wi\"4&=!\\ZQ(F/$ \"3'R$*efL-m2$F/7$$!3`DJ_2))*3t(F/$\"3+b\\v/\"[#Fas7$$!3\"HWy\\T]n#zF /$!3Au#f\\[0,3\"F/7$$!3\"=%G!QP_#RxF/$!3RG/iAk#e-#F/7$$!3p@lUq(41T(F/$ !3(y![0H)yP,$F/7$$!3i?'yc!*[$RpF/$!3I@G;4Fj!)RF/7$$!3]RR-?MDL5F/$!3v9ZX%f$*H$zF /7$$!3w.V\\Vn!fW#Fas$!3O&))Q&4E'***zF/7$$\"3D%y0Hgy,.\"F/$!3@)zngZ$RLz F/7$$\"3#f9jl2LS/#F/$!3j(yuNqjWt(F/7$$\"3Q2jW3(Q'oIF/$!3Qjlt`<1)Q(F/7$ $\"3A!4y3YSE+%F/$!3CC[W)3ym#pF/7$$\"3q*f!f>.[()[F/$!3Bj\"zM$eWLjF/7$$ \"3/$3u\"4.WzcF/$!3Bd3Z[w6oH,q%RpF/$!3ybQ1y:U!)RF/7$$\"39nEdi2+!Q(F/$!3]cdOh[(z3$F/7$$\"3&GHy `4^Ps(F/$!3'y/S<2ZT3#F/7$$\"3fjCxGWcDzF/$!3Mzv\"\\%Qx)3\"F/7$Fa[l$\"35 (f!\\7(4Lc'FhzFiz-%)POLYGONSG6V7&7$$\"\")!\"\"F+F'7$$\"+8q9@**!#5$\"+O BL`7F\\[m7$$\"+5w\"p$zF\\[m$\"+pem-5F\\[m7&F_[mFijl7$$\"+6;$eo*F\\[m$ \"+t))*o[#F\\[m7$$\"+*Gl'[xF\\[m$\"+)4>&*)>F\\[m7&Fj[mFe[m7$$\"+e[w(H* F\\[m$\"+GbC\"o$F\\[m7$$\"+')=@QuF\\[m$\"+Ak*\\%HF\\[m7&Fe\\mF`\\m7$$ \"+*zmIw)F\\[m$\"+Vn`<[F\\[m7$$\"+RMX5qF\\[m$\"+%RHS&QF\\[m7&F`]mF[]m7 $$\"+U*p,4)F\\[m$\"+DD&y(eF\\[m7$$\"+af8skF\\[m$\"+??G-ZF\\[m7&F[^mFf] m7$$\"+sio*G(F\\[m$\"+i5ZXoF\\[m7$$\"+=!\\<$eF\\[m$\"+]oPwaF\\[m7&Ff^m Fa^m7$$\"+%*)RUP'F\\[m$\"+IC80xF\\[m7$$\"+:>R*4&F\\[m$\"+Wf5khF\\[m7&F a_mF\\_m7$$\"+VzEe`F\\[m$\"+g#zKW)F\\[m7$$\"+aVh'G%F\\[m$\"+3MianF\\[m 7&F\\`mFg_m7$$\"+4HzdUF\\[m$\"+G0F[!*F\\[m7$$\"+FVB1MF\\[m$\"+AkhQsF\\ [m7&Fg`mFb`m7$$\"+Q*p,4$F\\[m$\"+l^c5&*F\\[m7$$\"+]f8sCF\\[m$\"+K@X3wF \\[m7&FbamF]am7$$\"+UJ\"Q(=F\\[m$\"+3D(G#)*F\\[m7$$\"+900*\\\"F\\[m$\" +1!)HeyF\\[m7&F]bmFham7$$\"+D>0zi!#6$\"+%Gn-)**F\\[m7$$\"+S:CB]Ffbm$\" +FQ@%)zF\\[m7&FibmFcbm7$$!+m?0ziFfbmFgbm7$$!+`;CB]FfbmF\\cm7&FbcmF_cm7 $$!+cJ\"Q(=F\\[m$\"+0D(G#)*F\\[m7$$!+D00*\\\"F\\[m$\"+/!)HeyF\\[m7&F[d mFfcm7$$!+_*p,4$F\\[m$\"+g^c5&*F\\[m7$$!+if8sCF\\[m$\"+G@X3wF\\[m7&Ffd mFadm7$$!+AHzdUF\\[m$\"+A0F[!*F\\[m7$$!+QVB1MF\\[m$\"+=khQsF\\[m7&Faem F\\em7$$!+bzEe`F\\[m$\"+_#zKW)F\\[m7$$!+kVh'G%F\\[m$\"+-MianF\\[m7&F\\ fmFgem7$$!+3*RUP'F\\[m$\"+>C80xF\\[m7$$!+E>R*4&F\\[m$\"+Nf5khF\\[m7&Fg fmFbfm7$$!+$G'o*G(F\\[m$\"+]5ZXoF\\[m7$$!+E!\\<$eF\\[m$\"+SoPwaF\\[m7& FbgmF]gm7$$!+]*p,4)F\\[m$\"+9D&y(eF\\[m7$$!+gf8skF\\[m$\"+6?G-ZF\\[m7& F]hmFhgm7$$!+0o1j()F\\[m$\"+Kn`<[F\\[m7$$!+WMX5qF\\[m$\"+'QHS&QF\\[m7& FhhmFchm7$$!+i[w(H*F\\[m$\"+>bC\"o$F\\[m7$$!+!*=@QuF\\[m$\"+:k*\\%HF\\ [m7&FcimF^im7$$!+;;$eo*F\\[m$\"+b))*o[#F\\[m7$$!+$Hl'[xF\\[m$\"+%3>&*) >F\\[m7&F^jmFiim7$$!+:q9@**F\\[m$\"+?BL`7F\\[m7$$!+7w\"p$zF\\[m$\"+cem -5F\\[m7&FijmFdjm7$$FhjlF*$!+in?59F/7$$!\")Fhjl$!+5a;G6F/7&Fc[nF_[n7$$ !+7q9@**F\\[m$!+[BL`7F\\[m7$$!+5w\"p$zF\\[m$!+yem-5F\\[m7&F^\\nFi[n7$$ !+3;$eo*F\\[m$!+$)))*o[#F\\[m7$$!+'Gl'[xF\\[m$!+1\">&*)>F\\[m7&Fi\\nFd \\n7$$!+_[w(H*F\\[m$!+XbC\"o$F\\[m7$$!+#)=@QuF\\[m$!+Ok*\\%HF\\[m7&Fd] nF_]n7$$!+#zmIw)F\\[m$!+dn`<[F\\[m7$$!+MMX5qF\\[m$!+1%HS&QF\\[m7&F_^nF j]n7$$!+M*p,4)F\\[m$!+PD&y(eF\\[m7$$!+Zf8skF\\[m$!+I?G-ZF\\[m7&Fj^nFe^ n7$$!+kio*G(F\\[m$!+r5ZXoF\\[m7$$!+6!\\<$eF\\[m$!+doPwaF\\[m7&Fe_nF`_n 7$$!+())RUP'F\\[m$!+PC80xF\\[m7$$!+5>R*4&F\\[m$!+]f5khF\\[m7&F``nF[`n7 $$!+JzEe`F\\[m$!+n#zKW)F\\[m7$$!+XVh'G%F\\[m$!+9MianF\\[m7&F[anFf`n7$$ !+'*GzdUF\\[m$!+M0F[!*F\\[m7$$!+*RUP'F\\[m$!+5C80xF\\[m7$$\"+N>R*4&F\\[m$!+Gf5khF\\[m7&FignF dgn7$$\"+$H'o*G(F\\[m$!+S5ZXoF\\[m7$$\"+M!\\<$eF\\[m$!+KoPwaF\\[m7&Fdh nF_hn7$$\"+f*p,4)F\\[m$!+-D&y(eF\\[m7$$\"+nf8skF\\[m$!+-?G-ZF\\[m7&F_i nFjhn7$$\"+7o1j()F\\[m$!+?n`<[F\\[m7$$\"+]MX5qF\\[m$!+w$HS&QF\\[m7&Fji nFein7$$\"+n[w(H*F\\[m$!+1bC\"o$F\\[m7$$\"+%*=@QuF\\[m$!+0k*\\%HF\\[m7 &FejnF`jn7$$\"+>;$eo*F\\[m$!+U))*o[#F\\[m7$$\"+&Hl'[xF\\[m$!+u!>&*)>F \\[m7&F`[oF[[o7$$\"+pUwF \\[m$\"+a;;kBF\\[m7%7$$\"+29F/qF\\[m$\"+4\"QI`#F\\[mFg]o7$$\"+_b86sF\\ [m$\"+nDIk=F\\[mF_]o-F$6%7$7$$\"+CJ0*e%F\\[m$\"+BhIlQF\\[m7$$\"+)\\P(= hF\\[m$\"+)\\TP:&F\\[m7%7$$\"+!)ocmaF\\[m$\"+UNm'*\\F\\[mF__o7$$\"+#\\ (4`eF\\[m$\"+I#ex`%F\\[mF_]o-F$6%7$7$$\"+[i5y\"*F\\[m$\"+YAhIxF\\[m7$$ \"+t=U[wF\\[m$\"+soFXFX-%* LINESTYLEG6#FY-F$6%7$7$F/FTF2FVFZ-F$6%7$7$$!3))**************HF-F(F'FV FZ-F$6%7$7$F_oF+F*FVFZ-F$6%7$7$$\"'f\"*GFB$F>F>7$F(F[p7%7$$\")+xu')!\" *$!+++++8!#5F\\p7$F_p$!+++++q!#6FL-F$6%7$7$$\"'f\"R$FBF[p7$F@F[p7%7$$ \"++BV'>'FapFfpF_q7$FbqFbpFL-%%TEXTG6%7$$\"'fTJFBF[pQ&2~~~r6\"-%%FONTG 6$%*HELVETICAG\"#5-Ffq6%7$$!#XFI$\"\"\"F>Q\"rF\\rF]r-Ffq6%FhqQ\"pF\\r- F^r6$%'SYMBOLGFar-Ffq6%7$$!#`FIFgrQ\"DF\\rF]s-%+AXESLABELSG6%Q!F\\rFis -F^r6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$F\\tF\\t" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv e 10" "Curve 11" "Curve 12" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Since the inner circumference o f the ring " }{XPPEDIT 18 0 "2*Pi*r" "6#*(\"\"#\"\"\"%#PiGF%%\"rGF%" } {TEXT -1 38 " is less than the outer circumference " }{XPPEDIT 18 0 "2 *Pi*(r+Delta*r);" "6#*(\"\"#\"\"\"%#PiGF%,&%\"rGF%*&%&DeltaGF%F(F%F%F% " }{TEXT -1 41 ", this process involves some distortion. " }}{PARA 0 " " 0 "" {TEXT -1 30 "The area of this rectangle is " }{XPPEDIT 18 0 "2* Pi*r*Delta;" "6#**\"\"#\"\"\"%#PiGF%%\"rGF%%&DeltaGF%" }{TEXT 281 1 "r " }{TEXT -1 48 ", which is the approximate value for the change " } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 282 1 "A" }{TEXT -1 52 " in the area of the plate obtained by approximating " }{XPPEDIT 18 0 "Del ta;" "6#%&DeltaG" }{TEXT 283 1 "A" }{TEXT -1 7 " using " }{XPPEDIT 18 0 "``(dA/dr)*Delta;" "6#*&-%!G6#*&%#dAG\"\"\"%#drG!\"\"F)%&DeltaGF)" } {TEXT 284 1 "r" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Average and instantaneous rates of change " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Given a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " and two numbers " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x[2]" "6#&%\"xG6#\"\"#" }{TEXT -1 11 " such that " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " is defined on th e interval " }{XPPEDIT 18 0 "[x[1],x[2]]" "6#7$&%\"xG6#\"\"\"&F%6#\"\" #" }{TEXT -1 15 ", the quotient " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(f(x[2]) - f(x[1]))/(x[2]-x[1])" "6#*&,&-%\"fG6#&%\"xG6 #\"\"#\"\"\"-F&6#&F)6#F,!\"\"F,,&&F)6#F+F,&F)6#F,F1F1" }{TEXT -1 13 " \+ ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 14 "is called the " }{TEXT 259 22 "average rate of change" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x )" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[x[1],x[2]]" "6#7$&%\"xG6#\"\"\"&F%6#\"\"#" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 29 "If the value of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " is denoted by the variabl e " }{TEXT 300 1 "y" }{TEXT -1 49 ", and if the specific values associ ated with the " }{TEXT 299 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "x=x[1]" "6#/%\"xG&F$6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x =x[2]" "6#/%\"xG&F$6#\"\"#" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "y[1]=f (x[1])" "6#/&%\"yG6#\"\"\"-%\"fG6#&%\"xG6#F'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y[2]=f(x[2])" "6#/&%\"yG6#\"\"#-%\"fG6#&%\"xG6#F'" } {TEXT -1 31 ", then the qotient (i) becomes " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(y[2]-y[1])/(x[2]-x[1])" "6#*&,&&%\"yG6 #\"\"#\"\"\"&F&6#F)!\"\"F),&&%\"xG6#F(F)&F/6#F)F,F," }{TEXT -1 15 " -- ----- (ii). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 345 239 239 {PLOTDATA 2 "69-%'CURVESG6$7&7$$\"3))**************H!#=$\"3u****** ******\\MF*7$$\"3A+++++++!*F*$\"3%************\\I\"!#<7$F.F+F'-%'COLOU RG6&%$RGBG$\"\"!F9F8$\"*++++\"!\")-F$6%7$7$F(F8F'-F56&F7F9F9F9-%*LINES TYLEG6#\"\"$-F$6%7$7$F.F8F3FAFC-F$6%7$7$F8F+F'FAFC-F$6%7$7$F8F0F-FAFC- F$6$7S7$F8F87$$\"3h*******\\ech#!#>$\"3#36Vp%o')\\EFZ7$$\"3-+++v*G:*[F Z$\"3qAjc`U;6]FZ7$$\"3u******\\L)4X(FZ$\"3:'))4W6p&GxFZ7$$\"3)******\\ MSF+\"F*$\"3)f`u\\v9I0\"F*7$$\"3#)****\\Fy:f7F*$\"3!fsF#\\ZIu\"F*$\"3))Hc8uy& \\*=F*7$$\"3u****\\xOi(*>F*$\"3%3VX`p[r>#F*7$$\"3#)****\\FPQ^AF*$\"3h* eA>;?[]#F*7$$\"3/+++IrS7DF*$\"3*>uV$4m,GGF*7$$\"3p*****\\o;Bu#F*$\"3$) o9/D$Gd'>bF*7$$\"3 **)*****fs#3u%F*$\"3%R&zX:%*fkeF*7$$\"3!)****\\<#Q'**\\F*$\"39GVzKtX\\ iF*7$$\"33++]_u3Y_F*$\"3:(*QYI\"f@i'F*7$$\"3[*****\\PJK]&F*$\"3l=#RL:4 v,(F*7$$\"3%*****\\n(p$RdF*$\"3[\\Y!R.)Q'Q(F*7$$\"3A*****\\#p2%*fF*$\" 3qr39M[_!z(F*7$$\"3o****\\xgkeiF*$\"3A1.n8'yr@)F*7$$\"3g****\\-V&*)['F *$\"3]im\\*p!G%f)F*7$$\"3.+++&\\$pPnF*$\"3%eEGm<>v+*F*7$$\"37******>am %*pF*$\"34(=()ojK4W*F*7$$\"3?*****\\JigC(F*$\"3_cJkoKLr)*F*7$$\"3G**** \\PGKH\"4]57F27$$\"3e******4zj_()F*$\"3 %e%y\"H92$e7F27$$\"3u)***\\<3;%**)F*$\"3Ah_!es!*QI\"F27$$\"3]****\\Z=i Y#*F*$\"3S@LqiA;_8F27$$\"39)*****\\'[M\\*F*$\"3wL4N,F(**R\"F27$$\"3y)* **\\PM&=v*F*$\"3i-C3r&y1X\"F27$$\"3*******fzs++\"F2$\"3$*G'\\YfX,]\"F2 7$$\"3(*****\\5Q_D5F2$\"3u>7X`LP^:F27$$\"3-++vxSw]5F2$\"3./\\H^m\"Gg\" F27$$\"3'******>EdR2\"F2$\"39o7Ii$\\1l\"F27$$\"3&*****\\o#R05\"F2$\"3c d_D4E81`9V7\"F2$\"3E*=La6cjv\"F27$$\"3)****\\<#Rm\\6F 2$\"3*\\zO#)[F0\"=F27$$\"3%****\\A_ER<\"F2$\"3#)*)42i#yH'=F27$$\"3%*** ***********>\"F2$\"3$*************>>F2-F56&F7F:F8F8-F$6&7$F'F--%'SYMBO LG6#%'CIRCLEGFA-%&STYLEG6#%&POINTG-F$6&Ff]l-Fh]l6#%(DIAMONDGFAF[^l-F$6 &Ff]l-Fh]l6#%&CROSSGFAF[^l-%%TEXTG6&7$$\"$0\"!\"#$\"#>!\"\"Q)y~=~f(x)6 \"Fb]l-%%FONTG6$%*HELVETICAG\"#5-Fj^l6&7$$\"#GF__l$\"#VF__lQ\"PFd_lFAF e_l-Fj^l6&7$$\"#))F__l$\"#9Fb_lQ\"QFd_lFAFe_l-Fj^l6&7$$FFFb_l$!\"&F__l Q\"xFd_lFAFe_l-Fj^l6&7$$\"\"*Fb_lF^alF`alFAFe_l-Fj^l6&7$$!\"%F__l$\"++ ++]O!#5Q\"yFd_lFAFe_l-Fj^l6&7$Fial$\"++++D8!\"*F^blFAFe_l-Fj^l6&7$$\"$ B$!\"$$Fb_lFb_lQ\"1Fd_lFA-Ff_l6$Fh_l\"\")-Fj^l6&7$$\"$D*FjblF[clQ\"2Fd _lFAF]cl-Fj^l6&7$$F2Fjbl$\"++++]JF]blF\\clFAF]cl-Fj^l6&7$$!#:Fjbl$\"++ ++v7FdblFeclFAF]cl-%+AXESLABELSG6%Q!Fd_lFfdl-Ff_l6#%(DEFAULTG-%*AXESST YLEG6#%%NONEG-%%VIEWG6$;F^al$\"#7Fb_l;F[cl$\"++++?>Fdbl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+ 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We say that the quotient (ii) gives the " }{TEXT 259 26 "average rate of change of " }{TEXT 301 1 "y" }{TEXT 259 17 " with respect to " }{TEXT 302 1 "x" } {TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[x[1],x[2]]" "6#7$&% \"xG6#\"\"\"&F%6#\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "For example, suppose that an object \+ is moving so that its position " }{TEXT 304 1 "s" }{TEXT -1 58 " measu red along a line is described as a function of time " }{TEXT 303 1 "t " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "s=s[1]" "6#/%\"sG&F$6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "s=s [2]" "6#/%\"sG&F$6#\"\"#" }{TEXT -1 41 " be the positions of the objec t at times " }{XPPEDIT 18 0 "t=t[1]" "6#/%\"tG&F$6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t=t[2]" "6#/%\"tG&F$6#\"\"#" }{TEXT -1 33 " \+ respectively, then the quotient " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(s[2]-s[1])/(t[2]-t[1])" "6#*&,&&%\"sG6#\"\"#\"\"\"&F&6 #F)!\"\"F),&&%\"tG6#F(F)&F/6#F)F,F," }{TEXT -1 14 " ------- (iii)" }} {PARA 0 "" 0 "" {TEXT -1 46 "is the average rate of change of the posi tion " }{TEXT 308 1 "s" }{TEXT -1 22 " with respect to time " }{TEXT 307 1 "t" }{TEXT -1 5 ", or " }{TEXT 259 16 "average velocity" }{TEXT -1 46 " of the object over the interval of time from " }{XPPEDIT 18 0 "t=t[1]" "6#/%\"tG&F$6#\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "t=t[ 2]" "6#/%\"tG&F$6#\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 142 "The magnitude of the quotient (iii) is just the distance travelle d divided by the time taken, which is the usual definition of average \+ speed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Average velocities computed over successively smaller intervals " }{XPPEDIT 18 0 "[t[1],t[2]]" "6#7$&%\"tG6#\"\"\"&F%6#\"\"#" }{TEXT -1 8 " where " }{XPPEDIT 18 0 "t[2]" "6#&%\"tG6#\"\"#" }{TEXT -1 10 " te nds to " }{XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\"\"\"" }{TEXT -1 25 " appro ach the derivative " }{XPPEDIT 18 0 "ds/dt" "6#*&%#dsG\"\"\"%#dtG!\"\" " }{TEXT -1 14 " evaluated at " }{XPPEDIT 18 0 "t=t[1]" "6#/%\"tG&F$6# \"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "The derivative \+ " }{XPPEDIT 18 0 "ds/dt" "6#*&%#dsG\"\"\"%#dtG!\"\"" }{TEXT -1 14 " ev aluated at " }{XPPEDIT 18 0 "t=t[1]" "6#/%\"tG&F$6#\"\"\"" }{TEXT -1 15 " is called the " }{TEXT 259 22 "instantaneous velocity" }{TEXT -1 23 " of the object at time " }{XPPEDIT 18 0 "t=t[1]" "6#/%\"tG&F$6#\" \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "More generally, \+ the derivative " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 15 " of a function " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6 #%\"xG" }{TEXT -1 11 " gives the " }{TEXT 259 28 "instantaneous rate o f change" }{TEXT -1 17 " of the variable " }{TEXT 305 1 "y" }{TEXT -1 17 " with respect to " }{TEXT 306 1 "x" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "Examples involving average and instantaneous rates of cha nge" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 323 8 "Questi on" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 65 "A spherical tank o f radius 2 metres holds a liquid to a depth of " }{TEXT 322 1 "x" } {TEXT -1 9 " metres. " }}{PARA 0 "" 0 "" {TEXT -1 32 "If the volume of this liquid is " }{TEXT 340 1 "V" }{TEXT -1 32 " cubic meters, then t he formula " }{XPPEDIT 18 0 "V = Pi*(2*x^2-x^3/3);" "6#/%\"VG*&%#PiG\" \"\",&*&\"\"#F'*$%\"xGF*F'F'*&F,\"\"$F.!\"\"F/F'" }{TEXT -1 26 " descr ibes how the volume " }{TEXT 341 1 "V" }{TEXT -1 24 " changes with the depth " }{TEXT 342 1 "x" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "0<=x " "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=4" "6#1%!G\"\"%" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "(a) Find the average rate of change o f the volume " }{TEXT 327 1 "V" }{TEXT -1 27 " with respect to the dep th " }{TEXT 328 1 "x" }{TEXT -1 24 " over the interval from " } {XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 56 "(b) Find the instantaneous rate of change of the volume " }{TEXT 329 1 "V" }{TEXT -1 41 " of the liquid with respect to the depth " } {TEXT 330 1 "x" }{TEXT -1 19 " when the depth is:" }}{PARA 0 "" 0 "" {TEXT -1 49 " (i) 1 metre (ii) 2 metres (iii) 3 metres. " }} {PARA 0 "" 0 "" {TEXT -1 22 "(c) Sketch a graph of " }{TEXT 325 1 "V" }{TEXT -1 9 " against " }{TEXT 326 1 "x" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 324 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 9 "(a) When " }{XPPEDIT 18 0 "x=1 " "6#/%\"xG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "V=Pi*(2-2/3)" "6#/ %\"VG*&%#PiG\"\"\",&\"\"#F'*&F)F'\"\"$!\"\"F,F'" }{XPPEDIT 18 0 "``=5* Pi/3" "6#/%!G*(\"\"&\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 10 " and when \+ " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "V=Pi*(18-9)" "6#/%\"VG*&%#PiG\"\"\",&\"#=F'\"\"*!\"\"F'" }{XPPEDIT 18 0 "``=9*Pi" "6#/%!G*&\"\"*\"\"\"%#PiGF'" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 96 "Hence the average rate of change of the volume \+ with respect to the depth over the interval from " }{XPPEDIT 18 0 "x=1 " "6#/%\"xG\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\" \"$" }{TEXT -1 5 ". is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(9*Pi-5*Pi/3)/(3-1) = ``(22*Pi/3)/2" "6#/*&,&*&\"\"*\"\"\"%#PiGF (F(*(\"\"&F(F)F(\"\"$!\"\"F-F(,&F,F(F(F-F-*&-%!G6#*(\"#AF(F)F(F,F-F(\" \"#F-" }{XPPEDIT 18 0 "``=11*Pi/3" "6#/%!G*(\"#6\"\"\"%#PiGF'\"\"$!\" \"" }{TEXT -1 26 " cubic metres per metre. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "(b) Given " }{XPPEDIT 18 0 "V = Pi*(2*x^2-x^3/3)" "6#/%\"VG*&%#PiG\"\"\",&*&\"\"#F'*$%\"xGF*F'F'*&F, \"\"$F.!\"\"F/F'" }{TEXT -1 38 ", the instantaneous rate of change of \+ " }{TEXT 337 1 "V" }{TEXT -1 17 " with respect to " }{TEXT 338 1 "x" } {TEXT -1 29 " is given by the derivative: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dV/dx = Pi*(4*x-x^2);" "6#/*&%#dVG\"\"\"%#dxG !\"\"*&%#PiGF&,&*&\"\"%F&%\"xGF&F&*$F.\"\"#F(F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The insta ntaneous rate of change of the volume " }{TEXT 339 1 "V" }{TEXT -1 27 " with respect to the depth " }{TEXT 331 1 "x" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 10 " (i) when " }{XPPEDIT 18 0 "x=1" "6#/%\"x G\"\"\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "3*Pi" "6#*&\"\"$\"\"\"%#Pi GF%" }{TEXT -1 25 " cubic metres per metre, " }}{PARA 0 "" 0 "" {TEXT -1 11 " (ii) when " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "4*Pi;" "6#*&\"\"%\"\"\"%#PiGF%" }{TEXT -1 25 " c ubic metres per metre, " }}{PARA 0 "" 0 "" {TEXT -1 12 " (iii) when " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "3*Pi" "6#*&\"\"$\"\"\"%#PiGF%" }{TEXT -1 25 " cubic metres per metr e. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plot(Pi*(2*x^2-x^3/3),x=0..4,labels=[`depth in metre s, x`,`volume in cu. metres, V`],\n labeldirections=[horizontal,vert ical]);" }}{PARA 13 "" 1 "" {GLPLOT2D 307 249 249 {PLOTDATA 2 "6%-%'CU RVESG6$7S7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>$\"3'>G4W`ypq%F-7$$\"3RLLLe'4 0j\"!#=$\"3y(HC&z&H]i\"F37$$\"3mmmm;6m$[#F3$\"3x;x6I5R:PF37$$\"3fmmm;y YULF3$\"3)G2u;k$eGmF37$$\"3%HLL$eF>(>%F3$\"3OeC#3XV%H5!#<7$$\"3Qmmm\"> K'*)\\F3$\"3Xz69*p,UV\"FE7$$\"3P*****\\Kd,\"eF3$\"3gK0*H.xc\">FE7$$\"3 -mmm\"fX(emF3$\"3-%**HCs=nZ#FE7$$\"3.*****\\U7Y](F3$\"3Up;4H\"Qg4$FE7$ $\"3'QLLLV!pu$)F3$\"3o*y1'y\\l\"z$FE7$$\"3xmmm;c0T\"*F3$\"3iJVy'e$H]WF E7$$\"3#*******H,Q+5FE$\"3-;\"G2&3dR_FE7$$\"3)*******\\*3q3\"FE$\"3uO6 @-;7zgFE7$$\"3)*******p=\\q6FE$\"3\"Q:%e!Gk*GpFE7$$\"3mmm;fBIY7FE$\"3` %Q>!R*pAt(FE7$$\"3GLLLj$[kL\"FE$\"3#*)=zcnwEs)FE7$$\"3?LLL`Q\"GT\"FE$ \"3_a0e@>P)e*FE7$$\"3!*****\\s]k,:FE$\"3id%*f+gAi5!#;7$$\"39LLL`dF!e\" FE$\"3AJVgV#=e:\"Faq7$$\"33++]sgam;FE$\"3e>#R)Q#o.E\"Faq7$$\"3/++]FE$\"3sS#>5@7kc\"Faq7$$\"3immmTc-)*>FE$\"37#Gm3.NIn\"Faq7$$\"3Mmm ;f`@'3#FE$\"3AvRB)Q!z$y\"Faq7$$\"3y****\\nZ)H;#FE$\"33#[rNRv)z=Faq7$$ \"3YmmmJy*eC#FE$\"3O]W!eRjH)>Faq7$$\"3')******R^bJBFE$\"33uRU*yV$)3#Fa q7$$\"3f*****\\5a`T#FE$\"3w)pjRfh**=#Faq7$$\"3o****\\7RV'\\#FE$\"3]0Nb _9a'G#Faq7$$\"3k*****\\@fke#FE$\"3)QJ&*[7g8R#Faq7$$\"3/LLL`4NnEFE$\"3] !o?ZI5I[#Faq7$$\"3#*******\\,s`FFE$\"3E#*QkQ%Hyd#Faq7$$\"3[mm;zM)>$GFE $\"3CJd#>m42m#Faq7$$\"3$*******pfa%Qp(H5$Faq7$$\"3bmmm,FT=MFE$\"37IXa;w5fJFaq7$$\"3FLL$ e#pa-NFE$\"31ODd(oR%3KFaq7$$\"3!*******Rv&)zNFE$\"3?0CM[z)yC$Faq7$$\"3 ILLLGUYoOFE$\"3qtR&[-'y&G$Faq7$$\"3_mmm1^rZPFE$\"3Jfk>wEs7LFaq7$$\"34+ +]sI@KQFE$\"3$p:e:6QQL$Faq7$$\"34++]2%)38RFE$\"3s$H)eH[NYLFaq7$$\"\"%F )$\"398\"HQ;K5N$Faq-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6'%3d epth~in~metres,~xG%8volume~in~cu.~metres,~VG-%%FONTG6#%(DEFAULTG%+HORI ZONTALG%)VERTICALG-%%VIEWG6$;F(FdzFg[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 4 "Note" }{TEXT -1 15 ": The graph of " }{XPPEDIT 18 0 "V \+ = Pi*(2*x^2-x^3/3)" "6#/%\"VG*&%#PiG\"\"\",&*&\"\"#F'*$%\"xGF*F'F'*&F, \"\"$F.!\"\"F/F'" }{TEXT -1 66 " is approximately linear (a straight l ine) over the interval from " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 162 " . This means that the average rate of change of the volume with depth \+ over this interval is approximately equal to the instantaneous rate of change at a value of " }{TEXT 332 1 "x" }{TEXT -1 53 " in this interv al, and this holds in particular when " }{XPPEDIT 18 0 "x=2" "6#/%\"xG \"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 92 "This correspond s to the observation that the tangent line to the graph at the point M where " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 14 " has gradi ent " }{XPPEDIT 18 0 "4*Pi" "6#*&\"\"%\"\"\"%#PiGF%" }{TEXT -1 114 ", \+ which is only slightly greater (in relative terms) than the gradient o f the secant line through P and Q, namely " }{XPPEDIT 18 0 "11*Pi/3" " 6#*(\"#6\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 324 298 298 {PLOTDATA 2 "6--%'CURVESG6$7S7$$ \"\"!F)F(7$$\"3Hmmmm;')=()!#>$\"3'>G4W`ypq%F-7$$\"3RLLLe'40j\"!#=$\"3y (HC&z&H]i\"F37$$\"3mmmm;6m$[#F3$\"3x;x6I5R:PF37$$\"3fmmm;yYULF3$\"3)G2 u;k$eGmF37$$\"3%HLL$eF>(>%F3$\"3OeC#3XV%H5!#<7$$\"3Qmmm\">K'*)\\F3$\"3 Xz69*p,UV\"FE7$$\"3P*****\\Kd,\"eF3$\"3gK0*H.xc\">FE7$$\"3-mmm\"fX(emF 3$\"3-%**HCs=nZ#FE7$$\"3.*****\\U7Y](F3$\"3Up;4H\"Qg4$FE7$$\"3'QLLLV!p u$)F3$\"3o*y1'y\\l\"z$FE7$$\"3xmmm;c0T\"*F3$\"3iJVy'e$H]WFE7$$\"3#**** ***H,Q+5FE$\"3-;\"G2&3dR_FE7$$\"3)*******\\*3q3\"FE$\"3uO6@-;7zgFE7$$ \"3)*******p=\\q6FE$\"3\"Q:%e!Gk*GpFE7$$\"3mmm;fBIY7FE$\"3`%Q>!R*pAt(F E7$$\"3GLLLj$[kL\"FE$\"3#*)=zcnwEs)FE7$$\"3?LLL`Q\"GT\"FE$\"3_a0e@>P)e *FE7$$\"3!*****\\s]k,:FE$\"3id%*f+gAi5!#;7$$\"39LLL`dF!e\"FE$\"3AJVgV# =e:\"Faq7$$\"33++]sgam;FE$\"3e>#R)Q#o.E\"Faq7$$\"3/++]FE$\"3s S#>5@7kc\"Faq7$$\"3immmTc-)*>FE$\"37#Gm3.NIn\"Faq7$$\"3Mmm;f`@'3#FE$\" 3AvRB)Q!z$y\"Faq7$$\"3y****\\nZ)H;#FE$\"33#[rNRv)z=Faq7$$\"3YmmmJy*eC# FE$\"3O]W!eRjH)>Faq7$$\"3')******R^bJBFE$\"33uRU*yV$)3#Faq7$$\"3f***** \\5a`T#FE$\"3w)pjRfh**=#Faq7$$\"3o****\\7RV'\\#FE$\"3]0Nb_9a'G#Faq7$$ \"3k*****\\@fke#FE$\"3)QJ&*[7g8R#Faq7$$\"3/LLL`4NnEFE$\"3]!o?ZI5I[#Faq 7$$\"3#*******\\,s`FFE$\"3E#*QkQ%Hyd#Faq7$$\"3[mm;zM)>$GFE$\"3CJd#>m42 m#Faq7$$\"3$*******pfa%Qp(H5$Faq7$$\"3bmmm,FT=MFE$\"37IXa;w5fJFaq7$$\"3FLL$e#pa-NFE$\"3 1ODd(oR%3KFaq7$$\"3!*******Rv&)zNFE$\"3?0CM[z)yC$Faq7$$\"3ILLLGUYoOFE$ \"3qtR&[-'y&G$Faq7$$\"3_mmm1^rZPFE$\"3Jfk>wEs7LFaq7$$\"34++]sI@KQFE$\" 3$p:e:6QQL$Faq7$$\"34++]2%)38RFE$\"3s$H)eH[NYLFaq7$$\"\"%F)$\"398\"HQ; K5N$Faq-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7$7$$\"\"\"F)$\"3$)))H)fv( )fB&FE7$$\"\"$F)$\"3\"Q\"3B)QLu#GFaq-Fiz6&F[[lF(F($\"*++++\"!\")-F$6$7 $7$$\"3a**************pF3$\"37:WJ%>!z)=%F37$$\"3#)*************H$FE$\" 35^StgU94LFaq-%&COLORG6&F[[lF)$\"\"(F^[l$\"\"#F^[l-F$6&7%Fb[l7$$Fd]lF) $\"3AcX\">3;bn\"FaqFg[l-%'SYMBOLG6#%'CIRCLEG-Fiz6&F[[lF)F)F)-%&STYLEG6 #%&POINTG-F$6&Fg]l-F]^l6#%(DIAMONDGF`^lFb^l-F$6&Fg]l-F]^l6#%&CROSSGF`^ lFb^l-%%TEXTG6%7$$\"#$*!\"#$\"#jF^[lQ\"P6\"F`^l-Fa_l6%7$$\"#JF^[l$\"$y #F^[lQ\"QFj_lF`^l-Fa_l6%7$$\"#>F^[l$\"$z\"F^[lQ\"MFj_lF`^l-%+AXESLABEL SG6'%3depth~in~metres,~xG%8volume~in~cu.~metres,~VG-%%FONTG6#%(DEFAULT G%+HORIZONTALG%)VERTICALG-%%VIEWG6$;F(FdzFcal" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 333 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 62 "When a stone is dropped from the top of a cliff, its distance " }{TEXT 335 1 "s" }{TEXT -1 41 " metres below th e top of the cliff after " }{TEXT 336 1 "t" }{TEXT -1 19 " secs. is gi ven by " }{XPPEDIT 18 0 "s=4.9*t^2" "6#/%\"sG*&-%&FloatG6$\"#\\!\"\"\" \"\"*$%\"tG\"\"#F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 66 "(a ) Find the average velocity of the stone over the interval from " } {XPPEDIT 18 0 "t=2" "6#/%\"tG\"\"#" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "t=6" "6#/%\"tG\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 80 "( b) Find the instantaneous velocity of the stone 4 seconds after it is \+ dropped. " }}{PARA 0 "" 0 "" {TEXT -1 58 "(c) Show that the average ve locity over any interval from " }{XPPEDIT 18 0 "t=t[1]" "6#/%\"tG&F$6# \"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "t=t[2]" "6#/%\"tG&F$6#\"\"# " }{TEXT -1 60 " is the same as the instantaneous velocity at the mid \+ point " }{XPPEDIT 18 0 "t=(t[1]+t[2])/2" "6#/%\"tG*&,&&F$6#\"\"\"F)&F$ 6#\"\"#F)F)F,!\"\"" }{TEXT -1 18 " of the interval. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 334 8 "Solution" }{TEXT -1 2 " : " }}{PARA 0 "" 0 "" {TEXT -1 9 "(a) When " }{XPPEDIT 18 0 "t=2" "6#/ %\"tG\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "s = 4.9*`.`*2^2;" "6#/%\" sG*(-%&FloatG6$\"#\\!\"\"\"\"\"%\".GF+\"\"#F-" }{TEXT -1 16 "=19.6, an d when " }{XPPEDIT 18 0 "t=4" "6#/%\"tG\"\"%" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "s = 4.9*`.`*6^2;" "6#/%\"sG*(-%&FloatG6$\"#\\!\"\"\"\" \"%\".GF+\"\"'\"\"#" }{XPPEDIT 18 0 "`` = 176.4;" "6#/%!G-%&FloatG6$\" %kF*F*F+,&\"\"'F+\"\"#F*F*-F'6$\"$#RF*" }{TEXT -1 17 " m etres per sec. " }}{PARA 0 "" 0 "" {TEXT -1 10 "(b) Given " }{XPPEDIT 18 0 "s=4.9*t^2" "6#/%\"sG*&-%&FloatG6$\"#\\!\"\"\"\"\"*$%\"tG\"\"#F+ " }{TEXT -1 56 ", the instantaneous velocity is given by the derivativ e " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ds/dt=9.8*t" "6 #/*&%#dsG\"\"\"%#dtG!\"\"*&-%&FloatG6$\"#)*F(F&%\"tGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "t=4" "6#/%\"tG \"\"%" }{TEXT -1 32 ", the instantaneous velocity is " }{XPPEDIT 18 0 "9.8*`.`*4 = 39.2;" "6#/*(-%&FloatG6$\"#)*!\"\"\"\"\"%\".GF*\"\"%F*-F& 6$\"$#RF)" }{TEXT -1 20 " metres per second. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "(c) The average velocity \+ over the interval " }{XPPEDIT 18 0 "[t[1], t[2]];" "6#7$&%\"tG6#\"\"\" &F%6#\"\"#" }{TEXT -1 4 " is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(4.9*``(t[2])^2-4.9*``(t[1])^2)/(t[2]-t[1]) = 4.9*(t[2] +t[1])*(t[2]-t[1])/(t[2]-t[1]);" "6#/*&,&*&-%&FloatG6$\"#\\!\"\"\"\"\" *$-%!G6#&%\"tG6#\"\"#F4F,F,*&-F(6$F*F+F,*$-F/6#&F26#F,F4F,F+F,,&&F26#F 4F,&F26#F,F+F+**-F(6$F*F+F,,&&F26#F4F,&F26#F,F,F,,&&F26#F4F,&F26#F,F+F ,,&&F26#F4F,&F26#F,F+F+" }{XPPEDIT 18 0 "``=4.9*(t[2]+t[1])" "6#/%!G*& -%&FloatG6$\"#\\!\"\"\"\"\",&&%\"tG6#\"\"#F+&F.6#F+F+F+" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 64 "On the other hand, the instantaneou s velocity at the mid-point " }{XPPEDIT 18 0 "t=(t[1]+t[2])/2" "6#/% \"tG*&,&&F$6#\"\"\"F)&F$6#\"\"#F)F)F,!\"\"" }{TEXT -1 17 " of the inte rval " }{XPPEDIT 18 0 "[t[1], t[2]];" "6#7$&%\"tG6#\"\"\"&F%6#\"\"#" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "9.8*`.`*``((t[1]+t[2])/2)=4.9*(t[2] +t[1])" "6#/*(-%&FloatG6$\"#)*!\"\"\"\"\"%\".GF*-%!G6#*&,&&%\"tG6#F*F* &F26#\"\"#F*F*F6F)F**&-F&6$\"#\\F)F*,&&F26#F6F*&F26#F*F*F*" }{TEXT -1 8 ", also. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The following diagram illustrates the situation for " } {XPPEDIT 18 0 "t[1]=2" "6#/&%\"tG6#\"\"\"\"\"#" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "t[2] =6" "6#/&%\"tG6#\"\"#\"\"'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 39 "The tangent line at the point M, where " }{XPPEDIT 18 0 "t=4" "6#/%\"tG\"\"%" }{TEXT -1 82 ", is parallel to th e secant line joining the points P and Q on the curve given by " } {XPPEDIT 18 0 "t=2" "6#/%\"tG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t=6" "6#/%\"tG\"\"'" }{TEXT -1 3 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 355 440 440 {PLOTDATA 2 "6--%'CURVESG6$7S7$$\"\"!F )F(7$$\"3gmmm\"z+e_\"!#=$\"3<2oOZLvS6F-7$$\"3sLL3->R`GF-$\"3OD'o*>U]*) RF-7$$\"3mmm;apSYVF-$\"3M'*=]rTrc#*F-7$$\"3Onm;z'=$\\eF-$\"3Qc>^@>^w;! #<7$$\"3!RL$3Ft3XtF-$\"3#*H:G%3lNk#F?7$$\"3tmmTNj&=t)F-$\"39`a0Q/-OPF? 7$$\"33+](=`xn,\"F?$\"3+.lp\"4*zl]F?7$$\"3#omT&y/Gl6F?$\"3c]W)*3^g`mF? 7$$\"3++]PurI88F?$\"3=\\x*y4,9X)F?7$$\"3aLL$e#3dl9F?$\"3sh:JW*pC0\"!#; 7$$\"3ymm\"Ht%o*f\"F?$\"3Ang\"*4d!RD\"Fhn7$$\"3K++]F_m]F?$\"3m;g&)y67t;JlFhn7$$\"3?+]7VLA &y$F?$\"3G=e4@(y1-(Fhn7$$\"3'pm;a?@.$RF?$\"3y#['4T\"Q#pvFhn7$$\"3)**** **\\\\@-3%F?$\"3WL8!Q)Q+ \"!#:7$$\"3%GLL$oT'ym%F?$\"3k,l!)Q)ew1\"Fbu7$$\"3'3++DE5!>[F?$\"34ORfN ,#z8\"Fbu7$$\"3Mm;a)3rf&\\F?$\"3c#e#4A3_.7Fbu7$$\"3*4++vW0d5&F?$\"3EG \\rxJMx7Fbu7$$\"3;L$3-\"QfY_F?$\"3M.l&Qe5)[8Fbu7$$\"3C+]PWF'QR&F?$\"3' Rn_*4SfD9Fbu7$$\"3[LL$e/Xy`&F?$\"3+z8$)f'=F]\"Fbu7$$\"3m**\\(=<\"e)o&F ?$\"3YE:qJyj&e\"Fbu7$$\"3%ymmm(zvLeF?$\"3!ox&QuQgn;Fbu7$$\"3-nm\"zAAA) fF?$\"3]K[Uc@c`%4%=Fbu7$$\"3#4++]p] ZE'F?$\"3&okaA'z5B>Fbu7$$\"3$QL$e*R7)>kF?$\"3G_j/rb[>?Fbu7$$\"3'pmmmV, &elF?$\"35yJk8Jo2@Fbu7$$\"3<+](o(GP1nF?$\"3t[0-@kz.AFbu7$$\"3g+]78Z!z% oF?$\"3MS>/\\hz(H#Fbu7$$\"\"(F)$\"3B++++++,CFbu-%'COLOURG6&%$RGBG$\"#5 !\"\"F(F(-F$6$7$7$$\"\"#F)$\"39++++++g>Fhn7$$\"\"'F)$\"31++++++kR F?7$$\"3M+++++++kF?$\"3=+++++![s\"Fbu-%&COLORG6&F[[lF)$FezF^[l$Fd[lF^[ l-F$6&7%Fb[l7$$\"\"%F)$\"3d++++++SyFhnFg[l-%'SYMBOLG6#%'CIRCLEG-Fiz6&F [[lF)F)F)-%&STYLEG6#%&POINTG-F$6&Fe]l-F\\^l6#%(DIAMONDGF_^lFa^l-F$6&Fe ]l-F\\^l6#%&CROSSGF_^lFa^l-%%TEXTG6%7$$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 94 "Example to introduce the notions of differentiability a nd non-differentiability of a function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the functi on " }{XPPEDIT 18 0 "f(x)=abs(x^2-1)" "6#/-%\"fG6#%\"xG-%$absG6#,&*$F' \"\"#\"\"\"F.!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "In the interval " }{XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"%\"xG" } {XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\"" }{TEXT -1 15 ", the value of " }{XPPEDIT 18 0 "``(x^2-1);" "6#-%!G6#,&*$%\"xG\"\"#\"\"\"F*!\"\"" } {TEXT -1 35 " is negative, and so the value of " }{XPPEDIT 18 0 "abs( x^2-1)" "6#-%$absG6#,&*$%\"xG\"\"#\"\"\"F*!\"\"" }{TEXT -1 36 " is obt ained by changing the sign of" }{XPPEDIT 18 0 "``(x^2-1);" "6#-%!G6#,& *$%\"xG\"\"#\"\"\"F*!\"\"" }{TEXT -1 41 ". This can be achieved by mul tiplying by " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "abs(x^2-1)=1-x^2" "6#/-%$absG6#,&*$%\"xG\"\"#\"\"\"F+!\"\",&F+F+*$F)F*F," }{TEXT -1 6 " \+ for " }{XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\"" }{TEXT -1 38 ", and over this interval the \+ graph of " }{XPPEDIT 18 0 "f(x)=abs(x^2-1)" "6#/-%\"fG6#%\"xG-%$absG6# ,&*$F'\"\"#\"\"\"F.!\"\"" }{TEXT -1 26 " is the reflection in the " } {TEXT 288 1 "x" }{TEXT -1 23 " axis of the graph of " }{XPPEDIT 18 0 "y=x^2-1" "6#/%\"yG,&*$%\"xG\"\"#\"\"\"F)!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 21 "Outside the interval " }{XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\" " }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "x>=1" "6#1\"\"\"%\"xG" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "x<=-1" "6#1%\"xG,$\"\"\"!\"\"" } {TEXT -1 14 ", the value of" }{XPPEDIT 18 0 "``(x^2-1);" "6#-%!G6#,&*$ %\"xG\"\"#\"\"\"F*!\"\"" }{TEXT -1 21 " is positive, and so " } {XPPEDIT 18 0 "abs(x^2-1)" "6#-%$absG6#,&*$%\"xG\"\"#\"\"\"F*!\"\"" } {TEXT -1 22 " has the same value as" }{XPPEDIT 18 0 "``(x^2-1);" "6#-% !G6#,&*$%\"xG\"\"#\"\"\"F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "abs(x^2-1) = x^2-1;" "6#/-%$absG6# ,&*$%\"xG\"\"#\"\"\"F+!\"\",&*$F)F*F+F+F," }{TEXT -1 5 " for " } {XPPEDIT 18 0 "x>=1" "6#1\"\"\"%\"xG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x<=-1" "6#1%\"xG,$\"\"\"!\"\"" }{TEXT -1 26 ", and for these values of " }{TEXT 289 1 "x" }{TEXT -1 14 " the graph of " }{XPPEDIT 18 0 "f (x)=abs(x^2-1)" "6#/-%\"fG6#%\"xG-%$absG6#,&*$F'\"\"#\"\"\"F.!\"\"" } {TEXT -1 30 " coincides with the graph of " }{XPPEDIT 18 0 "y=x^2-1" "6#/%\"yG,&*$%\"xG\"\"#\"\"\"F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 74 "This can be summarised by giving the following \"piece wise\" description of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) =PIECEWISE([x^2-1,x<=-1],[1-x^2,-1=1])" "6#/-%\"f G6#%\"xG-%*PIECEWISEG6%7$,&*$F'\"\"#\"\"\"F/!\"\"1F',$F/F07$,&F/F/*$F' F.F032,$F/F0F'2F'F/7$,&*$F'F.F/F/F01F/F'" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "p1 : = plot(abs(x^2-1),x=-1.8..1.8):\np2 := plot(x^2-1,x=-1..1,color=black, linestyle=2):\nplots[display]([p1,p2],tickmarks=[3,4]);" }}{PARA 13 " " 1 "" {GLPLOT2D 331 301 301 {PLOTDATA 2 "6'-%'CURVESG6$7[p7$$!3/+++++ ++=!#<$\"3A++++++SAF*7$$!31++]A^wgF*7$$!3++](y=#R(o\"F*$\"3Wz:q&R#HZ=F*7$$!3;++vITD`;F*$\" 3yQ>%3A\\Kt\"F*7$$!36+]7:B'[h\"F*$\"3)\\(3\"oH!y2;F*7$$!31++]*\\qkd\"F *$\"3szPzN#f_[\"F*7$$!3.+++)>Cy`\"F*$\"3CVV&RE.\\O\"F*7$$!3+++]'*y<*\\ \"F*$\"3ikTNlV`Z7F*7$$!3-+]7u_rg9F*$\"3zt?17\"*oL6F*7$$!3/++v^EDA9F*$ \"332*4ag-G-\"F*7$$!3-++DxGf'Q\"F*$\"3'fL/C2)RE#*!#=7$$!3+++v-J$4N\"F* $\"3Q?u0\"[--D)F^o7$$!35++vSe3x7F*$\"3qX8WYC[4jF^o7$$!3/++v'*Gr+7F*$\" 3Dix@/Y6Eh\"y\"F^o7$$!33+++hyFY5F*$\"3!)\\N(=CO(p%*!#>7$$!3.+]PWY.H5F*$ \"3ky#)RK*H7*eF]q7$$!3(****\\xU\"z65F*$\"3]O$QoK*=sBF]q7$$!37]PfB1[25F *$\"33r^o\"p?<]\"F]q7$$!30+vV>)pJ+\"F*$\"3;gjEImo\\j!#?7$$!3%)*\\7G:!f ))**F^o$\"3%odr8fn1G#Fbr7$$!3E++D6@[X**F^o$\"3AKi![d&Q(3\"F]q7$$!3)*** \\7GgEf)*F^o$\"3S!HgmQt[z#F]q7$$!3!3++]%*\\Ix*F^o$\"3-$[NDx%\\([%F]q7$ $!33++](Q9[Q*F^o$\"3y*H<7*e_#>\"F^o7$$!3Y+++I)yl**)F^o$\"3*>efb$p:1>F^ o7$$!3!4+++X>p@)F^o$\"31o6AvMA[KF^o7$$!3U******pJdluF^o$\"3[;wLC<_EWF^ o7$$!3e++]n(yKy'F^o$\"3Mn$Qi\"Hr)R&F^o7$$!3;,++IZ'>(fF^o$\"3T,OOEPcLkF ^o7$$!3s+++?`n%G&F^o$\"3)))G=in?s?(F^o7$$!37,+]ZV>&[%F^o$\"3g.a^mJI))z F^o7$$!3O-++?#=vx$F^o$\"3o,ov4c.t&)F^o7$$!3+,+]Z`3,IF^o$\"37!3-Pn[$*4* F^o7$$!33****\\UwthAF^o$\"3?a)\\OGa%)[*F^o7$$!3y*****\\(eI!\\\"F^o$\"3 1[S*)R))*yx*F^o7$$!3?1++vp!*=yF]q$\"36'H;Ppk)Q**F^o7$$!3]m,++D#px\"Fbr $\"3a*GkaUo*****F^o7$$\"3%\\(***\\K#QfxF]q$\"3WxW$f)>zR**F^o7$$\"3E(** *\\2H'oY\"F^o$\"3ql+1@8$[y*F^o7$$\"3a)****\\[!38AF^o$\"3g<7pwuA5&*F^o7 $$\"3;******fi*R)HF^o$\"3`,1.Kmd4\"*F^o7$$\"3P*****\\%p=QPF^o$\"3&p:Bk $ef-')F^o7$$\"3])***\\7_!zY%F^o$\"3+N:@,By.!)F^o7$$\"3C(****\\$H8y_F^o $\"3\"4$o/s789sF^o7$$\"39'*****z&eh+'F^o$\"3/V#*=6fg#R'F^o7$$\"3=+++]8 [$y'F^o$\"3%y@?u2Q%)R&F^o7$$\"3#z***\\78&y[(F^o$\"3G1#*=s#3KR%F^o7$$\" 3i,++IP\"zD)F^o$\"3!fu(y#3'o!=$F^o7$$\"3+)***\\_C[#)*)F^o$\"3_iH0**3]J >F^o7$$\"3c)***\\(*R@S@>]8&F]q7$$\"3L*\\PMfDC$)*F^o$\"3jR_^\\pSBLF]q7$$\"3U)*\\PWc) \\#**F^o$\"3A&>]*e*fY\\\"F]q7$$\"3SZP%)pcEr**F^o$\"324_&RT4'QdFbr7$$\" 3v\\7`pXv,5F*$\"3mt%Rnde#Q\")e!G\"F]q 7$$\"3x**\\iu0,65F*$\"3T@sN+#QU@#F]q7$$\"3\")*\\7[eA&H5F*$\"33v5!REv;* fF]q7$$\"3i*****\\fM![5F*$\"3R@8oJ7lP)*F]q7$$\"3m**\\78`z'3\"F*$\"3[gY IE0C6=F^o7$$\"3q***\\7.cb7\"F*$\"3q-D$[zj(oEF^o7$$\"3\"******zQ=-?\"F* $\"3(H=L*)yT_S%F^o7$$\"33++]J9dw7F*$\"3I$f>s>YjH'F^o7$$\"3E++DLAH_8F*$ \"3sJF3TG%pG)F^o7$$\"31+]i4?3(Q\"F*$\"3`)G`U,l*R#*F^o7$$\"3')*****fyr= U\"F*$\"3NHG#eP><-\"F*7$$\"3#)***\\dzWBj8F*7$$\"3 y*****ffVHd\"F*$\"3M5u>c::u9F*7$$\"3&***\\i!on4h\"F*$\"3k\"H=!oo@&f\"F *7$$\"39++Dl<**[;F*$\"3[6B'=%Q<>F*7$$\"3!***\\P$y*)3w\"F*$\"3D aW>HGt+@F*7$$\"3/+++++++=F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FfdlFed l-F$6%7S7$$FddlFfdlFedl7$$!3ommm;p0k&*F^o$!3]%\\gZH:)G&)F]q7$$!3wKL$3< XZ=*F^o$!3ix`o9c/k:F^o7$$!3mmmmT%p\"e()F^o$!3D>c5.oWHBF^o7$$!3:mmm\"4m (G$)F^o$!3%pNI!Rb;jIF^o7$$!3\"QLL3i.9!zF^o$!3a)=o?3#ycPF^o7$$!3\"ommT! R=0vF^o$!3/fPYc9AnVF^o7$$!3u****\\P8#\\4(F^o$!3'4so974i'\\F^o7$$!3+nm; /siqmF^o$!3weJIqKF]bF^o7$$!3[++](y$pZiF^o$!3Y!RjPBKm4'F^o7$$!33LLL$yaE \"eF^o$!3y6#zpV/8i'F^o7$$!3hmmm\">s%HaF^o$!3WO=*><$3_qF^o7$$!3Q+++]$*4 )*\\F^o$!3MdHv)G+>](F^o7$$!39+++]_&\\c%F^o$!3%Ru\\lN=h\"zF^o7$$!31+++] 1aZTF^o$!3!yvfcl!zz#)F^o7$$!3umm;/#)[oPF^o$!3')oc]l'\\)z&)F^o7$$!3hLLL $=exJ$F^o$!3!)pCpj![#**))F^o7$$!3*RLLLtIf$HF^o$!3Azo!H2J!Q\"*F^o7$$!3] ++]PYx\"\\#F^o$!3yt6f:f5z$*F^o7$$!3EMLLL7i)4#F^o$!3!yC+>*)y&f&*F^o7$$! 3c****\\P'psm\"F^o$!3ohqe&>@?s*F^o7$$!3')****\\74_c7F^o$!3><]k>b6U)*F^ o7$$!3)3LLL3x%z#)F]q$!3belAf-XJ**F^o7$$!3KMLL3s$QM%F]q$!37*\\2$y58\")* *F^o7$$!3]^omm;zr)*!#@$!2#HHxa-******F*7$$\"3%pJL$ezw5VF]q$!3%pS4'zsT \")**F^o7$$\"3s*)***\\PQ#\\\")F]q$!3'zU2R\"**eL**F^o7$$\"3GKLLe\"*[H7F ^o$!3%*yS&4kN)[)*F^o7$$\"3I*******pvxl\"F^o$!3p]*[G(zx$Rb&F^o7$$\"3wKLL3N1#4(F^o$!3ypmP>NEq\\F^o7$$\"3N mm;HYt7vF^o$!3!*)*> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 68 " changes direction abruptly at the points where the graph meets the " }{TEXT 290 1 "x" }{TEXT -1 12 " axis where " }{XPPEDIT 18 0 "x=``" "6#/%\"xG%!G" }{TEXT 291 1 "+" }{TEXT -1 3 "1. " }}{PARA 0 "" 0 "" {TEXT -1 66 "It is not possible to draw a unique tangent lin e at these points. " }}{PARA 0 "" 0 "" {TEXT -1 70 "Formally, if one a ttempts to determine the value of the derivative of " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=``" "6# /%\"xG%!G" }{TEXT 292 1 "+" }{TEXT -1 57 "1 from the limit definition, it turns out that the limit " }{TEXT 259 14 "does not exist" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The limit for " }{XPPEDIT 18 0 "`f '`(1)" "6#-%$f~'G6#\"\"\"" }{TEXT -1 4 " is " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((f(x)-f(1))/(x-1),x = 1);" "6#- %&LimitG6$*&,&-%\"fG6#%\"xG\"\"\"-F)6#F,!\"\"F,,&F+F,F,F/F//F+F," } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "Considering the limit a s " }{XPPEDIT 18 0 "x->1" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\" \"F*F*F*" }{TEXT -1 1 " " }{TEXT 259 14 "from the right" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "f(x)=abs(x^2-1)" "6#/-%\"fG6#%\"xG-%$absG6#,&*$F' \"\"#\"\"\"F.!\"\"" }{XPPEDIT 18 0 "``=x^2-1" "6#/%!G,&*$%\"xG\"\"#\" \"\"F)!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(1)=0" "6#/-%\"fG6#\"\" \"\"\"!" }{TEXT -1 15 ", and we have: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((x^2-1)/(x-1),x = 1,right) = Limit(``(x+1) ,x = 1,right);" "6#/-%&LimitG6%*&,&*$%\"xG\"\"#\"\"\"F,!\"\"F,,&F*F,F, F-F-/F*F,%&rightG-F%6%-%!G6#,&F*F,F,F,/F*F,F0" }{XPPEDIT 18 0 "``=2" " 6#/%!G\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "This corresponds to the fact that a secant line joining the point" }{XPPEDIT 18 0 "``(1,0)" "6#-%!G6$\"\"\"\"\"!" } {TEXT -1 24 " to a neighbouring point" }{XPPEDIT 18 0 "``(x,f(x))" "6# -%!G6$%\"xG-%\"fG6#F&" }{TEXT -1 34 " to the right approaches the line " }{XPPEDIT 18 0 "y=2*x-2" "6#/%\"yG,&*&\"\"#\"\"\"%\"xGF(F(F'!\"\"" }{TEXT -1 4 " as " }{TEXT 293 1 "x" }{TEXT -1 30 " approaches 1 from t he right. " }}{PARA 0 "" 0 "" {TEXT -1 25 "Considering the limit as " }{XPPEDIT 18 0 "x->1" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"\"F*F *F*" }{TEXT -1 1 " " }{TEXT 259 13 "from the left" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "f(x)=abs(x^2-1)" "6#/-%\"fG6#%\"xG-%$absG6#,&*$F'\"\"# \"\"\"F.!\"\"" }{XPPEDIT 18 0 "`` = 1-x^2;" "6#/%!G,&\"\"\"F&*$%\"xG\" \"#!\"\"" }{TEXT -1 7 ", when " }{TEXT 295 1 "x" }{TEXT -1 45 " is les s than 1 and close to 1, and we have: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((1-x^2)/(x-1),x = 1,left) = Limit(-(x+1),x = 1,left);" "6#/-%&LimitG6%*&,&\"\"\"F)*$%\"xG\"\"#!\"\"F),&F+F)F)F-F -/F+F)%%leftG-F%6%,$,&F+F)F)F)F-/F+F)F0" }{XPPEDIT 18 0 "`` = -2;" "6# /%!G,$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 65 "This corresponds to the fact that a secan t line joining the point" }{XPPEDIT 18 0 "``(1,0)" "6#-%!G6$\"\"\"\"\" !" }{TEXT -1 24 " to a neighbouring point" }{XPPEDIT 18 0 "``(x,f(x)) " "6#-%!G6$%\"xG-%\"fG6#F&" }{TEXT -1 33 " to the left approaches the \+ line " }{XPPEDIT 18 0 "y = 2-2*x;" "6#/%\"yG,&\"\"#\"\"\"*&F&F'%\"xGF' !\"\"" }{TEXT -1 4 " as " }{TEXT 294 1 "x" }{TEXT -1 29 " approaches 1 from the left. " }}{PARA 0 "" 0 "" {TEXT -1 21 "Neither of the lines \+ " }{XPPEDIT 18 0 "y=2*x-2" "6#/%\"yG,&*&\"\"#\"\"\"%\"xGF(F(F'!\"\"" } {TEXT -1 5 " nor " }{XPPEDIT 18 0 "y=2-2*x" "6#/%\"yG,&\"\"#\"\"\"*&F& F'%\"xGF'!\"\"" }{TEXT -1 40 " is a genuine tangent line to the graph \+ " }{XPPEDIT 18 0 "y=abs(x^2-1)" "6#/%\"yG-%$absG6#,&*$%\"xG\"\"#\"\"\" F,!\"\"" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(1,0)" "6#-%!G 6$\"\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "f := x -> abs(x^2-1):\n'f(x )'=f(x);\nLimit(('f'(x)-'f'(1))/(x-1),x=1,right);\n``=value(%);\nLimit (('f'(x)-'f'(1))/(x-1),x=1,left);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$absG6#,&*$)F'\"\"#\"\"\"F/F/!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6%*&,&-%\"fG6#%\"xG\"\"\"-F)6 #F,!\"\"F,,&F+F,F,F/F//F+F,%&rightG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6%*&,&-%\"fG6#% \"xG\"\"\"-F)6#F,!\"\"F,,&F+F,F,F/F//F+F,%%leftG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "plot([abs(x^2-1),2*x-2,2-2*x],x=-1 .8..1.8,y=-1..2,\n color=[red,blue,COLOR(RGB,0,.7,.2)],tickm arks=[3,4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 355 370 370 {PLOTDATA 2 "6( -%'CURVESG6$7[p7$$!3/+++++++=!#<$\"3A++++++SAF*7$$!31++]A^wgF*7$$!3++](y=#R(o\"F*$\"3Wz:q&R #HZ=F*7$$!3;++vITD`;F*$\"3yQ>%3A\\Kt\"F*7$$!36+]7:B'[h\"F*$\"3)\\(3\"o H!y2;F*7$$!31++]*\\qkd\"F*$\"3szPzN#f_[\"F*7$$!3.+++)>Cy`\"F*$\"3CVV&R E.\\O\"F*7$$!3+++]'*y<*\\\"F*$\"3ikTNlV`Z7F*7$$!3-+]7u_rg9F*$\"3zt?17 \"*oL6F*7$$!3/++v^EDA9F*$\"332*4ag-G-\"F*7$$!3-++DxGf'Q\"F*$\"3'fL/C2) RE#*!#=7$$!3+++v-J$4N\"F*$\"3Q?u0\"[--D)F^o7$$!35++vSe3x7F*$\"3qX8WYC[ 4jF^o7$$!3/++v'*Gr+7F*$\"3Dix@/Y6Eh\"y\"F^o7$$!33+++hyFY5F*$\"3!)\\N(=CO (p%*!#>7$$!3.+]PWY.H5F*$\"3ky#)RK*H7*eF]q7$$!3(****\\xU\"z65F*$\"3]O$Q oK*=sBF]q7$$!37]PfB1[25F*$\"33r^o\"p?<]\"F]q7$$!30+vV>)pJ+\"F*$\"3;gjE Imo\\j!#?7$$!3%)*\\7G:!f))**F^o$\"3%odr8fn1G#Fbr7$$!3E++D6@[X**F^o$\"3 AKi![d&Q(3\"F]q7$$!3)***\\7GgEf)*F^o$\"3S!HgmQt[z#F]q7$$!3!3++]%*\\Ix* F^o$\"3-$[NDx%\\([%F]q7$$!33++](Q9[Q*F^o$\"3y*H<7*e_#>\"F^o7$$!3Y+++I) yl**)F^o$\"3*>efb$p:1>F^o7$$!3!4+++X>p@)F^o$\"31o6AvMA[KF^o7$$!3U***** *pJdluF^o$\"3[;wLC<_EWF^o7$$!3e++]n(yKy'F^o$\"3Mn$Qi\"Hr)R&F^o7$$!3;,+ +IZ'>(fF^o$\"3T,OOEPcLkF^o7$$!3s+++?`n%G&F^o$\"3)))G=in?s?(F^o7$$!37,+ ]ZV>&[%F^o$\"3g.a^mJI))zF^o7$$!3O-++?#=vx$F^o$\"3o,ov4c.t&)F^o7$$!3+,+ ]Z`3,IF^o$\"37!3-Pn[$*4*F^o7$$!33****\\UwthAF^o$\"3?a)\\OGa%)[*F^o7$$! 3y*****\\(eI!\\\"F^o$\"31[S*)R))*yx*F^o7$$!3?1++vp!*=yF]q$\"36'H;Ppk)Q **F^o7$$!3]m,++D#px\"Fbr$\"3a*GkaUo*****F^o7$$\"3%\\(***\\K#QfxF]q$\"3 WxW$f)>zR**F^o7$$\"3E(***\\2H'oY\"F^o$\"3ql+1@8$[y*F^o7$$\"3a)****\\[! 38AF^o$\"3g<7pwuA5&*F^o7$$\"3;******fi*R)HF^o$\"3`,1.Kmd4\"*F^o7$$\"3P *****\\%p=QPF^o$\"3&p:Bk$ef-')F^o7$$\"3])***\\7_!zY%F^o$\"3+N:@,By.!)F ^o7$$\"3C(****\\$H8y_F^o$\"3\"4$o/s789sF^o7$$\"39'*****z&eh+'F^o$\"3/V #*=6fg#R'F^o7$$\"3=+++]8[$y'F^o$\"3%y@?u2Q%)R&F^o7$$\"3#z***\\78&y[(F^ o$\"3G1#*=s#3KR%F^o7$$\"3i,++IP\"zD)F^o$\"3!fu(y#3'o!=$F^o7$$\"3+)*** \\_C[#)*)F^o$\"3_iH0**3]J>F^o7$$\"3c)***\\(*R@S@>]8&F]q7$$\"3L*\\PMfDC$)*F^o$\"3jR_ ^\\pSBLF]q7$$\"3U)*\\PWc)\\#**F^o$\"3A&>]*e*fY\\\"F]q7$$\"3SZP%)pcEr** F^o$\"324_&RT4'QdFbr7$$\"3v\\7`pXv,5F*$\"3mt%Rnde#Q\")e!G\"F]q7$$\"3x**\\iu0,65F*$\"3T@sN+#QU@#F]q7$$\"3\") *\\7[eA&H5F*$\"33v5!REv;*fF]q7$$\"3i*****\\fM![5F*$\"3R@8oJ7lP)*F]q7$$ \"3m**\\78`z'3\"F*$\"3[gYIE0C6=F^o7$$\"3q***\\7.cb7\"F*$\"3q-D$[zj(oEF ^o7$$\"3\"******zQ=-?\"F*$\"3(H=L*)yT_S%F^o7$$\"33++]J9dw7F*$\"3I$f>s> YjH'F^o7$$\"3E++DLAH_8F*$\"3sJF3TG%pG)F^o7$$\"31+]i4?3(Q\"F*$\"3`)G`U, l*R#*F^o7$$\"3')*****fyr=U\"F*$\"3NHG#eP><-\"F*7$$\"3#)***\\dzWBj8F*7$$\"3y*****ffVHd\"F*$\"3M5u>c::u9F*7$$\"3&***\\i!o n4h\"F*$\"3k\"H=!oo@&f\"F*7$$\"39++Dl<**[;F*$\"3[6B'=%Q<>F*7$$ \"3!***\\P$y*)3w\"F*$\"3DaW>HGt+@F*7$$\"3/+++++++=F*F+-%'COLOURG6&%$RG BG$\"*++++\"!\")$\"\"!FfdlFedl-F$6$7S7$F($!3k*************f&F*7$F3$!3e +++!\\gIW&F*7$F=$!3K++]h#3lI&F*7$FG$!39+++**4%H:&F*7$FQ$!3)******Hzb$) *\\F*7$Fen$!33++].`]W[F*7$F`o$!3,++]0i'=q%F*7$Feo$!3u****\\\"orTb%F*7$ Fjo$!3j****\\$zD9S%F*7$F_p$!3K++]j(p\"\\UF*7$Fip$!3;+++Adb#4%F*7$Fcs$! 3;+++*)*4Y&RF*7$F]t$!3K+++mdJ*z$F*7$Fbt$!3'********)QQVOF*7$Fgt$!3))** ***Rj9J\\$F*7$F\\u$!37++]`dlcLF*7$Fau$!3A+++YHR%>$F*7$Ffu$!39+++k]$p0$ F*7$F[v$!3A++]p)Qq*GF*7$F`v$!3[+++WO]bFF*7$Fev$!3?++]pq@+EF*7$Fjv$!3#) ****\\GvM_CF*7$F_w$!3'******\\<++]\"*o -_F]q7$Fg^l$\"3%\\#*******=pg*F]q7$Fa_l$\"3+%****\\i?6^#F^o7$Ff_l$\"3G )*****fxO/SF^o7$F[`l$\"3^,++I'G9`&F^o7$F``l$\"310++lY%e/(F^o7$Fj`l$\"3 ?(*****>dVP%)F^o7$Fdal$\"3e*****4hNK+\"F*7$F^bl$\"3c*****>>()e9\"F*7$F hbl$\"3E++]IN)zH\"F*7$Fbcl$\"3`****\\L\"fNW\"F*7$F\\dl$\"33+++++++;F*- F_dl6&FadlFedlFedlFbdl-F$6$7S7$F($\"3k*************f&F*7$F3$\"3e+++!\\ gIW&F*7$F=$\"3K++]h#3lI&F*7$FG$\"39+++**4%H:&F*7$FQ$\"3)******Hzb$)*\\ F*7$Fen$\"33++].`]W[F*7$F`o$\"3,++]0i'=q%F*7$Feo$\"3u****\\\"orTb%F*7$ Fjo$\"3j****\\$zD9S%F*7$F_p$\"3K++]j(p\"\\UF*7$Fip$\"3;+++Adb#4%F*7$Fc s$\"3;+++*)*4Y&RF*7$F]t$\"3K+++mdJ*z$F*7$Fbt$\"3'********)QQVOF*7$Fgt$ \"3))*****Rj9J\\$F*7$F\\u$\"37++]`dlcLF*7$Fau$\"3A+++YHR%>$F*7$Ffu$\"3 9+++k]$p0$F*7$F[v$\"3A++]p)Qq*GF*7$F`v$\"3[+++WO]bFF*7$Fev$\"3?++]pq@+ EF*7$Fjv$\"3#)****\\GvM_CF*7$F_w$\"3'******\\<++]\"*o-_F]q7$Fg^l$!3%\\#*******=pg*F]q7$Fa_l$! 3+%****\\i?6^#F^o7$Ff_l$!3G)*****fxO/SF^o7$F[`l$!3^,++I'G9`&F^o7$F``l$ !310++lY%e/(F^o7$Fj`l$!3?(*****>dVP%)F^o7$Fdal$!3e*****4hNK+\"F*7$F^bl $!3c*****>>()e9\"F*7$Fhbl$!3E++]IN)zH\"F*7$Fbcl$!3`****\\L\"fNW\"F*7$F \\dl$!33+++++++;F*-%&COLORG6&FadlFfdl$\"\"(!\"\"$\"\"#Fjgm-%+AXESLABEL SG6$Q\"x6\"Q\"yFahm-%*AXESTICKSG6$\"\"$\"\"%-%%VIEWG6$;$F^oFjgm$\"#=Fj gm;$FjgmFfdl$F\\hmFfdl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We say that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 259 18 "not differentiable" }{TEXT -1 7 " wh ere " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 42 ", that is, t he derivative is not defined. " }}{PARA 0 "" 0 "" {TEXT -1 34 "A simil ar conclusion applies when " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"! \"\"" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 29 " is not differentiable where " }{XPPEDIT 18 0 "x=-1" "6#/ %\"xG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "At all other values of " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 16 " the derivative " }{XPPEDIT 18 0 "`f '`(a) = Limit((f(x)-f(a))/(x- a),x = a);" "6#/-%$f~'G6#%\"aG-%&LimitG6$*&,&-%\"fG6#%\"xG\"\"\"-F.6#F '!\"\"F1,&F0F1F'F4F4/F0F'" }{TEXT -1 16 " exists, and so " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 " " }{TEXT 259 17 "is differ entiable" }{TEXT -1 25 " for all other values of " }{TEXT 296 1 "x" } {TEXT -1 12 " apart from " }{XPPEDIT 18 0 "x=``" "6#/%\"xG%!G" }{TEXT 297 1 "+" }{TEXT -1 3 "1. " }}{PARA 0 "" 0 "" {TEXT -1 29 "Using the p iecewise formula: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=PIECEWISE([x^2-1,x<=-1],[1-x^2,-1=1])" "6#/ -%\"fG6#%\"xG-%*PIECEWISEG6%7$,&*$F'\"\"#\"\"\"F/!\"\"1F',$F/F07$,&F/F /*$F'F.F032,$F/F0F'2F'F/7$,&*$F'F.F/F/F01F/F'" }{TEXT -1 4 " , " }} {PARA 0 "" 0 "" {TEXT -1 10 "we obtain " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = PIECEWISE([2*x, x < 1],[undefined, x \+ = -1],[-2*x, -1 < x and x < 1],[undefined, x = 1],[2*x, 1 < x]);" "6#/ -%$f~'G6#%\"xG-%*PIECEWISEG6'7$*&\"\"#\"\"\"F'F.2F'F.7$%*undefinedG/F' ,$F.!\"\"7$,$*&F-F.F'F.F432,$F.F4F'2F'F.7$F1/F'F.7$*&F-F.F'F.2F.F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 348 "f := x -> abs(x^2-1):\n'f(x)'=f(x);\n'f(x)'=con vert(f(x),piecewise);\nDiff('f(x)',x)=convert(diff(f(x),x),piecewise); \ng := unapply(rhs(%),x):\np1 := plot([f(x),g(x)],x=-1.8..1.8,color=[r ed,blue],discont=true):\np2 := plot([[-1,-2],[-1,2],[1,-2],[1,2]],styl e=point,\n color=black,symbol=circle,symbolsize=15):\nplots[displ ay]([p1,p2],tickmarks=[3,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" fG6#%\"xG-%$absG6#,&*$)F'\"\"#\"\"\"F/F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$,&*$)F'\"\"#\"\"\"F0F0! \"\"1F'F17$,&F0F0F-F12F'F07$F,1F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%%DiffG6$-%\"fG6#%\"xGF*-%*PIECEWISEG6'7$,$*&\"\"#\"\"\"F*F2F22F*! \"\"7$%*undefinedG/F*F47$,$*&F1F2F*F2F42F*F27$F6/F*F27$F/2F2F*" }} {PARA 13 "" 1 "" {GLPLOT2D 343 453 453 {PLOTDATA 2 "6(-%'CURVESG6$7[p7 $$!3/+++++++=!#<$\"3A++++++SAF*7$$!31++]A^wgF*7$$!3++](y=#R(o\"F*$\"3Wz:q&R#HZ=F*7$$!3;++vI TD`;F*$\"3yQ>%3A\\Kt\"F*7$$!36+]7:B'[h\"F*$\"3)\\(3\"oH!y2;F*7$$!31++] *\\qkd\"F*$\"3szPzN#f_[\"F*7$$!3.+++)>Cy`\"F*$\"3CVV&RE.\\O\"F*7$$!3++ +]'*y<*\\\"F*$\"3ikTNlV`Z7F*7$$!3-+]7u_rg9F*$\"3zt?17\"*oL6F*7$$!3/++v ^EDA9F*$\"332*4ag-G-\"F*7$$!3-++DxGf'Q\"F*$\"3'fL/C2)RE#*!#=7$$!3+++v- J$4N\"F*$\"3Q?u0\"[--D)F^o7$$!35++vSe3x7F*$\"3qX8WYC[4jF^o7$$!3/++v'*G r+7F*$\"3Dix@/Y6Eh\"y\"F^o7$$!33+++hyFY5F*$\"3!)\\N(=CO(p%*!#>7$$!3.+]PW Y.H5F*$\"3ky#)RK*H7*eF]q7$$!3(****\\xU\"z65F*$\"3]O$QoK*=sBF]q7$$!37]P fB1[25F*$\"33r^o\"p?<]\"F]q7$$!30+vV>)pJ+\"F*$\"3;gjEImo\\j!#?7$$!3%)* \\7G:!f))**F^o$\"3%odr8fn1G#Fbr7$$!3E++D6@[X**F^o$\"3AKi![d&Q(3\"F]q7$ $!3)***\\7GgEf)*F^o$\"3S!HgmQt[z#F]q7$$!3!3++]%*\\Ix*F^o$\"3-$[NDx%\\( [%F]q7$$!33++](Q9[Q*F^o$\"3y*H<7*e_#>\"F^o7$$!3Y+++I)yl**)F^o$\"3*>efb $p:1>F^o7$$!3!4+++X>p@)F^o$\"31o6AvMA[KF^o7$$!3U******pJdluF^o$\"3[;wL C<_EWF^o7$$!3e++]n(yKy'F^o$\"3Mn$Qi\"Hr)R&F^o7$$!3;,++IZ'>(fF^o$\"3T,O OEPcLkF^o7$$!3s+++?`n%G&F^o$\"3)))G=in?s?(F^o7$$!37,+]ZV>&[%F^o$\"3g.a ^mJI))zF^o7$$!3O-++?#=vx$F^o$\"3o,ov4c.t&)F^o7$$!3+,+]Z`3,IF^o$\"37!3- Pn[$*4*F^o7$$!33****\\UwthAF^o$\"3?a)\\OGa%)[*F^o7$$!3y*****\\(eI!\\\" F^o$\"31[S*)R))*yx*F^o7$$!3?1++vp!*=yF]q$\"36'H;Ppk)Q**F^o7$$!3]m,++D# px\"Fbr$\"3a*GkaUo*****F^o7$$\"3%\\(***\\K#QfxF]q$\"3WxW$f)>zR**F^o7$$ \"3E(***\\2H'oY\"F^o$\"3ql+1@8$[y*F^o7$$\"3a)****\\[!38AF^o$\"3g<7pwuA 5&*F^o7$$\"3;******fi*R)HF^o$\"3`,1.Kmd4\"*F^o7$$\"3P*****\\%p=QPF^o$ \"3&p:Bk$ef-')F^o7$$\"3])***\\7_!zY%F^o$\"3+N:@,By.!)F^o7$$\"3C(****\\ $H8y_F^o$\"3\"4$o/s789sF^o7$$\"39'*****z&eh+'F^o$\"3/V#*=6fg#R'F^o7$$ \"3=+++]8[$y'F^o$\"3%y@?u2Q%)R&F^o7$$\"3#z***\\78&y[(F^o$\"3G1#*=s#3KR %F^o7$$\"3i,++IP\"zD)F^o$\"3!fu(y#3'o!=$F^o7$$\"3+)***\\_C[#)*)F^o$\"3 _iH0**3]J>F^o7$$\"3c)***\\(*R@S@>]8&F]q7$$\"3L*\\PMfDC$)*F^o$\"3jR_^\\pSBLF]q7$$\"3 U)*\\PWc)\\#**F^o$\"3A&>]*e*fY\\\"F]q7$$\"3SZP%)pcEr**F^o$\"324_&RT4'Q dFbr7$$\"3v\\7`pXv,5F*$\"3mt%Rnde#Q\" )e!G\"F]q7$$\"3x**\\iu0,65F*$\"3T@sN+#QU@#F]q7$$\"3\")*\\7[eA&H5F*$\"3 3v5!REv;*fF]q7$$\"3i*****\\fM![5F*$\"3R@8oJ7lP)*F]q7$$\"3m**\\78`z'3\" F*$\"3[gYIE0C6=F^o7$$\"3q***\\7.cb7\"F*$\"3q-D$[zj(oEF^o7$$\"3\"****** zQ=-?\"F*$\"3(H=L*)yT_S%F^o7$$\"33++]J9dw7F*$\"3I$f>s>YjH'F^o7$$\"3E++ DLAH_8F*$\"3sJF3TG%pG)F^o7$$\"31+]i4?3(Q\"F*$\"3`)G`U,l*R#*F^o7$$\"3') *****fyr=U\"F*$\"3NHG#eP><-\"F*7$$\"3#)***\\dzWBj 8F*7$$\"3y*****ffVHd\"F*$\"3M5u>c::u9F*7$$\"3&***\\i!on4h\"F*$\"3k\"H= !oo@&f\"F*7$$\"39++Dl<**[;F*$\"3[6B'=%Q<>F*7$$\"3!***\\P$y*)3w \"F*$\"3DaW>HGt+@F*7$$\"3/+++++++=F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\") $\"\"!FfdlFedl-F$6&7S7$F($!33+++++++OF*7$$!3ObQvwAc#y\"F*$!3u5x]`X7lNF *7$$!3E%Q'*p!)*Qn1Dj\" F*$!3![gTVQ7]E$F*7$$!3Is2e)))yrh\"F*$!3gW:;xxNMKF*7$$!3=!Q+](R#**f\"F* $!3Qg2+]z%)*>$F*7$$!3%*3q36#)f#e\"F*$!3'y,u@U'>lJF*7$$!3)>\\qri,fc\"F* $!3&R)4MaK!=8$F*7$$!37pHTH&R2b\"F*$!3AQf#)e!z95$F*7$$!3&=yp'G.rK:F*$!3 qj&Rtl?a1$F*7$$!3BZhuIsV<:F*$!3X%H#\\hW([.$F*7$$!39X;+()4n*\\\"F*$!3G! H.S(>M**HF*7$$!314O\"4&[%R[\"F*$!37=s#=q*)y'HF*7$$!3+Yl;(y!pm9F*$!3*>4 LVd\"QLHF*7$$!3!fp[#Q3E]9F*$!3z\"R(\\w;_+HF*7$$!3\"Qun,\"z6L9F*$!3i([N .#eBmGF*7$$!3gcku]`P<9F*$!3>8H\\,2vMGF*7$$!3H#pkO([R+9F*$!3d%QHtu*y+GF *7$$!3;#)GDIpv#Q\"F*$!3Kkd]gQ^lFF*7$$!3!\\)Hm[ISn8F*$!3!)pfK(41[t#F*7$ $!3SkD\"fV?3N\"F*$!3!)G^#=(3k,FF*7$$!3?b:Lu*)oL8F*$!3S5Jm[zPnEF*7$$!3J a`T\"=HpJ\"F*$!3i32$GOeQj#F*7$$!3(RV'**>Kr+8F*$!3$z'G**RkU,EF*7$$!37fk ef\"3FG\"F*$!3C=H<>jTlDF*7$$!3E%o+?\")HlE\"F*$!3`o8+C'fI`#F*7$$!3-?Pvs fD\\7F*$!3/Su]X>^)\\#F*7$$!3H]')*pI.OB\"F*$!3e+t*Rh1sY#F*7$$!38Yv\"*33 \\;7F*$!3E#4Nyh\")HV#F*7$$!3/(QJ=G*Q+7F*$!34uFmj&y2S#F*7$$!3;2AeY&eN= \"F*$!3K9W;$4#F*7$$!3ed)zb\\GS3\"F*$!3;:(f 6*p0o@F*7$$!3a(z,!erIm5F*$!33&f.gJ9E8#F*7$$!3!>Q9C)pX]5F*$!3!QwG['R\"4 5#F*7$$!3>8AL*QdN.\"F*$!3QEWmyZ6n?F*7$$!3Y)38CK#Q<5F*$!3#pF*7$$!3/_&Qv*o0k&*F^o$\"3T5x]z8\"G\">F*7$$!3QUQY_^u%=*F^o$\"3ZoF\\I !\\p$=F*7$$!3=F.:C%p\"e()F^o$\"3Wl+$[)Qj^F1n'F^o$\"3A #3l\"Qa7M8F*7$$!3U7Y+vPpZiF^o$\"3[A4+b(Q&\\7F*7$$!3ZB!3g2+o)>'***F^o7$$!3'))3q3Cb\\c%F^o$\"3sx,u\"[5*H\"*F^o7$$!3m<\\qT1aZT F^o$\"3IN)4MG\"3&H)F^o7$$!39!pHm>)[oPF^o$\"3H!QfKRwp`(F^o7$$!3.%F^o7$$! 3Ega;M'psm\"F^o$\"3^?4Lo#RXL$F^o7$$!3'y&p)*44_c7F^o$\"3u:R(*>=/8DF^o7$ $!3owVxmqZz#)F]q$\"3Mv[N8a*el\"F^o7$$!3T]ck*>PQM%F]q$\"3#3I\"H*Ruwo)F] q7$$!3YbP#pk\"zr)*!#@$\"35^ZQH$eV(>Fbr7$$\"3L-=r\\zw5VF]q$!3m/OU**e`@' )F]q7$$\"39F:qe$Q#\\\")F]q$!3V0.urw%)H;F^o7$$\"3#fNue:*[H7F^o$!3%=r[&4Nn'F^o7$$\"3u)ziCu+'oPF^o$!3\\(f D\\[,s`(F^o7$$\"3g(\\8vQ<*fTF^o$!3?&*p-vZ$)>$)F^o7$$\"3QSX#3%)Hxe%F^o$ !3w!3\\;ofa<*F^o7$$\"3GHh=>!o-*\\F^o$!3ceAPQg`!)**F^o7$$\"3xFzn,k.6aF^ o$!3c&eN.G2A3\"F*7$$\"3QP=-0WTAeF^o$!3[nV+\")G[k6F*7$$\"3m>Q**\\!*3`iF ^o$!3%Rw)**4yh]7F*7$$\"3ctv**>*zym'F^o$!3r9&**R)fdL8F*7$$\"3m1#\\T\\j? 4(F^o$!3LT)H))p7%=9F*7$$\"3q(>TThMF^(F^o$!3aR#GG#pa-:F*7$$\"3yC9?%o(G* *yF^o$!3'\\GSo`d)z:F*7$$\"30D?)\\7@BM)F^o$!3+0k*\\Ak%o;F*7$$\"3'G=ce^v &Q()F^o$!3eO7<.^rZF*7$$\"2%******z********F*$!3))*****f*******>F*7S 7$$\"36+++/+++5F*$\"3A+++3+++?F*7$$\"3zWhCFxV<5F*$\"3d*G#\\aa([.#F*7$$ \"3*eh.q>5E.\"F*$\"3yJs+%R?_1#F*7$$\"3Gn\\3EKn\\5F*$\"3aM*p@XY$*4#F*7$ $\"3q'3***f$\\o1\"F*$\"3St\")**>()pL@F*7$$\"3UZpueQ%R3\"F*$\"3'[*Q\\#F*7$$\"3S%)*=*\\J?;6F*$\"3zoz$ )*H1CB#F*7$$\"3$)eu;:\\H=\"Q\\n6F*$\"3^&ReOi()\\L#F*7$$\"3%yA>a6@G=\"F* $\"3qb%Q3BUcO#F*7$$\"3'*>'***Gg2+7F*$\"3#*R#**z0_,S#F*7$$\"3A\"*H\"Hz, u@\"F*$\"3W#)f#ee.[V#F*7$$\"3=3&HoP)4M7F*$\"3O;!fOv'>oCF*7$$\"3/Jqeu/E \\7F*$\"33iS<\\4_)\\#F*7$$\"3I=-Lv'*Gn7F*$\"3gO/m]$zX`#F*7$$\"3#H&QDtF c#G\"F*$\"3'eq2laD^c#F*7$$\"3,b$)*p,H.I\"F*$\"3.5n*R.e1g#F*7$$\"34\"R' 3`^0;8F*$\"3>#yshI5@j#F*7$$\"3;aM$o@4LL\"F*$\"3K3pmL%=mm#F*7$$\"3E/8vl \"R(\\8F*$\"3]3E]J$y%*p#F*7$$\"3McA$Q4#)oO\"F*$\"3o7Xm(=kPt#F*7$$\"3cV ND`Yi#Q\"F*$\"37(32lI\\_w#F*7$$\"3'yIN.80'*R\"F*$\"3u:1ng-@*z#F*7$$\"3 +=rutIC<9F*$\"3*fB%\\Zh[MGF*7$$\"3D:qLbpfK9F*$\"3]ISn5R>lGF*7$$\"3wNu3 o&z\"\\9F*$\"3]r[>qM`\"F*$\"3yJ')*RQSp1$F* 7$$\"39!GY7.W2b\"F*$\"3EgD\\i!)[,JF*7$$\"3')\\8+(p'Rm:F*$\"3u*p-SR$zKJ F*7$$\"3-aC3&>4Ne\"F*$\"3/3\\;!R=q;$F*7$$\"358'o@s5'*f\"F*$\"3?EsLW9A* >$F*7$$\"3*Hz:nm\"F*$\"3&[ ^**fRIML$F*7$$\"3u?\\\"4a#o$o\"F*$\"3]T)H=3ltO$F*7$$\"3-?Tm&Q40q\"F*$ \"3/S#G8x=5S$F*7$$\"3dU,U3:(fr\"F*$\"38&GSo,V>V$F*7$$\"3i-#)*f%GpLp&QnMF*7$$\"3E=ce@Ia\\ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 30 "Find an approximate value for " }{XPPEDIT 18 0 "sqrt(63) " "6#-%%sqrtG6#\"#j" }{TEXT -1 71 " by using an appropriate derivative to estimate the difference between " }{XPPEDIT 18 0 "sqrt(63)" "6#-%% sqrtG6#\"#j" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sqrt(64)" "6#-%%sqrtG 6#\"#k" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "8-1/16=7" "6#/,&\"\")\"\"\"*&F&F&\"#;!\"\"F)\"\"(" } {XPPEDIT 18 0 "15/16" "6#*&\"#:\"\"\"\"#;!\"\"" }{TEXT -1 1 " " } {TEXT 343 1 "~" }{TEXT -1 8 " 7.9375 " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_________________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 84 "A stone is thrown vertically downwards from the top of a cliff so that its dista nce " }{TEXT 344 1 "s" }{TEXT -1 41 " metres below the top of the clif f after " }{TEXT 345 1 "t" }{TEXT -1 19 " secs. is given by " } {XPPEDIT 18 0 "s = 2*t+4.9*t^2;" "6#/%\"sG,&*&\"\"#\"\"\"%\"tGF(F(*&-% &FloatG6$\"#\\!\"\"F(*$F)F'F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 66 "(a) Find the average velocity of the stone over the inter val from " }{XPPEDIT 18 0 "t=2" "6#/%\"tG\"\"#" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "t=6" "6#/%\"tG\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 80 "(b) Find the instantaneous velocity of the stone 4 second s after it is dropped. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 51 " (a) \+ 41.2 metres per sec. (b) 41.2 metres per sec. " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Q3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 107 "If a metal rod is \+ homogeneous, then its (uniform) linear density is defined to be the ma ss per unit length." }}{PARA 0 "" 0 "" {TEXT -1 83 "Suppose, however, \+ that a length of rod is not homogeneous, but that its total mass " } {TEXT 346 1 "m" }{TEXT -1 45 ", measured from the left-hand end to a p oint " }{TEXT 350 1 "x" }{TEXT -1 34 " units from the end, is given by : " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m = f(x);" "6#/ %\"mG-%\"fG6#%\"xG" }{XPPEDIT 18 0 "``=(x+sqrt(x))/3" "6#/%!G*&,&%\"xG \"\"\"-%%sqrtG6#F'F(F(\"\"$!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 347 1 "x" }{TEXT -1 27 " is measured in met res and " }{TEXT 349 1 "m" }{TEXT -1 18 " is in kilograms. " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 708 101 101 {PLOTDATA 2 "64-%'CU RVESG6$7'7$$\"\"!F)F(7$F($\"35+++++++?!#=7$$\"\"&F)F+7$F/F(F'-%'COLOUR G6&%$RGBGF)F)F)-F$6$7$7$$\"\"$F)F(7$F:F+F2-F$6$7$7$$\"\"%F)F(7$FAF+F2- %)POLYGONSG6%7&F'7$F($\"\"#!\"\"7$F/FIF1-%&COLORG6&F5$\"#&)!\"#FP$\"#( )FR-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$F*7$F($\"3U+++++++!)F-F2-%*LINESTYL EG6#F;-F$6%7$7$$FJF)F+7$F`oFgnF2Fin-F$6%7$7$FP$F0FK7$F(Ffo7%7$$\"*+++] )!#5$\"+++++XF\\pFgo7$Fjo$\"+++++bF\\pFU-F$6%7$7$$\"$:\"FRFfo7$$\"$+#F RFfo7%7$$\"++++:>!\"*F`pFhp7$F]qF]pFU-%%TEXTG6&7$$\"\"\"F)$\"#bFRQ\"x6 \"F2-%%FONTG6$%*HELVETICAG\"#5-Fbq6&7$F:$!#:FRFiqF2F[r-Fbq6&7$FAFcrFiq F2F[r-Fbq6&7$FeqFeqQ1This~part~of~rodFjqF2-F\\r6$F^r\"\")-Fbq6&7$Feq$F ^sFKQ.has~mass~f(x)FjqF2F\\s-Fbq6&7$$\"$0$FR$!#BFRQ\"lFjqF2F\\s-Fbq6&7 $$\"$0%FRFisQ\"2FjqF2F\\s-%+AXESLABELSG6%Q!FjqFet-F\\r6#%(DEFAULTG-%*A XESSTYLEG6#%%NONEG-%%VIEWG6$FhtFht" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 59 "The average linear density for the section of wire b etween " }{XPPEDIT 18 0 "x=x[1]" "6#/%\"xG&F$6#\"\"\"" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "x=x[2]" "6#/%\"xG&F$6#\"\"#" }{TEXT -1 5 " is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(f(x[2])-f(x[1]))/( x[2]-x[1])" "6#*&,&-%\"fG6#&%\"xG6#\"\"#\"\"\"-F&6#&F)6#F,!\"\"F,,&&F) 6#F+F,&F)6#F,F1F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 66 "The instantaneous rate of change of the mass per unit length, or " } {TEXT 259 20 "local linear density" }{TEXT -1 28 " is given by the der ivative " }{XPPEDIT 18 0 "dm/dx = `f '`(x);" "6#/*&%#dmG\"\"\"%#dxG!\" \"-%$f~'G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 "(a) Find the average linear density for \+ " }{TEXT 348 1 "x" }{TEXT -1 43 " between 1 and 1.2 in kilograms per m etre. " }}{PARA 0 "" 0 "" {TEXT -1 101 "(b) Find the instantaneous rat e of change of the mass per unit length, or local linear density, when " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 61 " (a) 0.49241 kilograms per metre (b) 0.5 \+ kilograms per metre " }}}{PARA 0 "" 0 "" {TEXT -1 37 "________________ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 5 "Q4 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 52 " be the function defined in a piecewise fashion by: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=PIECEWISE([- x,x<0],[x^2,0<=x and x<1],[1/x,x>=1])" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6 %7$,$F'!\"\"2F'\"\"!7$*$F'\"\"#31F/F'2F'\"\"\"7$*&F6F6F'F-1F6F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "The graph " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " is as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "f := x -> piecewise(x<0,-x,x<1,x^2,1/x):\n'f(x)'=f(x);\nplot(f(x),x=-1. 4..3,color=red,tickmarks=[4,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"fG6#%\"xG-%*PIECEWISEG6%7$,$F'!\"\"2F'\"\"!7$*$)F'\"\"#\"\"\"2F'F47 $*&F4F4F'F-%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 470 181 181 {PLOTDATA 2 "6'-%'CURVESG6#7^o7$$!3!**************R\"!#<$\"3!********* *****R\"F*7$$!3cmmm@D4/8F*$\"3cmmm@D4/8F*7$$!37LLePRk?7F*$\"37LLePRk?7 F*7$$!3cmm;xszE6F*$\"3cmm;xszE6F*7$$!3kmm;S&GB.\"F*$\"3kmm;S&GB.\"F*7$ $!3;KL$e'z3$Q*!#=$\"3;KL$e'z3$Q*FD7$$!3llm;*e/9^)FD$\"3llm;*e/9^)FD7$$ !3!))***\\Up#)3wFD$\"3!))***\\Up#)3wFD7$$!3%\\mm\"\\)z`n'FD$\"3%\\mm\" \\)z`n'FD7$$!3&))***\\Kj#\\u&FD$\"3&))***\\Kj#\\u&FD7$$!3+JLLB0%yy%FD$ \"3+JLLB0%yy%FD7$$!3@lmm@)Q[%RFD$\"3@lmm@)Q[%RFD7$$!3G(*****p&=e*HFD$ \"3G(*****p&=e*HFD7$$!32)*****\\:!H/#FD$\"32)*****\\:!H/#FD7$$!36)**** *H%*eC6FD$\"36)*****H%*eC6FD7$$!3K?L$eR$\"3K?L$eR$4qFap$\"34Q7^Rm08\\Faq7$$\"3`pmm\">U47\"FD$\"3Y w%eqR6lD\"Fap7$$\"3npmm'Q_4a\"FD$\"3E0P(zDMXP#Fap7$$\"3z,+](z&4=DFD$\" 3UqrQXk!3M'Fap7$$\"3#zmmmGLIQ$FD$\"3on%p=U\"\\W6FD7$$\"3<.+](z1?L%FD$ \"3OB'e$*GGm(=FD7$$\"3_/+]#*RlN_FD$\"3gB\"=HF27u#FD7$$\"33qmmT]^yhFD$ \"3\\H,,7[Sk&FD7$$\"3qMLLe?GyzFD$ \"3CfBBg%)HljFD7$$\"3WMLe/bKj%)FD$\"3uuIlfyyirFD7$$\"3;ML$3&*o$[*)FD$ \"352Q-)oIt+)FD7$$\"3%)omm'p+1P*FD$\"3'e)pjTd\"3y)FD7$$\"3].+]UC$Gz*FD $\"3#z0)oCn&**e*FD7$$\"3+q\"Hd!z$o!**FD$\"3LK#R!HPa9)*FD7$$\"3lLe*oL%3 -5F*$\"3ey(z_)**>z**FD7$$\"3>](=K)[[85F*$\"3!o'4k$RXp')*FD7$$\"3&pmT&H a)[-\"F*$\"3!H,(*4=)=d(*FD7$$\"3C+v=AloZ5F*$\"3#**HzdyR[a*FD7$$\"3aLL$ [h([q5F*$\"3a4rt:t`T$*FD7$$\"3#pm;W8*f<6F*$\"3c(oo#)4_x%*)FD7$$\"3G+++ a1rk6F*$\"3A+v'3TBee)FD7$$\"3;++]:&*)oD\"F*$\"3)3ij;z[h&zFD7$$\"3Y++v. t2Y8F*$\"3e1>pTQ**GuFD7$$\"31++]O^5X9F*$\"3PWjFS<\"*>pFD7$$\"3ammm[g3M :F*$\"3IqK0r\"R&=lFD7$$\"3K+++l@4H;F*$\"33qx*[7)QQhFD7$$\"3mLL3F==:F*$\"3%Re\")=KoB%FD7$$\"3AnmT=;!GX#F*$\"3U A=(\\cqp2%FD7$$\"3U+++%HVy`#F*$\"3)=Tgeh`.%RFD7$$\"3Inm;^1JNEF*$\"3W* \\d,0>Yz$FD7$$\"3>MLL " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "(a) Describe the derivative " } {XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 67 " in a piecew ise fashion with particular attention to any values of " }{TEXT 351 1 "x" }{TEXT -1 38 " where the derivative is not defined. " }}{PARA 0 " " 0 "" {TEXT -1 40 "(b) Sketch the graph of the derivative. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 5 " (a) " } }{PARA 257 "" 0 "" {XPPEDIT 18 0 "`f '`(x) = PIECEWISE([-1, x < 0],[un defined, x = 0],[2*x, 0 < x and x < 1],[undefined, x = 1],[-1/(x^2), 1 < x]);" "6#/-%$f~'G6#%\"xG-%*PIECEWISEG6'7$,$\"\"\"!\"\"2F'\"\"!7$%*u ndefinedG/F'F07$*&\"\"#F-F'F-32F0F'2F'F-7$F2/F'F-7$,$*&F-F-*$F'F6F.F.2 F-F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "f := x -> piecewise(x<0,-x,x<1,x^2,1/x):\ng := D(f):\nDiff('f(x)',x)=conve rt(D(f)(x),piecewise);\np1 := plot(g(x),x=-1.5..3,color=blue,discont=t rue):\np2 := plot([[0,-1],[0,0],[1,2],[1,-1]],style=point,\n colo r=black,symbol=circle,symbolsize=15):\nplots[display]([p1,p2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*-%*PIECEWISE G6'7$!\"\"2F*\"\"!7$%*undefinedG/F*F17$,$*&\"\"#\"\"\"F*F9F92F*F97$F3/ F*F97$,$*&F9F9*$)F*F8F9F/F/2F9F*" }}{PARA 13 "" 1 "" {GLPLOT2D 491 256 256 {PLOTDATA 2 "6&-%'CURVESG6&7S7$$!3++++++++:!#<$!\"\"\"\"!7$$!3 W\"R:)oUIn9F*F+7$$!38)yMz)e&)Q9F*F+7$$!3eu7J3F'oS\"F*F+7$$!35&oDrXdYP \"F*F+7$$!3R*Gx=F0EM\"F*F+7$$!3OAn=$z))GJ\"F*F+7$$!3uh#[25>@G\"F*F+7$$ !3'e!>\"3/(H]7F*F+7$$!3%fMDY.x&=7F*F+7$$!3#=5y$4\"\\f=\"F*F+7$$!3Kz01: /@d6F*F+7$$!3?&G+?Xd[7\"F*F+7$$!3qc_cW;P#4\"F*F+7$$!3#*oyi\\b1h5F*F+7$ $!3pEsC;mjK5F*F+7$$!3=jL_Z'=$))**!#=F+7$$!3$R5'fg![>q*FhnF+7$$!3%*QtQ* )4$)o$*FhnF+7$$!3mo?&o$f'R2*FhnF+7$$!3)e4C1C_/v)FhnF+7$$!3`=-\\(p!RU%) FhnF+7$$!3my!3]zg47)FhnF+7$$!3>CM(\\!yyDyFhnF+7$$!3B$>N(eQS2vFhnF+7$$! 3C:;x=CpwrFhnF+7$$!3'eQ(4Qr!)))oFhnF+7$$!3[LU4[J)yd'FhnF+7$$!3'[m'[U#o mD'FhnF+7$$!3\\c^OC@UUfFhnF+7$$!3Cb#[o%GPQcFhnF+7$$!3HX%[J'zx+`FhnF+7$ $!3;M^+XRV(*\\FhnF+7$$!3M,H:e%\\Nn%FhnF+7$$!3?x[Ou>1!Q%FhnF+7$$!3Y$f\" QME?fSFhnF+7$$!3Y./h+!*HdPFhnF+7$$!39b:u8FsTMFhnF+7$$!3PAOB6#*=LJFhnF+ 7$$!3cMY]FK=5GFhnF+7$$!3o>=+v+4*\\#FhnF+7$$!3'f4)Q%R_4=#FhnF+7$$!3e,TR a!\\a'=FhnF+7$$!3EJ*[=DMbd\"FhnF+7$$!3=#[j7F+7$$!3Mwfmo+,#H'F`tF+7$$!3qN;B7]=fKF`tF+7$$!3)***************H!#E F+7S7$$\"37+++++++?Fjt$\"3C+++++++SFjt7$$\"3s/y%zg:(z@F`t$\"3W4c*e@J%f VF`t7$$\"3_O#G&HVFwSF`t$\"3/tk0f'[D:)F`t7$$\"3Lb+$o'H:4iF`t$\"326gO$fI =C\"Fhn7$$\"33))>C3(phN)F`t$\"3i(R[;%RBr;Fhn7$$\"3]ShQ0#)H\\5Fhn$\"3)4 Gs2T'f)4#Fhn7$$\"3EMq#H13uC\"Fhn$\"3_oS&e7;[\\#Fhn7$$\"3[U)RaMRDX\"Fhn $\"3'\\oz3py]!HFhn7$$\"3+@zD6kok;Fhn$\"3*>%e^AGPHLFhn7$$\"3a(QX(=J:w=F hn$\"33v2\\PiI_PFhn7$$\"3yU'e*>En$4#Fhn$\"3e&G<*R_M(=%Fhn7$$\"3+6c-:RE &G#Fhn$\"3)>A^+$y_qXFhn7$$\"3o)>'*\\L]4]#Fhn$\"3N(R#**p1!>+&Fhn7$$\"35 5*HTQAvr#Fhn$\"3>?)f#oZ/NaFhn7$$\"3E\"3&H$oHi#HFhn$\"3_i,fm$fC&eFhn7$$ \"3nUOX0fv:JFhn$\"3M&G24\"=^JiFhn7$$\"3y\\)o\\\"47TLFhn$\"3c*pP*H=C#o' Fhn7$$\"3*[>0#RY.KNFhn$\"3y*Q5%y#pS1(Fhn7$$\"3))[NB'o7Tv$Fhn$\"3w(4nCP D#3vFhn7$$\"3Uv0`(Q*o]RFhn$\"3$3:h]xy8!zFhn7$$\"3_RXe%=lj;%Fhn$\"31z!p \"p.tK$)Fhn7$$\"31UIEY&RGP3#G,***Fhn7$$\"3G8X0(RQb@&Fhn$\"3l-4Tzw5V5F*7$$\"37_,7<>Y 2aFhn$\"3UISU$Q#\\\"3\"F*7$$\"3n)o2ndWZh&Fhn$\"3uP:M:*[H7\"F*7$$\"3[[W oYy))GeFhn$\"3q*)oLpvxl6F*7$$\"3BfkMe_QQgFhn$\"3&=Hp;0xw?\"F*7$$\"3Qhc GwZ3TiFhn$\"3GKrDbp@[7F*7$$\"333ajJ![hY'Fhn$\"3i\"3FjgHKH\"F*7$$\"35B) fmPx$omFhn$\"3ik>LvanL8F*7$$\"3%*)ziuO+V)oFhn$\"3yfD\\t+'oP\"F*7$$\"3Q Jof*oe*zqFhn$\"3Gm$>zt\"*fT\"F*7$$\"3cSX#e\"\\'QH(Fhn$\"353\\;$)Hxe9F* 7$$\"3+(z-Y+M^\\(Fhn$\"3Sf0#4!o-*\\\"F*7$$\"3EFzU&>=bq(Fhn$\"3X&e&3RO5 T:F*7$$\"3Z/&)o'>27\"zFhn$\"3!4qP$R9C#e\"F*7$$\"3A@Qu=XaE\")Fhn$\"3Dk( [P!*3`i\"F*7$$\"3w24L`*RRL)Fhn$\"3a\"=m1*zym;F*7$$\"3GSD)*R<.Y&)Fhn$\" 313l*zM1#4F*7$$\"3o;\">'** 4s#y*Fhn$\"3MBQ#**>Wl&>F*7$$\"2%******z********F*$\"3))*****f*******>F *7S7$$\"30+++1+++5F*$!2!******z)*******F*7$$\"3b]D?9VfV5F*$!3?g\\@k_)> =*Fhn7$$\"3?!4s')[D:3\"F*$!3c4kqz(>#\\&)Fhn7$$\"3<%yg91$=C6F*$!3E!\\r. KAF\"zFhn7$$\"3;j>L'RBr;\"F*$!3G+>ua(*=TtFhn7$$\"3%y3(GV'f)47F*$!3u^i] $Q><$oFhn7$$\"3i))[$[h\"[\\7F*$!3&o(z)zc6`S'Fhn7$$\"3Mw%y8(y]!H\"F*$!3 eJ1ns`_/gFhn7$$\"3a@Xe%GPHL\"F*$!3#[P,3:V$GcFhn7$$\"373V7E1Bv8F*$!3oKq '4L#[(G&Fhn7$$\"3;j/TEXt=9F*$!3AMzG*f%=o\\Fhn7$$\"35v@Y&y_qX\"F*$!3/2Y BN$4.r%Fhn7$$\"3!*H%*\\p+>+:F*$!3.Rjr?$=LW%Fhn7$$\"3w'[p$zW]V:F*$!3KU$ *RVsV(>%Fhn7$$\"3QiUCRfC&e\"F*$!3/C*os<+$zRFhn7$$\"3!*zQr$=^Ji\"F*$!3[ *f')p79cz$Fhn7$$\"39%*>m&=C#o;F*$!3YJl'z$4G$f$Fhn7$$\"33YuaIpS1F*$!37v**)o1)e?FFhn7$$\"3+#)p/J;cc>F*$!30 tQ)\\:RAh#Fhn7$$\"3)\\HOQ#G,**>F*$!3(*)[J)o(pC]#Fhn7$$\"3k5SX#o2J/#F*$ !3sKk'=b8#Fhn7$$\"33pp([0xw? #F*$!3ibaL=6x^?Fhn7$$\"33\\`]ep@[AF*$!3aXR&\\3V%y>Fhn7$$\"3I6.i4'HKH#F *$!3btf+1w`,>Fhn7$$\"3&*RcmyanLBF*$!3hAF(y:'>O=Fhn7$$\"3!*>%po2goP#F*$ !3=+)H=[z+x\"Fhn7$$\"3Ve`LT<*fT#F*$!3*Gq'Q;T?8W!o-*\\#F*$!3*G&35=kC,;Fhn7$$\"3J*oEEk.6a#F*$ !31>**G4ql[:Fhn7$$\"3GU*>HWTAe#F*$!3k%o]tSP2*3`i#F* $!3I$='=jc!4X\"Fhn7$$\"3apHL%*zymEF*$!3bn%ff/AhS\"Fhn7$$\"3L9dq^j?4FF* $!3#)>n$*=^Vi8Fhn7$$\"3?YGmjMF^FF*$!3dh3XJ-4@8Fhn7$$\"3g8-jq(G**y#F*$! 3uFH)3jNZG\"Fhn7$$\"3SqRm9@BMGF*$!3#>05\"=[)[C\"Fhn7$$\"3)4w6PbdQ(GF*$ !3M&fEw%>z57Fhn7$$\"3]!o,l`1h\"HF*$!3c0r,N='f<\"Fhn7$$\"3wn.)Q?Wl&HF*$ !3fr)>lv8S9\"Fhn7$$\"\"$F-$!3066666666Fhn-%'COLOURG6&%$RGBG$F-F-F[dm$ \"*++++\"!\")-F$6&7&7$F[dmF+7$F[dmF[dm7$$\"\"\"F-$\"\"#F-7$FedmF+-Fhcm 6&FjcmF-F-F--%'SYMBOLG6$%'CIRCLEG\"#:-%&STYLEG6#%&POINTG-%+AXESLABELSG 6%Q\"x6\"Q!Fiem-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#:F,FccmF^fm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________ ________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pic tures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 14 "small changes " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 638 "fn := x -> exp(x): x : = 'x':\na := .5: b := .7:\nfa := fn(a): fb := fn(b): \np1 := plot([[a, fa],[b,fb],[b,fa],[a,fa]],linestyle=1,\n color=blue,thickness=2):\n p2 := plot(fn(x),x=0..1.2,color=red):\np3 := plot([[[a,fa],[b,fb]]$3], style=point,\n symbol=[circle,diamond,cross],color=black):\n t1 := plots[textplot]([1,3.2,`y = f(x)`],\n font=[HELVETIC A,10],color=red):\nt2 := plots[textplot]([[.59,1.57,`D`],[.74,1.85,`D` ]],\n font=[SYMBOL,10],color=black):\nt3 := plots[textplot] ([[.42,1.7,`(x,y)`],[.62,1.57,`x`],\n [.77,1.85,`y`]],font=[HELVE TICA,10],color=black):\nplots[display]([p1,p2,p3,t1,t2,t3],axes=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 761 "h := evalf(Pi/25):\na := .8 :\np1 := plot([[cos(t),sin(t),t=0..2*Pi],\n [a*cos(t),a*sin(t),t= 0..2*Pi]],color=black):\np2 := plots[polygonplot]([seq([[a*cos((i-1)*h ),a*sin((i-1)*h)],\n [cos((i-1)*h),sin((i-1)*h)],[cos(i*h),sin(i*h)] ,\n [a*cos(i*h),a*sin(i*h)]],i=1..50)],\n style=patchnogrid,color= COLOR(RGB,.85,.85,.87)):\np3 := plottools[arrow]([0,0],[a*cos(.3),a*si n(.3)],0,.07,.07,arrow):\np4 := plottools[arrow]([.6*cos(.7),.6*sin(.7 )],\n [a*cos(.7),a*sin(.7)],0,.06,.3,arrow):\np5 := plottools [arrow]([1.2*cos(.7),1.2*sin(.7)],\n [cos(.7),sin(.7)],0,.06, .3,arrow):\nt1 := plots[textplot]([[.35,.2,`r`],[1,.62,`r`]],font=[HEL VETICA,10]):\nt2 := plots[textplot]([.93,.62,`D`],font=[SYMBOL,10]):\n plots[display]([p1,p2,p3,p4,p5,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 681 "p1 := plot([[0,0],[0,.2],[6.28318,.2],[6.28318,0] ,[0,0]],color=black):\np2 := plots[polygonplot]([[0,0],[0,.2],[6.28318 ,.2],[6.28318,0]],\n style=patchnogrid,color=COLOR(RGB,.85,.85,.87) ):\np3 := plot([[[0,-.2],[0,0]],[[6.28318,-.2],[6.28318,0]],\n [[ -.3,0],[0,0]],[[-.3,.2],[0,.2]]],\n color=COLOR(RGB,.3,.3,.3),lin estyle=3):\np4 := plottools[arrow]([3.14159-.25,-.1],[0,-.1],0,.06,.03 ,arrow):\np5 := plottools[arrow]([3.14159+.25,-.1],[6.28318,-.1],0,.06 ,.03,arrow):\nt1 := plots[textplot]([[3.14159,-.1,`2 r`],[-.45,.1,`r `]],font=[HELVETICA,10]):\nt2 := plots[textplot]([[3.14159,-.1,`p`],[- .53,.1,`D`]],font=[SYMBOL,10]):\nplots[display]([p1,p2,p3,p4,p5,t1,t2] ,axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 23 "ave rage rate of change " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 860 "fn := x -> x^2/2+x: x := 'x':\na := .3: \+ b := .9:\nfa := fn(a): fb := fn(b): \np1 := plot([[a,fa],[b,fb],[b,fa] ,[a,fa]],color=blue):\np2 := plot([[[a,0],[a,fa]],[[b,0],[b,fa]],\n \+ [[0,fa],[a,fa]],[[0,fb],[b,fb]]],color=black,linestyle=3):\np3 \+ := plot(fn(x),x=0..1.2,color=red):\np4 := plot([[[a,fa],[b,fb]]$3],sty le=point,\n symbol=[circle,diamond,cross],color=black):\nt1 \+ := plots[textplot]([1.05,1.9,`y = f(x)`],\n font=[HELVETIC A,10],color=red):\nt2 := plots[textplot]([[.28,.43,`P`],[.88,1.4,`Q`], \n [a,-.05,`x`],[b,-.05,`x`],[-.04,fa+.02,`y`],\n [-. 04,fb+.02,`y`]],font=[HELVETICA,10],color=black):\nt3 := plots[textplo t]([[a+.023,-.1,`1`],[b+.025,-.1,`2`],\n [-.017,fa-.03,`1`],[-.015 ,fb-.03,`2`]],\n font=[HELVETICA,8],color=black):\nplots[displa y]([p1,p2,p3,p4,t1,t2,t3],view=[-.05..1.2,-.1..fn(1.2)],axes=none);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 519 "f := x -> Pi*(2*x^2-x^3/3): \ng := x -> Pi*(4*x-8/3):\np1 := plot(f(x),x=0..4):\np2 := plot([[1,f( 1)],[3,f(3)]],color=blue):\np3 := plot([[.7,g(.7)],[3.3,g(3.3)]],color =COLOR(RGB,0,.7,.2)):\np4 := plot([[[1,f(1)],[2,f(2)],[3,f(3)]]$3],sty le=point,\n symbol=[circle,diamond,cross],color=black):\nt1 := pl ots[textplot]([[.93,6.3,`P`],[3.1,27.8,`Q`],\n [1.9,17.9,`M`] ],color=black):\nplots[display]([p1,p2,p3,p4,t1],\n labels=[`depth in metres, x`,`volume in cu. metres, V`],\n labeldirections=[horizonta l,vertical]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 507 "f := t -> 4.9*t^2:\ng := t -> 39.2*t-78.4:\np1 \+ := plot(f(t),t=0..7):\np2 := plot([[2,f(2)],[6,f(6)]],color=blue):\np3 := plot([[2.1,g(2.1)],[6.4,g(6.4)]],color=COLOR(RGB,0,.7,.2)):\np4 := plot([[[2,f(2)],[4,f(4)],[6,f(6)]]$3],style=point,\n symbol=[cir cle,diamond,cross],color=black):\nt1 := plots[textplot]([[1.7,30.3,`P` ],[5.7,186,`Q`],\n [4.3,75.6,`M`]],color=black):\nplots[displ ay]([p1,p2,p3,p4,t1],\n labels=[`time in secs., t`,`distance in metre s, s`],\n labeldirections=[horizontal,vertical]);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 258 "" 0 "" {TEXT -1 16 "f(x)=abs(x^2-1) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "p1 := \+ plot(abs(x^2-1),x=-1.8..1.8):\np2 := plot(x^2-1,x=-1..1,color=black,li nestyle=2):\nplots[display]([p1,p2],tickmarks=[3,4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 258 "" 0 "" {TEXT -1 17 "density of wire " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 709 "p1 : = plot([[[0,0],[0,.2],[5,.2],[5,0],[0,0]],\n [[3,0],[3,.2] ],[[4,0],[4,.2]]],color=black):\np2 := plots[polygonplot]([[0,0],[0,.2 ],[5,.2],[5,0]],\n style=patchnogrid,color=COLOR(RGB,.85,.85,.87)): \np3 := plot([[[0,.2],[0,.8]],[[2,.2],[2,.8]]],color=black,linestyle=3 ):\np4 := plottools[arrow]([.85,.5],[0,.5],0,.1,.1,arrow):\np5 := plot tools[arrow]([1.15,.5],[2,.5],0,.1,.1,arrow):\nt1 := plots[textplot]([ [1,.55,`x`],[3,-.15,`x`],[4,-.15,`x`]],\n font=[HELVETICA, 10],color=black):\nt2 := plots[textplot]([[1,1,`This part of rod`],\n \+ [1,.8,`has mass f(x)`],[3.05,-.23,`l`],[4.05,-.23,`2`]],\n \+ font=[HELVETICA,8],color=black):\nplots[display]([p1,p2,p3,p4,p5,t 1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }