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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 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 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 "Tim es" 1 14 0 0 0 1 2 1 1 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 261 1 {CSTYLE "" -1 -1 " " 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 262 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 263 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 264 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 265 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 269 1 {CSTYLE "" -1 -1 " " 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 270 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 271 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 273 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 274 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 276 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 277 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 257 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 289 4 "Math" }{TEXT 376 1 " " } {TEXT 457 3 "233" }{TEXT 375 21 " \nExam 2 Fall 2005" }}{PARA 0 "" 0 "" {TEXT 256 2 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 4 "Load" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(plots): with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning , new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ma gnitude := r -> sqrt(r[1]^2+r[2]^2+r[3]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*magnitudeGR6#%\"rG6\"6$%)operatorG%&arrowGF(-%%sqrtG 6#,(*$)&9$6#\"\"\"\"\"#\"\"\"F5*$)&F36#F6F6F7F5*$)&F36#\"\"$F6F7F5F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "velocity := t -> subs (u=t, map(z->diff(z,u),r(u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)v elocityGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%%subsG6$/%\"uG9$-%$mapG6$R 6#%\"zGF(F)F(-%%diffG6$F1F0F(F(F(-%\"rG6#F0F(F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 57 "acceleration := t -> subs(u=t, map(z->diff(z ,u$2),r(u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accelerationGR6#% \"tG6\"6$%)operatorG%&arrowGF(-%%subsG6$/%\"uG9$-%$mapG6$R6#%\"zGF(F)F (-%%diffG6$F1-%\"$G6$F0\"\"#F(F(F(-%\"rG6#F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "speed := t -> magnitude(velocity(t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&speedGR6#%\"tG6\"6$%)operatorG%&arr owGF(-%*magnitudeG6#-%)velocityG6#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "unitTangent := t -> map(z->z/magnitude(velocity(t)) ,velocity(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,unitTangentGR6#% \"tG6\"6$%)operatorG%&arrowGF(-%$mapG6$R6#%\"zGF(F)F(*&9$\"\"\"-%*magn itudeG6#-%)velocityG6#T$!\"\"F(F(6$F'F3-F96#F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "principalUnitNormal := t -> simplify(sub s(u=t,map(w-> w/magnitude(map(z->diff(z,u),unitTangent(u))),\n\nmap(z- >diff(z,u) ,unitTangent(u)) )));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%4principalUnitNormalGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%)simplifyG6 #-%%subsG6$/%\"uG9$-%$mapG6$R6#%\"wGF(F)F(*&F4\"\"\"-%*magnitudeG6#-F6 6$R6#%\"zGF(F)F(-%%diffG6$F4F3F(F(F(-%,unitTangentG6#F3!\"\"F(F(F(-F66 $RFCF(F)F(FEF(F(F(FHF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "curvature := t -> magnitude(crossprod(velocity(t),acceleration(t)) )/speed(t)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*curvatureGR6#%\"tG 6\"6$%)operatorG%&arrowGF(*&-%*magnitudeG6#-%*crossprodG6$-%)velocityG 6#9$-%-accelerationGF5\"\"\"*$)-%&speedGF5\"\"$F9!\"\"F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 259 47 "The path of a particle parameter ized by time " }{XPPEDIT 258 1 "t;" "6#%\"tG" }{TEXT 481 10 " is \+ " }{XPPEDIT 258 1 "r(t) = `<`*t^2,2*t,t*`>`;" "6%/-%\"rG6#%\"tG*&% \"GF*" }{TEXT 256 3 " . " }} {PARA 276 "" 0 "" {TEXT -1 79 "The first ten questions of this exam co ncern this parameterized curve at time " }{TEXT 515 5 "t = 1" }{TEXT -1 353 ". The questions require you to compute the speed, the acceler ation, the unit tangent vector, the principal unit mormal vector, the \+ tangential and normal components of acceleration, the curvature, the o sculating plane, and the center of curvature. Calculations used for on e question may be useful for another question, so organize your work a ccordingly." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 399 39 "1. What is the s peed of the particle at" }{TEXT 402 2 " " }{XPPEDIT 403 1 "t=1" "6#/% \"tG\"\"\"" }{TEXT 401 1 " " }{TEXT 404 2 "? " }{TEXT 405 2 " " }} {PARA 270 "" 0 "" {TEXT 400 4 "a) " }{XPPEDIT 19 1 "1" "6#\"\"\"" } {TEXT 416 14 " b) " }{XPPEDIT 19 1 "2" "6#\"\"#" }{TEXT 415 14 " c) " }{XPPEDIT 19 1 "3" "6#\"\"$" }{TEXT 414 14 " \+ d) " }{XPPEDIT 19 1 "4" "6#\"\"%" }{TEXT 413 18 " e) \+ " }{XPPEDIT 19 1 "5" "6#\"\"&" }{TEXT 412 15 " \nf) " } {XPPEDIT 19 1 "6" "6#\"\"'" }{TEXT 411 14 " g) " }{XPPEDIT 19 1 "7" "6#\"\"(" }{TEXT 410 14 " h) " }{XPPEDIT 19 1 "8" " 6#\"\")" }{TEXT 409 14 " i) " }{XPPEDIT 19 1 "9" "6#\"\"*" } {TEXT 408 17 " j) " }{XPPEDIT 19 1 "10" "6#\"#5" }{TEXT 407 1 " " }{TEXT 406 7 " \n" }}{PARA 0 "" 0 "" {TEXT 417 12 "Solu tion: c" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x := t -> t^2: y := t -> 2*t: z := t -> t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "r \+ := t -> [x(t),y(t),z(t)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGR6# %\"tG6\"6$%)operatorG%&arrowGF(7%-%\"xG6#9$-%\"yGF/-%\"zGF/F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "velocity(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$%\"tG\"\"#F&\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "speed(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sq rtG6#,&*$)%\"tG\"\"#\"\"\"\"\"%\"\"&\"\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "speed(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 261 "" 0 "" {TEXT -1 1 "2" }{TEXT 455 1 "." }{TEXT -1 2 " " }{TEXT 441 5 "Let " }{TEXT 452 1 " " }{XPPEDIT 451 1 "`<`*a,b,c* `>`" "6%*&%\"GF%" }{TEXT 450 50 " de note the unit tangent vector to this curve at" }{TEXT 447 2 " " } {XPPEDIT 448 1 "t=1" "6#/%\"tG\"\"\"" }{TEXT 440 3 ". " }{TEXT -1 8 " What is " }{TEXT 453 1 " " }{XPPEDIT 454 1 "b/c;" "6#*&%\"bG\"\"\"%\"c G!\"\"" }{TEXT -1 3 " ? " }}{PARA 272 "" 0 "" {TEXT 423 4 "a) " } {XPPEDIT 418 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT 419 13 " " } {TEXT -1 3 "b) " }{TEXT 442 1 " " }{XPPEDIT 424 0 "-2" "6#,$\"\"#!\"\" " }{TEXT 425 14 " c) " }{XPPEDIT 426 0 "-3" "6#,$\"\"$!\"\" " }{TEXT 427 8 " " }{TEXT -1 8 " d) " }{XPPEDIT 428 0 "-4" "6#,$\"\"%!\"\"" }{TEXT 429 7 " " }{TEXT -1 6 " e)" }{TEXT 445 2 " " }{XPPEDIT 18 0 "-5" "6#,$\"\"&!\"\"" }{TEXT -1 3 " " } {TEXT 432 2 " " }}{PARA 271 "" 0 "" {TEXT 421 4 "f) " }{TEXT 420 2 " " }{XPPEDIT 422 0 "1" "6#\"\"\"" }{TEXT 431 17 " g) " } {XPPEDIT 430 0 "2" "6#\"\"#" }{TEXT 434 17 " h) " } {XPPEDIT 433 0 "3" "6#\"\"$" }{TEXT 436 19 " i) " } {XPPEDIT 443 0 "4" "6#\"\"%" }{TEXT 444 18 " j) " } {XPPEDIT 446 0 "5" "6#\"\"&" }{TEXT 439 1 " " }{TEXT 438 2 " " }} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 449 12 "Solution : g" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "unitTangent(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7% ,$*&%\"tG\"\"\"*$-%%sqrtG6#,&*$)F&\"\"#F'\"\"%\"\"&\"\"\"F'!\"\"F/,$*& F'F'*$-F*6#F,F'F3F/F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "un itTangent(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"#\"\"$F$#\"\" \"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "unitTangent(1)[2]/u nitTangent(1)[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 57 "3. What is the magnitude of the acceleration \+ vector at " }{XPPEDIT 258 1 "t=1" "6#/%\"tG\"\"\"" }{TEXT 256 2 "? " }}{PARA 262 "" 0 "" {TEXT 264 5 "a) " }{XPPEDIT 19 1 "1;" "6#\"\"\" " }{TEXT 288 20 " b) " }{XPPEDIT 19 1 "sqrt(2);" "6#-% %sqrtG6#\"\"#" }{TEXT 287 19 " c) " }{XPPEDIT 19 1 "3; " "6#\"\"$" }{TEXT 286 23 " d) " }{XPPEDIT 19 1 "2; " "6#\"\"#" }{TEXT 285 18 " e) " }{XPPEDIT 19 1 "sqrt(5) ;" "6#-%%sqrtG6#\"\"&" }{TEXT 284 16 " \n f) " }{XPPEDIT 19 1 "sqrt(6);" "6#-%%sqrtG6#\"\"'" }{TEXT 283 19 " g) " } {XPPEDIT 19 1 "sqrt(7);" "6#-%%sqrtG6#\"\"(" }{TEXT 282 19 " \+ h) " }{XPPEDIT 19 1 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"# F%" }{TEXT 281 16 " i) " }{XPPEDIT 19 1 "3;" "6#\"\"$" } {TEXT 280 19 " j) " }{XPPEDIT 19 1 "3*sqrt(2);" "6#*&\" \"$\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 279 8 " \n" }}{PARA 0 "" 0 " " {TEXT 377 12 "Solution: d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ac celeration(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"#\"\"!F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "magnitude(acceleration(t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 9 "4. Let " }{TEXT 505 1 " " }{XPPEDIT 261 1 "`<`*p,q,r*`>`;" "6%*&%\"GF%" }{TEXT 260 58 " denote the principal unit normal vecto r to this curve at" }{TEXT 508 3 " " }{XPPEDIT 259 1 "t=1" "6#/%\"tG \"\"\"" }{TEXT 256 11 ". What is " }{XPPEDIT 483 1 "pq;" "6#%#pqG" } {TEXT 482 2 " ?" }}{PARA 273 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 19 1 "-1/3;" "6#,$*&\"\"\"\"\"\"\"\"$!\"\"F(" }{TEXT -1 16 " b) \+ " }{XPPEDIT 19 1 "-2/3;" "6#,$*&\"\"#\"\"\"\"\"$!\"\"F(" }{TEXT -1 14 " c) " }{XPPEDIT 19 1 "-1/9;" "6#,$*&\"\"\"\"\"\"\"\"*!\" \"F(" }{TEXT -1 13 " d) " }{XPPEDIT 19 1 "-2/9;" "6#,$*&\"\"# \"\"\"\"\"*!\"\"F(" }{TEXT -1 18 " e) " }{XPPEDIT 19 1 " -4/9;" "6#,$*&\"\"%\"\"\"\"\"*!\"\"F(" }}{PARA 274 "" 0 "" {TEXT -1 5 "f) " }{XPPEDIT 19 1 "1/3;" "6#*&\"\"\"\"\"\"\"\"$!\"\"" }{TEXT -1 20 " g) " }{XPPEDIT 19 1 "2/3;" "6#*&\"\"#\"\"\"\"\"$! \"\"" }{TEXT -1 18 " h) " }{XPPEDIT 19 1 "1/9;" "6#*&\" \"\"\"\"\"\"\"*!\"\"" }{TEXT -1 18 " i) " }{XPPEDIT 19 1 "2/9;" "6#*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 21 " j) \+ " }{XPPEDIT 19 1 "4/9;" "6#*&\"\"%\"\"\"\"\"*!\"\"" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 456 12 "Solution: e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "principalUnitNormal(1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*$-%%sqrtG6#\"\"&\"\"\"#\"\"\"\"\"$,$F%# !\"%\"#:,$F%#!\"#F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "dotp rod(principalUnitNormal(1),unitTangent(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "mag nitude(principalUnitNormal(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "principalUnitNormal (1)[1]*principalUnitNormal(1)[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6## !\"%\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 2 " " } {TEXT 516 2 "5." }{TEXT -1 1 " " }{TEXT 458 68 "What is the curvature \+ of the particle's path at the point for which " }{TEXT 460 1 " " } {XPPEDIT 461 1 "t=1" "6#/%\"tG\"\"\"" }{TEXT 459 1 "?" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 463 7 "a) " }{XPPEDIT 19 1 "sqrt(5)/3" "6#*&-%%sqrtG6#\"\"&\"\"\"\"\"$!\"\"" }{TEXT -1 5 " \+ " }{TEXT 464 11 " b) " }{XPPEDIT 19 1 "2*sqrt(5)/3" "6#*(\" \"#\"\"\"-%%sqrtG6#\"\"&F%\"\"$!\"\"" }{TEXT -1 5 " " }{TEXT 465 14 " c) " }{XPPEDIT 19 1 "sqrt(5)/27" "6#*&-%%sqrtG6#\"\"&\" \"\"\"#F!\"\"" }{TEXT -1 5 " " }{TEXT 466 13 " d) " } {XPPEDIT 19 1 "2*sqrt(5)/27" "6#*(\"\"#\"\"\"-%%sqrtG6#\"\"&F%\"#F!\" \"" }{TEXT -1 5 " " }{TEXT 467 13 " e) " }{XPPEDIT 19 1 " sqrt(5)/81" "6#*&-%%sqrtG6#\"\"&\"\"\"\"#\")!\"\"" }{TEXT -1 5 " \+ " }{TEXT 468 11 " \n\n f) " }{XPPEDIT 19 1 "2*sqrt(5)/81" "6#*(\" \"#\"\"\"-%%sqrtG6#\"\"&F%\"#\")!\"\"" }{TEXT 469 15 " g) \+ " }{XPPEDIT 19 1 "2*sqrt(5)/5" "6#*(\"\"#\"\"\"-%%sqrtG6#\"\"&F%\"\"&! \"\"" }{TEXT 470 18 " h) " }{XPPEDIT 19 1 "3*sqrt(5)/5" "6#*(\"\"$\"\"\"-%%sqrtG6#\"\"&F%\"\"&!\"\"" }{TEXT 471 14 " \+ i) " }{XPPEDIT 19 1 "6*sqrt(5)/5" "6#*(\"\"'\"\"\"-%%sqrtG6#\"\"&F%\" \"&!\"\"" }{TEXT 472 16 " j) " }{XPPEDIT 19 1 "12*sqrt(5)/ 5" "6#*(\"#7\"\"\"-%%sqrtG6#\"\"&F%\"\"&!\"\"" }{TEXT 473 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 378 13 "Solution: d \+ " }{TEXT 462 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "curvature(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$-%%sqrtG6#\"\"&\"\"\"F**$), &*$)%\"tG\"\"#F*\"\"%F)\"\"\"#\"\"$F1F*!\"\"F1" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "curvature(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"&\"\"\"#\"\"#\"#F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 435 1 " " }{TEXT 437 40 "6. What is the tangential component, " }{XPPEDIT 262 1 "a[T]" "6#&%\"aG6#%\"TG" }{TEXT 260 26 " , of acceleration at \+ at" }{TEXT 509 3 " " }{XPPEDIT 256 1 "t=1" "6#/%\"tG\"\"\"" }{TEXT 484 4 " ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT 485 174 "a) 1/3 b) 2/3 c) 1 \+ d) 4/3 e) 5/3\n\nf) 2 g) 7/3 \+ h) 8/3 i) 3 j) 10/3 " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 12 "Solution: d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "diff(speed(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"tG\"\"\"*$-%%sqrtG6#,&*$)F%\"\"#F&\"\"%\"\"& \"\"\"F&!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a[T] := \+ simplify(subs(t=1,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#%\" TG#\"\"%\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 486 3 " 7." }{TEXT -1 1 " " }{TEXT 263 1 " " }{TEXT 507 31 "W hat is the normal component " }{XPPEDIT 261 1 "a[N]" "6#&%\"aG6#%\"N G" }{TEXT 259 24 " of acceleration at " }{XPPEDIT 256 1 "t=1" "6#/ %\"tG\"\"\"" }{TEXT 258 3 " ?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 4 "a) " }{XPPEDIT 488 1 "sq rt(5)/2;" "6#*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" }{TEXT 258 15 " \+ b) " }{XPPEDIT 490 1 "sqrt(5);" "6#-%%sqrtG6#\"\"&" }{TEXT 257 13 " c)" }{TEXT 487 2 " " }{XPPEDIT 491 1 "3*sqrt(5)/2;" "6 #*(\"\"$\"\"\"-%%sqrtG6#\"\"&F%\"\"#!\"\"" }{TEXT 259 18 " \+ d) " }{XPPEDIT 492 1 "2*sqrt(5);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"&F% " }{TEXT 260 16 " e) " }{XPPEDIT 493 1 "5*sqrt(5)/2;" "6#* (\"\"&\"\"\"-%%sqrtG6#\"\"&F%\"\"#!\"\"" }{TEXT 261 12 " \n\nf) \+ " }{XPPEDIT 494 1 "sqrt(5)/3;" "6#*&-%%sqrtG6#\"\"&\"\"\"\"\"$!\"\"" } {TEXT 262 15 " g) " }{XPPEDIT 495 1 "2*sqrt(5)/3;" "6#*(\" \"#\"\"\"-%%sqrtG6#\"\"&F%\"\"$!\"\"" }{TEXT 263 16 " h) \+ " }{XPPEDIT 496 1 "3*sqrt(5);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"&F%" } {TEXT 264 17 " i) " }{XPPEDIT 497 1 "4*sqrt(5)/3;" "6#*( \"\"%\"\"\"-%%sqrtG6#\"\"&F%\"\"$!\"\"" }{TEXT 265 13 " " }{TEXT 498 2 "j)" }{TEXT 489 2 " " }{XPPEDIT 499 1 "5*sqrt(5)/3;" "6# *(\"\"&\"\"\"-%%sqrtG6#\"\"&F%\"\"$!\"\"" }{TEXT 266 11 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 500 13 "Solution: g" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a[N] := sqrt(2^2-( 4/3)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#%\"NG,$*$-%%sqrtG 6#\"\"&\"\"\"#\"\"#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "[seq(a[T]*unitTangent(1)[j]+a[N]*pr incipalUnitNormal(1)[j],j=1..3)]; #check" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"#\"\"!F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 501 29 "8. The osculating plane \+ at " }{XPPEDIT 503 1 "t = 1;" "6#/%\"tG\"\"\"" }{TEXT 257 2 " " } {TEXT 502 54 " has equation Ax + y + Cz = D. What is D-C ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 504 4 "a) " } {XPPEDIT 269 1 "1;" "6#\"\"\"" }{TEXT 258 19 " b) " } {XPPEDIT 271 1 "sqrt(5);" "6#-%%sqrtG6#\"\"&" }{TEXT 257 11 " c) " }{XPPEDIT 272 1 "2;" "6#\"\"#" }{TEXT 259 20 " d) \+ " }{XPPEDIT 273 1 "2*sqrt(5);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"&F%" } {TEXT 260 12 " e) " }{XPPEDIT 274 1 "3;" "6#\"\"$" }{TEXT 261 13 " \n\n\nf) " }{XPPEDIT 275 1 "3*sqrt(5);" "6#*&\"\"$\"\"\"-%% sqrtG6#\"\"&F%" }{TEXT 262 14 " g) " }{XPPEDIT 276 1 "4;" "6 #\"\"%" }{TEXT 263 16 " h) " }{XPPEDIT 277 1 "4*sqrt(5);" "6#*&\"\"%\"\"\"-%%sqrtG6#\"\"&F%" }{TEXT 264 14 " i) " } {XPPEDIT 278 1 "5;" "6#\"\"&" }{TEXT 265 19 " j) " } {XPPEDIT 279 1 "5*sqrt(5);" "6#*&\"\"&\"\"\"-%%sqrtG6#\"\"&F%" }{TEXT 266 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 506 13 "Solution: c" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Binormal := crossprod( unitTangent(1),principalUnitNormal(1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%)BinormalG-%'vectorG6#7%\"\"!,$*$-%%sqrtG6#\"\"&\" \"\"#\"\"\"F/,$F+#!\"#F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " r(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"\"\"#F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Binormal[2]*(y-2) + Binormal[3]*(z- 1) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%%sqrtG6#\"\"&\"\"\", &%\"yG\"\"\"!\"#F-F-#F-F)*&F&F*,&%\"zGF-!\"\"F-F-#F.F)\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(expand, % );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%%sqrtG6#\"\"&\"\"\"%\"yG\"\"\"#F,F)*& F&F*%\"zGF,#!\"#F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " map(z ->z*sqrt(5), % );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&*&-%%s qrtG6#\"\"&\"\"\"%\"yG\"\"\"#F-F**&F'F+%\"zGF-#!\"#F*F-F'F+\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn := map(expand, % );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&%\"yG\"\"\"%\"zG!\"#\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "rhs(eqn) - coeff(lhs(eqn), z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 510 15 "9. What is the " }{TEXT 511 1 "x" } {TEXT 512 39 "-component of the center of curvature? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 4 "a) " }{XPPEDIT 19 1 "11/2;" "6#*&\"#6\"\"\"\"\"#!\"\"" } {TEXT 258 15 " b) " }{XPPEDIT 19 1 "6;" "6#\"\"'" }{TEXT 257 13 " c) " }{XPPEDIT 19 1 "13/2;" "6#*&\"#8\"\"\"\"\"#!\" \"" }{TEXT 259 14 " d) " }{XPPEDIT 19 1 "16/3;" "6#*&\"#;\" \"\"\"\"$!\"\"" }{TEXT 260 13 " e) " }{XPPEDIT 19 1 "5;" "6# \"\"&" }{TEXT 261 12 " \n\nf) " }{XPPEDIT 19 1 "20/3;" "6#*&\"#? \"\"\"\"\"$!\"\"" }{TEXT 262 14 " g) " }{XPPEDIT 19 1 "17/4; " "6#*&\"#<\"\"\"\"\"%!\"\"" }{TEXT 263 12 " h) " }{XPPEDIT 19 1 "26/5;" "6#*&\"#E\"\"\"\"\"&!\"\"" }{TEXT 264 15 " i) \+ " }{XPPEDIT 19 1 "21/5;" "6#*&\"#@\"\"\"\"\"&!\"\"" }{TEXT 265 14 " \+ j) " }{XPPEDIT 19 1 "25/4;" "6#*&\"#D\"\"\"\"\"%!\"\"" }{TEXT 266 11 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 513 12 "Solution: a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "[seq(r(1)[j]+ (1/curvature(1))*prin cipalUnitNormal(1)[j],j=1..3)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%# \"#6\"\"##!\")\"\"&#!\"%F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 514 352 "10. \+ The distance the particle has moved along its trajectory in the time i nterval [0,1] is an integral that can be calculated in terms of a spec ial function known as the hypergeometric function. Instead, approximat e the arc length integral by using a Riemann sum in which only one sub interval is used with its midpoint selected as the choice of point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 256 5 "a) " }{XPPEDIT 19 1 "sqrt(6);" "6#-%%sqrtG6#\"\"'" } {TEXT 257 12 " b)" }{TEXT 517 3 " " }{XPPEDIT 19 1 "sqrt(7) ;" "6#-%%sqrtG6#\"\"(" }{TEXT 258 13 " c) " }{XPPEDIT 19 1 "s qrt(8);" "6#-%%sqrtG6#\"\")" }{TEXT 259 14 " d) " }{XPPEDIT 19 1 "3;" "6#\"\"$" }{TEXT 260 17 " e) " }{XPPEDIT 19 1 " 2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 261 8 " \n\n f) \+ " }{XPPEDIT 19 1 "3*sqrt(2);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F%" } {TEXT 262 13 " g) " }{XPPEDIT 19 1 "2*sqrt(3);" "6#*&\"\"#\" \"\"-%%sqrtG6#\"\"$F%" }{TEXT 263 11 " h) " }{XPPEDIT 19 1 "3*s qrt(3);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 264 13 " i) " }{XPPEDIT 19 1 "3*sqrt(5)/2;" "6#*(\"\"$\"\"\"-%%sqrtG6#\"\"&F%\" \"#!\"\"" }{TEXT 265 11 " j) " }{XPPEDIT 19 1 "5/2;" "6#*&\"\"& \"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 12 "Solution: a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "int(magnitude(velocity(t)),t=0..1); # exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%sqrtG6#\"\"&\"\"\"-%*hypergeomG6%7$#\" \"\"\"\"##!\"\"F/7##\"\"$F/#!\"%F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf( % ); # floating point approximation of exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&p)*e]#!\"*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "magnitude(velocity(1/2)); # answer" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"\"'\"\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 52 "evalf( % ); # floating point approximation o f answer" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V(*[\\C!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 269 "" 0 "" {TEXT -1 4 "11. " }{TEXT 384 75 "A particle is movin g along a space curve in such a way that its speed is " }{XPPEDIT 385 1 "1+2*t^3;" "6#,&\"\"\"\"\"\"*&\"\"#F%*$%\"tG\"\"$F%F%" }{TEXT 386 142 " at time t. The curvature of the particle's path is 2 \+ at time t = 1. What is the magnitude of its acceleration vector at t hat time? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 387 5 "a) " }{XPPEDIT 19 1 "3*sqrt(5 );" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"&F%" }{TEXT 389 18 " b) \+ " }{XPPEDIT 19 1 "4*sqrt(5);" "6#*&\"\"%\"\"\"-%%sqrtG6#\"\"&F%" } {TEXT 390 16 " c) " }{XPPEDIT 19 1 "5*sqrt(5);" "6#*&\"\"& \"\"\"-%%sqrtG6#\"\"&F%" }{TEXT 391 17 " d) " }{XPPEDIT 19 1 "6*sqrt(5);" "6#*&\"\"'\"\"\"-%%sqrtG6#\"\"&F%" }{TEXT 388 18 " \+ e) " }{XPPEDIT 19 1 "8*sqrt(5);" "6#*&\"\")\"\"\"-%%sqrtG 6#\"\"&F%" }{TEXT 392 12 " \n\nf) " }{XPPEDIT 19 1 "2*sqrt(10); " "6#*&\"\"#\"\"\"-%%sqrtG6#\"#5F%" }{TEXT 393 15 " g) " } {XPPEDIT 19 1 "3*sqrt(10);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"#5F%" }{TEXT 394 16 " h) " }{XPPEDIT 19 1 "4*sqrt(10);" "6#*&\"\"%\"\" \"-%%sqrtG6#\"#5F%" }{TEXT 395 14 " i) " }{XPPEDIT 19 1 "5*s qrt(10);" "6#*&\"\"&\"\"\"-%%sqrtG6#\"#5F%" }{TEXT 396 17 " \+ j) " }{XPPEDIT 19 1 "6*sqrt(10);" "6#*&\"\"'\"\"\"-%%sqrtG6#\"#5F% " }{TEXT 397 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 398 12 "Solution: j" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "v \+ := 1+ 2*t^3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG,&\"\"\"F&*$)% \"tG\"\"$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a_T := diff(v,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_TG,$*$)%\"tG\" \"#\"\"\"\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a_T := su bs(t = 1, a_T );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_TG\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a_N := kappa*v^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_NG*&%&kappaG\"\"\"),&F'F'*$)%\"tG\"\"$ \"\"\"\"\"#F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a_N := s ubs(\{t=1,kappa=2\}, a_N );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_NG \"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "acc := sqrt(a_T^2 + a_N^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$accG,$*$-%%sqrtG6#\"#5 \"\"\"\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 535 0 "" }{TEXT 256 23 "12. \+ For the function" }{TEXT 532 3 " " }{XPPEDIT 257 1 "f(x,y) = x^2+y ^3+2*x*y^2;" "6#/-%\"fG6$%\"xG%\"yG,(*$F'\"\"#\"\"\"*$F(\"\"$F,*(\"\"# F,F'F,F(\"\"#F," }{TEXT 258 1 " " }{TEXT 534 12 ", calculate" }{TEXT 533 12 " \n\n " }{XPPEDIT 259 1 "Diff(f(x,y),x,x)*Diff(f(x,y), y,y)-(Diff(f(x,y),x,y))^2" "6#,&*&-%%DiffG6%-%\"fG6$%\"xG%\"yGF+F+\"\" \"-F&6%-F)6$F+F,F,F,F-F-*$-F&6%-F)6$F+F,F+F,\"\"#!\"\"" }{TEXT 260 3 " \n\n" }{TEXT 519 22 "at the point (1,-1). " }{TEXT 520 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 156 "a) 86 b) 68 c) 44 \+ d) 21 e) 8 \nf) -8 g) -20 h ) -44 i) -68 j) -86\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 12 "Solution: g" }{TEXT 518 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f := (x,y) -> x^2 + y^3 + 2 *x*y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "A := diff(f(x,y) ,x,x)*diff(f(x,y),y,y)-(diff(f(x,y),x,y))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,(%\"yG\"#7%\"xG\"\")*$)F&\"\"#\"\"\"!#;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A := expand(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,(%\"yG\" #7%\"xG\"\")*$)F&\"\"#\"\"\"!#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(\{x=1,y=-1\},A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 264 "" 0 "" {TEXT -1 11 "13. If " }{XPPEDIT 19 1 "3*x+y^2+z^3 = 2*(1+x*y*z);" "6#/,( *&\"\"$\"\"\"%\"xGF'F'*$%\"yG\"\"#F'*$%\"zG\"\"$F'*&\"\"#F',&\"\"\"F'* (F(F'F*F'F-F'F'F'" }{TEXT -1 20 " then calculate " }{XPPEDIT 19 1 "diff( z ,x)" "6#-%%diffG6$%\"zG%\"xG" }{TEXT -1 10 " at " } {XPPEDIT 19 1 "``(3,2,1)" "6#-%!G6%\"\"$\"\"#\"\"\"" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 187 "a) \+ 1/6 b) 5/6 c) 2/3 d) 1/3 \+ e) 1/9 \nf) -1/6 g) -5/6 h) - 2/3 i) -1/3 j) -1/9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 379 13 "Solution: j " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqn0 := 3*x + y^2 + z^3 = 2*(1 + x*y*z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn0G/,(%\"xG\" \"$*$)%\"yG\"\"#\"\"\"\"\"\"*$)%\"zGF(F-F.,&F,F.*(F'F.F+F.F1F.F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(\{x=3,y=2,z=1\}, eqn0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"#9F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn1 := subs( z = z(x,y), eqn0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,(%\"xG\"\"$*$)%\"yG\"\"#\"\"\"\"\"\"*$)-% \"zG6$F'F+F(F-F.,&F,F.*(F'F.F+F.F1F.F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn2 := map(u -> diff(u,x),eqn1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%eqn2G/,&\"\"$\"\"\"*&)-%\"zG6$%\"xG%\"yG\"\"#\"\" \"-%%diffG6$F+F.F(F',&*&F/F(F+F(F0*(F.F(F/F1F2F1F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eqn3 := diff(z(x,y),x) = solve(eqn2, diff(z (x,y),x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%%diffG6$-%\" zG6$%\"xG%\"yGF,*&,&!\"$\"\"\"*&F-F1F)F1\"\"#\"\"\",&*$)F)F3F4\"\"$*&F ,F1F-F4!\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(\{ x=3,y=2\}, rhs(eqn3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&!\"$\"\" \"-%\"zG6$\"\"$\"\"#\"\"%\"\"\",&*$)F'F+F-F*!#7F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(z(3,2)=1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 263 13 "14. The plane" }{TEXT 521 2 " " }{XPPEDIT 292 1 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 291 2 " " }{TEXT 293 24 " intersects the graph of" }{TEXT 294 2 " " }{XPPEDIT 295 1 "z = x*y^3-4*x^2;" "6#/% \"zG,&*&%\"xG\"\"\"*$%\"yG\"\"$F(F(*&\"\"%F(*$F'\"\"#F(!\"\"" }{TEXT 290 2 " " }{TEXT 296 59 " in a curve. The tangent line to this cur ve at the point" }{XPPEDIT 298 1 "``(1, 2, 4);" "6#-%!G6%\"\"\"\"\"#\" \"%" }{TEXT 297 1 " " }{TEXT 299 27 " passes through the point " } {XPPEDIT 300 1 "``(1,3,c);" "6#-%!G6%\"\"\"\"\"$%\"cG" }{TEXT 301 12 " . What is " }{TEXT 303 1 " " }{XPPEDIT 304 1 "c" "6#%\"cG" }{TEXT 302 5 "? " }}{PARA 258 "" 0 "" {TEXT 305 5 "\na) " }{XPPEDIT 306 1 "8;" "6#\"\")" }{TEXT 307 10 " b) " }{XPPEDIT 308 1 "9;" "6#\" \"*" }{TEXT 309 10 " c) " }{XPPEDIT 310 1 "10;" "6#\"#5" }{TEXT 311 8 " d) " }{XPPEDIT 312 1 "11;" "6#\"#6" }{TEXT 313 10 " e ) " }{XPPEDIT 314 1 "12;" "6#\"#7" }{TEXT 315 11 " f) " } {XPPEDIT 316 1 "13;" "6#\"#8" }{TEXT 317 11 " g) " }{XPPEDIT 318 1 "14;" "6#\"#9" }{TEXT 319 9 " h) " }{XPPEDIT 320 1 "15;" "6 #\"#:" }{TEXT 321 9 " i) " }{XPPEDIT 322 1 "16;" "6#\"#;" }{TEXT 323 10 " j) " }{XPPEDIT 325 1 "17;" "6#\"#<" }{TEXT -1 1 " " } {TEXT 324 3 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 381 13 "Solution: i " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := (x,y) -> x*y^ 3-4*x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%) operatorG%&arrowGF),&*&9$\"\"\")9%\"\"$\"\"\"F0*$)F/\"\"#F4!\"%F)F)F) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "subs(\{x=1,y=2\}, diff( f(x,y),y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "r := t -> [1,2+t,4+t*subs(\{x=1,y=2\}, di ff(f(x,y),y))]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGR6#%\"tG6\"6 $%)operatorG%&arrowGF(7%\"\"\",&\"\"#F-9$F-,&\"\"%F-*&F0F--%%subsG6$<$ /%\"xGF-/%\"yGF/-%%diffG6$-%\"fG6$F9F;F;F-F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "t_0 := solve( r(t)[2]=3,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t_0G\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(t=t_0,r(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7%\"\"\"\"\"$\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 480 36 "15. The vector-valued function\n\n " } {TEXT 478 10 " " }{XPPEDIT 479 1 "r(u,v) = (u-v)*i+u*v*j+(u+v )*k;" "6#/-%\"rG6$%\"uG%\"vG,(*&,&F'\"\"\"F(!\"\"F,%\"iGF,F,*(F'F,F(F, %\"jGF,F,*&,&F'F,F(F,F,%\"kGF,F," }{TEXT 475 33 " \n \nof the two re al variables " }{TEXT 524 1 "u" }{TEXT 525 6 " and " }{TEXT 526 1 "v " }{TEXT 527 1 " " }{TEXT 477 1 " " }{TEXT 476 95 "defines a surface. \+ The plane that is tangent to this surface at the point (1,2,3) has equ ation " }{TEXT 522 16 "x + By + Cz = D" }{TEXT 523 15 ". What is D? \+ \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 5 " \na) " }{XPPEDIT 257 1 "-4" "6#,$\"\"%!\"\"" }{TEXT 258 10 " b) \+ " }{XPPEDIT 259 1 "-3" "6#,$\"\"$!\"\"" }{TEXT 260 11 " c) " } {XPPEDIT 261 1 "-2" "6#,$\"\"#!\"\"" }{TEXT 262 9 " d) " } {XPPEDIT 263 1 "-1" "6#,$\"\"\"!\"\"" }{TEXT 264 10 " e) " } {XPPEDIT 265 1 "0" "6#\"\"!" }{TEXT 266 10 " \nf) " }{XPPEDIT 267 1 "1" "6#\"\"\"" }{TEXT 268 17 " g) " }{XPPEDIT 269 1 "2" "6#\"\"#" }{TEXT 270 7 " " }{TEXT 528 3 " h)" }{TEXT 529 2 " " }{XPPEDIT 271 1 "3" "6#\"\"$" }{TEXT 272 11 " i) " } {XPPEDIT 273 1 "4" "6#\"\"%" }{TEXT 274 13 " j) " }{XPPEDIT 276 1 "5" "6#\"\"&" }{TEXT -1 1 " " }{TEXT 275 3 " " }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 474 0 "" }} {PARA 3 "" 0 "" {TEXT -1 14 "Solution: a \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "x :=(u,v) -> u-v: y := (u,v)-> u*v: z :=(u,v) - > u+v:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "r := (u,v) -> [x( u,v),y(u,v),z(u,v)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGR6$%\"uG %\"vG6\"6$%)operatorG%&arrowGF)7%-%\"xG6$9$9%-%\"yGF0-%\"zGF0F)F)F)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "r(u,v); r(2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&%\"uG\"\"\"%\"vG!\"\"*&F%F&F'F&,&F%F&F'F &" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "v1 := map( w -> diff(w,u), r(u,v) ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G7%\"\"\"%\"vGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "v2 := map(w->diff(w,v), r(u,v));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G7%!\"\"%\"uG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "linalg[crossprod](v1,v2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,&%\"vG\"\"\"%\"uG!\"\"!\"#,&F *F)F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "N := subs(\{u=2, v=1\},%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'vectorG6#7%!\"\" !\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "x := 'x': y := 'y': z := 'z':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn := N[1]*(x-1) + N[2]*(y-2) + N[3]*(z-3) = 0 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,*%\"xG!\"\"!\"%\"\"\"%\"yG!\"#%\"zG\"\"$\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "map(w->-w,eqn);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,*%\"xG\"\"\"\"\"%F&%\"yG\"\"#%\"zG! \"$\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 265 "" 0 " " {TEXT -1 46 "16. The plane that is tangent to the graph of " }{TEXT 326 1 " " }{XPPEDIT 19 1 "x^3*z^2-x^2+x*y^4 = 1;" "6#/,(*&%\"xG\"\"$% \"zG\"\"#\"\"\"*$F&\"\"#!\"\"*&F&F**$%\"yG\"\"%F*F*\"\"\"" }{TEXT 530 2 " " }{TEXT -1 5 " at " }{XPPEDIT 328 1 "`(`*1,1,1*`)`;" "6%*&%\"(G \"\"\"\"\"\"F%\"\"\"*&\"\"\"F%%\")GF%" }{TEXT 327 1 " " }{TEXT -1 16 " has equation \n" }{TEXT 531 15 "x + By + Cz = D" }{TEXT -1 12 ". Wha t is D?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT 329 4 "a) " }{XPPEDIT 330 1 "0" "6#\"\"!" }{TEXT 331 10 " b) " }{XPPEDIT 332 1 "1" "6#\"\"\"" }{TEXT 333 10 " c) " }{XPPEDIT 334 1 "2" "6#\"\"#" }{TEXT 335 8 " d) " }{XPPEDIT 336 1 "3" "6#\" \"$" }{TEXT 337 10 " e) " }{XPPEDIT 338 1 "4" "6#\"\"%" }{TEXT 339 13 " \nf) " }{XPPEDIT 340 1 "5" "6#\"\"&" }{TEXT 341 11 " \+ g) " }{XPPEDIT 342 1 "6" "6#\"\"'" }{TEXT 343 10 " h) " } {XPPEDIT 344 1 "7" "6#\"\"(" }{TEXT 345 9 " i) " }{XPPEDIT 346 1 "8" "6#\"\")" }{TEXT 347 10 " j) " }{XPPEDIT 349 1 "9" "6#\"\"* " }{TEXT -1 1 " " }{TEXT 348 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 382 12 "Solution: e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "F := (x,y,z) -> x^3*z^2-x^2+ x*y^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6%%\"xG%\"yG%\"zG6\"6 $%)operatorG%&arrowGF*,(*&)9$\"\"$\"\"\")9&\"\"#F3\"\"\"*$)F1F6F3!\"\" *&F1F7)9%\"\"%F3F7F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "N := subs(\{x=1,y=1,z=1\},linalg[grad](F(x,y,z),[x,y,z]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'vectorG6#7%\"\"#\"\"%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "eqn := N[1]*(x-1) + N[2]*(y-1) + N[ 3]*(z-1)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,*%\"xG\"\"#!\" )\"\"\"%\"yG\"\"%%\"zGF(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn := map(w->w/2,eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq nG/,*%\"xG\"\"\"!\"%F(%\"yG\"\"#%\"zGF(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn := map(w->w+4,eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(%\"xG\"\"\"%\"yG\"\"#%\"zGF(\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 0 {PARA 3 "" 0 "" {TEXT 350 8 "17. Let" }{TEXT 354 2 " " } {XPPEDIT 355 1 "f(x,y) = 6*sqrt(x-y^2);" "6#/-%\"fG6$%\"xG%\"yG*&\"\"' \"\"\"-%%sqrtG6#,&F'F+*$F(\"\"#!\"\"F+" }{TEXT 351 39 ". What is the linear approximation of" }{TEXT 356 1 " " }{XPPEDIT 357 1 "f(31/5,4/5 );" "6#-%\"fG6$*&\"#J\"\"\"\"\"&!\"\"*&\"\"%F(\"\"&F*" }{TEXT 352 1 " \+ " }{TEXT 358 6 " when " }{XPPEDIT 359 1 "``(5,1);" "6#-%!G6$\"\"&\"\" \"" }{TEXT 353 29 " is used as the base point? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 204 "a) 41/3 b) 44/3 c) 46/3 \+ d) 72/5 e) 74/5 \nf) 76/5 \+ g) 147/10 h) 149/10 i) 151/10 \+ j) 153/10" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 383 12 "Solution: d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f := (x,y) - > 6*sqrt(x-y^2); a := 5: b := 1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"fGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$-%%sqrtG6#,&9$\"\"\"*$) 9%\"\"#\"\"\"!\"\"\"\"'F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A := simplify( subs(\{u=a,v=b\}, diff(f(u,v),u)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG#\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "B := simplify( subs(\{u=a,v=b\}, diff(f(u,v),v)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "L := (x,y) -> f(a,b) + A*(x-a) + B*(y-b); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGR6$%\"xG%\"yG6\"6$%)operatorG%&a rrowGF),(-%\"fG6$%\"aG%\"bG\"\"\"*&%\"AGF3,&9$F3F1!\"\"F3F3*&%\"BGF3,& 9%F3F2F8F3F3F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "simpl ify(L(31/5,4/5)); \napprox := evalf(%); #for comparison in the next ex ecution group" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#s\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'approxG$\"++++S9!\")" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 149 "f(31/5,4/5); evalf(%); #to compare approxim ation with actual value\nperCentError := evalf( abs(L(31/5,4/5)-f(31/ 5,4/5)) / f(31/5,4/5) )*100*per_cent;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"$R\"\"\"\"#\"\"'\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M\"zZT\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-perCentEr rorG,$%)per_centG$\"+6;n#y\"!\"*" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 360 10 "18. Let " }{XPPEDIT 361 1 "z=x^3*y^4 " "6#/%\"zG*&%\"xG\"\"$ %\"yG\"\"%" }{TEXT 362 5 ", " }{XPPEDIT 363 1 "x=2*s^2+t-1" "6#/%\" xG,(*&\"\"#\"\"\"*$%\"sG\"\"#F(F(%\"tGF(\"\"\"!\"\"" }{TEXT 364 9 ", \+ and " }{XPPEDIT 365 1 "y = exp(s+2*t-3);" "6#/%\"yG-%$expG6#,(%\"sG \"\"\"*&\"\"#F*%\"tGF*F*\"\"$!\"\"" }{TEXT 366 13 ". Calculate" } {TEXT 370 1 " " }{XPPEDIT 371 1 "diff(z,s)" "6#-%%diffG6$%\"zG%\"sG" } {TEXT 367 1 " " }{TEXT 372 8 " when " }{XPPEDIT 373 1 "s=1" "6#/%\"s G\"\"\"" }{TEXT 368 8 " and " }{XPPEDIT 374 1 "t = 1;" "6#/%\"tG\" \"\"" }{TEXT 369 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 182 "a) 10 \+ b) 20 c) 30 d) 40 e) 50 \+ \nf) 60 g) 70 h) 80 i) 90 j) 100" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 380 12 "Solution: h" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Direct" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "z := x^3*y^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*&)%\"xG\" \"$\"\"\")%\"yG\"\"%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "s ubs(\{x=2*s^2+t-1,y=exp(s+2*t-3)\}, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&),(*$)%\"sG\"\"#\"\"\"F)%\"tG\"\"\"!\"\"F,\"\"$F*)-%$expG6#,(F (F,F+F)!\"$F,\"\"%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "dif f(%,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(),(*$)%\"sG\"\"#\"\"\"F *%\"tG\"\"\"!\"\"F-F*F+)-%$expG6#,(F)F-F,F*!\"$F-\"\"%F+F)F-\"#7*&)F& \"\"$F+F/F+F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(s ubs(\{s=1,t=1\}, %));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#!)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Chain Rule" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "z := x^3*y^4;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"zG*&)%\"xG\"\"$\"\"\")%\"yG\"\"%F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "A := diff(z,x)*Diff(x,s)+di ff(z,y)*Diff(y,s); \n#Comment: The capital D in \"Diff\" is used to de lay the differentiation until after the substitutions " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"AG,&*()%\"xG\"\"#\"\"\")%\"yG\"\"%F*-%%DiffG 6$F(%\"sG\"\"\"\"\"$*()F(F3F*)F,F3F*-F/6$F,F1F2F-" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 43 "A := subs(\{x=2*s^2+t-1,y=exp(s+2*t-3)\}, A) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"AG,&*(),(*$)%\"sG\"\"#\"\"\"F ,%\"tG\"\"\"!\"\"F/F,F-)-%$expG6#,(F+F/F.F,!\"$F/\"\"%F--%%DiffG6$F(F+ F/\"\"$*()F(F;F-)F2F;F--F96$F2F+F/F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "A := value(A);\n#Comment: The command \"value\" forc es Maple to perform the differentiations that we have delayed by using % Diff% instead of % diff\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"A G,&*(),(*$)%\"sG\"\"#\"\"\"F,%\"tG\"\"\"!\"\"F/F,F-)-%$expG6#,(F+F/F.F ,!\"$F/\"\"%F-F+F/\"#7*&)F(\"\"$F-F1F-F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A := simplify(subs(\{s=1,t=1\}, A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"#!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 536 9 "19. Let " }{XPPEDIT 537 1 "f(x,y) = 6*x^2+y^5;" "6#/-%\" fG6$%\"xG%\"yG,&*&\"\"'\"\"\"*$F'\"\"#F,F,*$F(\"\"&F," }{TEXT 538 11 " . Let u = " }{TEXT 543 1 "a" }{TEXT 544 4 "i + " }{TEXT 545 1 "b" } {TEXT 546 39 "j be a unit vector in the plane. Set " }{XPPEDIT 539 1 "psi(t) = f(1+a*t,1+b*t);" "6#/-%$psiG6#%\"tG-%\"fG6$,&\"\"\"\"\"\"* &%\"aGF-F'F-F-,&\"\"\"F-*&%\"bGF-F'F-F-" }{TEXT 540 35 " . What is the greatest value that " }{XPPEDIT 541 1 "D(psi)(0);" "6#--%\"DG6#%$psiG 6#\"\"!" }{TEXT 542 10 " can have?" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT -1 174 "a) 5 b) 6 c) 8 d) 10 e) 11 \nf) 12 g) 13 \+ h) 15 i) 16 j) 17" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 12 "Solution: g" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := (x,y) -> 6*x^2 + y^5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF),&*$)9$\"\"#\"\"\"\"\"'*$)9%\"\"&F2\"\"\"F) F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "gradient := subs(\{ x=1,y=1\},linalg[grad](f(x,y),[x,y]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)gradientG-%'vectorG6#7$\"#7\"\"&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "simplify(sqrt(gradient[1]^2 + gradient[2]^2));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 547 4 "20. " }{TEXT 256 1 " " }{TEXT 567 3 "Let" }{TEXT 568 3 " " } {TEXT 552 1 "u" }{TEXT 553 1 " " }{TEXT 554 2 " =" }{TEXT 555 1 " " } {TEXT -1 1 " " }{TEXT 257 1 "a" }{TEXT 550 1 "i" }{TEXT 551 4 " + b" } {TEXT 548 1 "j" }{TEXT 549 122 " be a unit vector that is tangent to the level curve of f at the point P = (3,4). \nIf the gradient of \+ f at P is 3" }{TEXT 569 1 "i" }{TEXT 570 23 " + 2j , then what is \+ " }{XPPEDIT 556 1 "a^2;" "6#*$%\"aG\"\"#" }{TEXT 258 3 " ? " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 278 "" 0 "" {TEXT -1 187 "a) 1/2 b) 4/5 c) 9/10 d) 16/17 \+ e) 25/26 \nf) 1/5 g) 9/13 h) 1 /10 i) 4/13 j) 16/25" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 566 13 "Solution: i" }{TEXT -1 39 "\nThe two unit vectors perpendicular \+ to " }{TEXT 256 1 " " }{TEXT 559 1 "3" }{TEXT 562 4 "i + " }{TEXT 560 1 "2" }{TEXT 561 1 "j" }{TEXT -1 9 " are " }{TEXT 256 1 " " } {XPPEDIT 19 1 "2/sqrt(13);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"#8!\"\"" } {TEXT 557 1 " " }{TEXT 563 4 "i -" }{TEXT 564 2 " " }{XPPEDIT 19 1 " 3/sqrt(13);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"#8!\"\"" }{TEXT 558 1 "j" } {TEXT -1 7 " and " }{XPPEDIT 19 1 "-2/sqrt(13);" "6#,$*&\"\"#\"\"\"- %%sqrtG6#\"#8!\"\"F+" }{TEXT 256 6 "i + " }{XPPEDIT 19 1 "3/sqrt(13) ;" "6#*&\"\"$\"\"\"-%%sqrtG6#\"#8!\"\"" }{TEXT 257 1 "j" }{TEXT -1 22 " . In either case, " }{XPPEDIT 19 1 "a^2 = 4/13;" "6#/*$%\"aG\"\"# *&\"\"%\"\"\"\"#8!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "17 15 0 0" 15 }{VIEWOPTS 1 1 0 1 1 1803 }