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"" {TEXT -1 80 "Warning, the protected names norm and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "magni tude := r -> sqrt(r[1]^2+r[2]^2+r[3]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*magnitudeGf*6#%\"rG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,(*$)& 9$6#\"\"\"\"\"#F5F5*$)&F36#F6F6F5F5*$)&F36#\"\"$F6F5F5F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "velocity := t -> subs(u=t, m ap(z->diff(z,u),r(u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocity Gf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%%subsG6$/%\"uG9$-%$mapG6$f*6#%\" zGF(F)F(-%%diffG6$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$2 ),r(u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accelerationGf*6#%\"tG 6\"6$%)operatorG%&arrowGF(-%%subsG6$/%\"uG9$-%$mapG6$f*6#%\"zGF(F)F(-% %diffG6$F1-%\"$G6$F0\"\"#F(F(F(-%\"rG6#F0F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "speed := t -> magnitude(velocity(t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&speedGf*6#%\"tG6\"6$%)operatorG%&ar rowGF(-%*magnitudeG6#-%)velocityG6#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "unitTangent := t -> subs(u=t,map(z->simplify(z/mag nitude(velocity(u))),velocity(u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%,unitTangentGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%%subsG6$/%\"uG9$- %$mapG6$f*6#%\"zGF(F)F(-%)simplifyG6#*&F1\"\"\"-%*magnitudeG6#-%)veloc ityG6#F0!\"\"F(F(F(F@F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "principalUnitNormal := t -> simplify(subs(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#>%4principalUnitNormalGf* 6#%\"tG6\"6$%)operatorG%&arrowGF(-%)simplifyG6#-%%subsG6$/%\"uG9$-%$ma pG6$f*6#%\"wGF(F)F(*&F4\"\"\"-%*magnitudeG6#-F66$f*6#%\"zGF(F)F(-%%dif fG6$F4F3F(F(F(-%,unitTangentG6#F3!\"\"F(F(F(-F66$f*FCF(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#>%*curvatureGf*6#%\"tG6\"6$%)operatorG %&arrowGF(*&-%*magnitudeG6#-%*crossprodG6$-%)velocityG6#9$-%-accelerat ionGF5\"\"\"-%&speedGF5!\"$F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 259 41 "A particle's trajectory is given by " }{XPPEDIT 258 1 "r(t) = `< `*t^2+2*t*` `,-2*t*` `,` `*t^2-t*`>`;" "6%/-%\"rG6#%\"tG,&*&%\"GF+F2" }{TEXT -1 3 " . " }}{PARA 262 "" 0 "" {TEXT -1 80 "The first nine questions of this exam concern this parameterized curv e at time " }{TEXT 478 4 "t = " }{TEXT -1 444 "0. The questions requ ire you to compute the speed, the acceleration, the unit tangent vecto r, the principal unit normal vector, the tangential and normal compone nts of acceleration, the curvature, the osculating plane, and the cent er of curvature. Calculations used for one question may be useful for \+ another question, so organize your work accordingly. Depending on your approach, you may want to do some of these nine questions out of orde r." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 371 39 "1. What is the speed of t he particle at" }{TEXT 374 2 " " }{XPPEDIT 375 1 "t=0" "6#/%\"tG\"\"! " }{TEXT 373 1 " " }{TEXT 376 2 "? " }{TEXT 377 2 " " }}{PARA 261 "" 0 "" {TEXT 372 4 "a) " }{XPPEDIT 19 1 "1" "6#\"\"\"" }{TEXT 388 15 " \+ b) " }{XPPEDIT 19 1 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT 387 13 " c) " }{XPPEDIT 19 1 "sqrt(3)" "6#-%%sqrtG6#\"\"$" } {TEXT 386 15 " d) " }{XPPEDIT 19 1 "2" "6#\"\"#" }{TEXT 385 15 " e) " }{XPPEDIT 19 1 "sqrt(5)" "6#-%%sqrtG6#\"\"&" }{TEXT 384 15 " \nf) " }{XPPEDIT 19 1 "sqrt(6)" "6#-%%sqrtG6 #\"\"'" }{TEXT 383 12 " g) " }{XPPEDIT 19 1 "sqrt(7)" "6#-%%sq rtG6#\"\"(" }{TEXT 382 13 " h) " }{XPPEDIT 19 1 "2*sqrt(2)" " 6#*&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 381 12 " i) " }{XPPEDIT 19 1 "3" "6#\"\"$" }{TEXT 380 14 " j) " }{XPPEDIT 19 1 "sqrt (10)" "6#-%%sqrtG6#\"#5" }{TEXT 379 1 " " }{TEXT 378 7 " \n" }} {PARA 0 "" 0 "" {TEXT 389 12 "Solution: i" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r := t -> [t^2+2*t , -2*t , t^2-t];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"tG6\"6$%)operatorG%&arrowGF(7%,&*$)9$\" \"#\"\"\"F2*&F1F2F0F2F2,$*&F1F2F0F2!\"\",&F.F2F0F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "velocity(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*&\"\"#\"\"\"%\"tGF'F'F&F'!\"#,&*&F&F'F(F'F'F'!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "speed(t);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*$,(*&\"\")\"\"\")%\"tG\"\"#F'F'*&\"\"%F'F)F'F' \"\"*F'#F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "speed(0);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 1 "2" }{TEXT 427 1 "." }{TEXT -1 2 " " }{TEXT 413 5 "Let " }{TEXT 424 1 " " }{XPPEDIT 423 1 "`<`*a,b,c*`>`" "6%*&%\"GF%" }{TEXT 422 50 " denote the unit tangent vector to thi s curve at" }{TEXT 419 2 " " }{XPPEDIT 420 1 "t=0" "6#/%\"tG\"\"!" } {TEXT 412 3 ". " }{TEXT -1 8 "What is " }{TEXT 425 1 " " }{XPPEDIT 426 1 "6*b" "6#*&\"\"'\"\"\"%\"bGF%" }{TEXT -1 3 " ? " }}{PARA 262 "" 0 "" {TEXT 395 4 "a) " }{XPPEDIT 390 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT 391 13 " " }{TEXT -1 3 "b) " }{TEXT 414 1 " " }{XPPEDIT 396 0 "-2" "6#,$\"\"#!\"\"" }{TEXT 397 14 " c) " }{XPPEDIT 398 0 "-3" "6#,$\"\"$!\"\"" }{TEXT 399 8 " " }{TEXT -1 8 " d) " }{XPPEDIT 400 0 "-4" "6#,$\"\"%!\"\"" }{TEXT 401 7 " " } {TEXT -1 6 " e)" }{TEXT 417 2 " " }{XPPEDIT 18 0 "-5" "6#,$\"\"&! \"\"" }{TEXT -1 3 " " }{TEXT 404 2 " " }}{PARA 257 "" 0 "" {TEXT 393 4 "f) " }{TEXT 392 2 " " }{XPPEDIT 394 0 "1" "6#\"\"\"" }{TEXT 403 17 " g) " }{XPPEDIT 402 0 "2" "6#\"\"#" }{TEXT 406 17 " h) " }{XPPEDIT 405 0 "3" "6#\"\"$" }{TEXT 408 19 " \+ i) " }{XPPEDIT 415 0 "4" "6#\"\"%" }{TEXT 416 18 " \+ j) " }{XPPEDIT 418 0 "5" "6#\"\"&" }{TEXT 411 1 " " }{TEXT 410 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 421 12 "Solution: d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "unitTangent(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*(\"\"#\"\"\",&%\"tGF'F'F'F',(*&\"\")F')F)F&F'F'*& \"\"%F'F)F'F'\"\"*F'#!\"\"F&F',$*&F&F'F*F1F2*&,&*&F&F'F)F'F'F'F2F'F*F1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "unitTangent(0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*(\"\"#\"\"\"\"\"*!\"\"F(#F'F&F',$ *(F&F'F(F)F(F*F),$*&F(F)F(F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify( 6*unitTangent(0)[2] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT 257 57 "3. What is t he magnitude of the acceleration vector at " }{XPPEDIT 258 1 "t=0" "6 #/%\"tG\"\"!" }{TEXT -1 2 "? " }}{PARA 261 "" 0 "" {TEXT 264 5 "a) \+ " }{XPPEDIT 19 1 "1;" "6#\"\"\"" }{TEXT 287 20 " b) " }{XPPEDIT 19 1 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT 286 19 " \+ c) " }{XPPEDIT 19 1 "3;" "6#\"\"$" }{TEXT 285 23 " \+ d) " }{XPPEDIT 19 1 "2;" "6#\"\"#" }{TEXT 284 18 " e ) " }{XPPEDIT 19 1 "sqrt(5);" "6#-%%sqrtG6#\"\"&" }{TEXT 283 16 " \+ \n f) " }{XPPEDIT 19 1 "sqrt(6);" "6#-%%sqrtG6#\"\"'" }{TEXT 282 17 " g) " }{XPPEDIT 19 1 "sqrt(7);" "6#-%%sqrtG6#\"\" (" }{TEXT 281 20 " h) " }{XPPEDIT 19 1 "2*sqrt(2);" "6 #*&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 280 16 " i) " } {XPPEDIT 19 1 "3;" "6#\"\"$" }{TEXT 279 19 " j) " } {XPPEDIT 19 1 "3*sqrt(2);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 278 8 " \n" }}{PARA 0 "" 0 "" {TEXT 350 12 "Solution: h" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "acceleration(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"#\"\"!F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "magnitude(acceleration(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"F%#F&F%F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 10 "4. Let " }{TEXT 645 1 " " }{TEXT 643 1 "p" }{TEXT 644 5 " i + " }{TEXT 641 1 "q" }{TEXT 642 5 " j + " }{TEXT 639 1 "r" }{TEXT 640 2 " k" }{TEXT 646 1 " " } {TEXT 595 1 " " }{TEXT 471 1 " " }{TEXT 596 59 "denote the principal u nit normal vector to this curve at " }{XPPEDIT 259 1 "t=0" "6#/%\"tG \"\"!" }{TEXT -1 2 ". " }{TEXT 593 7 "What is" }{TEXT -1 1 " " }{TEXT 594 1 " " }{XPPEDIT 451 1 "r/q;" "6#*&%\"rG\"\"\"%\"qG!\"\"" }{TEXT 450 2 " ?" }}{PARA 261 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 19 1 "1/3; " "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 16 " b) " }{XPPEDIT 19 1 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 14 " c) " }{XPPEDIT 19 1 "1/5;" "6#*&\"\"\"F$\"\"&!\"\"" }{TEXT -1 13 " \+ d) " }{XPPEDIT 19 1 "2/5;" "6#*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT -1 17 " e) " }{XPPEDIT 19 1 "3/5" "6#*&\"\"$\"\"\"\"\"&!\"\"" }} {PARA 261 "" 0 "" {TEXT -1 5 "f) " }{XPPEDIT 19 1 "3" "6#\"\"$" } {TEXT -1 16 " g) " }{XPPEDIT 19 1 "3/2" "6#*&\"\"$\"\"\"\" \"#!\"\"" }{TEXT -1 15 " h) " }{XPPEDIT 19 1 "5" "6#\"\"&" }{TEXT -1 15 " i) " }{XPPEDIT 19 1 "5/2" "6#*&\"\"&\"\"\"\" \"#!\"\"" }{TEXT -1 17 " j) " }{XPPEDIT 19 1 "5/3" "6#*& \"\"&\"\"\"\"\"$!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 428 12 "Solution: h" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "principalUnitNormal(0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*( \"\"(\"\"\"\"#^!\"\"\"#<#F'\"\"#F',$*(F,F'F(F)F*F+F',$*(\"#5F'F(F)F*F+ F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "dotprod(principalUnit Normal(0),unitTangent(0)); # A check" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "magnitude(princi palUnitNormal(0)); # Another check" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "principalUnitNorm al(0)[3]/principalUnitNormal(0)[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 2 " " }{TEXT 479 2 "5." }{TEXT -1 1 " " }{TEXT 430 68 "What is the curvature of the particle's path at the point for which " } {TEXT 432 1 " " }{XPPEDIT 433 1 "t=0" "6#/%\"tG\"\"!" }{TEXT 431 1 "? " }}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 435 7 "a) \+ " }{XPPEDIT 19 1 "sqrt(17)/3" "6#*&-%%sqrtG6#\"#<\"\"\"\"\"$!\"\" " }{TEXT -1 5 " " }{TEXT 436 11 " b) " }{XPPEDIT 19 1 "2*sq rt(17)/3" "6#*(\"\"#\"\"\"-%%sqrtG6#\"# " 0 "" {MPLTEXT 1 0 13 "curvature(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#<#\"\"\"\"\"#,(*$%\"tGF(\" \")F+\"\"%\"\"*F'#!\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "curvature(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"#<#\"\"\"\" \"##F(\"#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 407 1 " " }{TEXT 409 40 "6. What i s the tangential component, " }{XPPEDIT 262 1 "a[T]" "6#&%\"aG6#%\"T G" }{TEXT 260 29 " , of acceleration at at " }{XPPEDIT 256 1 "t=0 " "6#/%\"tG\"\"!" }{TEXT 452 4 " ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 453 174 "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 9 "Solution:" }{TEXT 490 2 " " }{TEXT 491 1 "b" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "acc := diff(speed(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$accG,$*&,(*$%\"tG\"\"#\"\")F)\"\"%\"\"*\"\"\"#!\"\"F *,&F)\"#;F,F.F.#F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a[T ] := simplify(subs(t=0,acc));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\" aG6#%\"TG#\"\"#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 " " 0 "" {TEXT 454 1 " " }{TEXT 585 3 "7. " }{TEXT 263 29 " What is the \+ normal component" }{TEXT 584 1 " " }{TEXT 586 3 " " }{XPPEDIT 261 1 "a[N]" "6#&%\"aG6#%\"NG" }{TEXT 259 1 " " }{TEXT 689 2 " " }{TEXT 589 2 " " }{TEXT 587 21 "of acceleration at " }{TEXT 651 1 "t" } {TEXT 652 4 " = 0" }{TEXT 653 1 " " }{TEXT 588 1 " " }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 647 4 "a) " } {XPPEDIT 455 1 "sqrt(17)/2;" "6#*&-%%sqrtG6#\"#<\"\"\"\"\"#!\"\"" } {TEXT 648 10 " b) " }{XPPEDIT 456 1 "sqrt(17);" "6#-%%sqrtG6#\"# <" }{TEXT 257 14 " c) " }{XPPEDIT 457 1 "3*sqrt(17)/2;" "6#* (\"\"$\"\"\"-%%sqrtG6#\"# " 0 "" {MPLTEXT 1 0 36 "a[N] := sqrt((2*sqrt(2))^2-( 2/3)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#%\"NG,$*$\"#<#\" \"\"\"\"##F-\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 85 "simplify([seq(a[T]*unitTangent(0)[j]+a[N]*pr incipalUnitNormal(0)[j],j=1..3)]); #check" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"#\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 467 29 "8. The osculatin g plane at " }{XPPEDIT 469 1 "t = 0" "6#/%\"tG\"\"!" }{TEXT 257 2 " \+ " }{TEXT 468 56 " has equation 2x + By + Cz = D. What is \+ B ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 470 4 "a ) " }{XPPEDIT 269 1 "1;" "6#\"\"\"" }{TEXT 654 17 " b) \+ " }{XPPEDIT 271 1 "sqrt(17);" "6#-%%sqrtG6#\"#<" }{TEXT 257 13 " \+ c) " }{XPPEDIT 272 1 "2;" "6#\"\"#" }{TEXT 259 15 " \+ " }{TEXT 690 3 " d)" }{TEXT 691 2 " " }{XPPEDIT 273 1 "2*sqrt(17);" " 6#*&\"\"#\"\"\"-%%sqrtG6#\"# " 0 "" {MPLTEXT 1 0 76 "Binormal := map(simplify,crossprod( unitTangent(0),principalUnitNormal(0)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)BinormalG-%'VECTORG6#7%,$*$\"#<#\"\"\"\"\"##!\"#F+,$ F*#!\"$F+,$F*#F.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r(0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "Binormal[1]*(x-0) + Binormal[2]*(y-0) + Binorm al[3]*(z-0) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&\"#<#\"\"\"\" \"#%\"xGF(#!\"#F&*&F&F'%\"yGF(#!\"$F&*&F&F'%\"zGF(#F)F&\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "map(z->expand(-sqrt(17)*z), \+ \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"xG\"\"#%\"yG\"\"$%\"zG! \"#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 473 15 "9. What \+ is the " }{TEXT 474 1 "y" }{TEXT 475 70 "-component of the center of c urvature for the parameterized curve at " }{TEXT 673 1 "t" }{TEXT 674 6 " = 0? " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 657 4 "a) " } {XPPEDIT 492 1 "7/15" "6#*&\"\"(\"\"\"\"#:!\"\"" }{TEXT 658 16 " \+ b) " }{XPPEDIT 493 1 "5/8" "6#*&\"\"&\"\"\"\"\")!\"\"" }{TEXT 257 14 " c) " }{XPPEDIT 494 1 "2/3" "6#*&\"\"#\"\"\"\"\"$!\" \"" }{TEXT 259 10 " " }{TEXT 696 2 "d)" }{TEXT 697 3 " " } {XPPEDIT 495 1 "3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 260 17 " \+ e) " }{XPPEDIT 496 1 "4/7" "6#*&\"\"%\"\"\"\"\"(!\"\"" }{TEXT 659 7 " \n" }{TEXT 497 5 "f) " }{XPPEDIT 498 1 "9/17" "6#*&\"\" *\"\"\"\"# " 0 "" {MPLTEXT 1 0 67 "[seq(r(0)[j]+ (1/c urvature(0))*principalUnitNormal(0)[j],j=1.. 3)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"#j\"#M#\"\"*\"#<#\"#XF)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 477 45 "10. What is the arc length of the space curve" }{TEXT 592 5 " r( " }{TEXT 681 1 "t" }{TEXT 682 7 " ) = ( " }{XPPEDIT 687 0 "t^3;" "6#*$%\"tG\"\"$" }{TEXT 686 2 "/ " }{TEXT 683 5 "3 + 3" }{TEXT 685 1 "t" }{TEXT 684 10 " ) i + 4" }{XPPEDIT 679 0 "t;" "6#%\"tG" } {TEXT 678 1 " " }{TEXT 680 4 "j +" }{TEXT 676 1 " " }{XPPEDIT 677 0 " t^2;" "6#*$%\"tG\"\"#" }{TEXT 675 4 " k " }{TEXT 590 13 " , 0 < t < \+ 1" }{TEXT 591 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 598 2 "a)" }{TEXT 700 2 " " }{TEXT 699 3 "4/3" }{TEXT 661 2 " " }{TEXT 257 88 " b) 8/3 c) 16/3 d ) 32/3 e) 11/6 \nf) " }{XPPEDIT 605 1 "8*sqrt(2);" " 6#*&\"\")\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 606 14 " g) " } {XPPEDIT 603 1 "10*sqrt(2);" "6#*&\"#5\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 604 12 " h) " }{XPPEDIT 602 1 "2*sqrt(3);" "6#*&\"\"#\"\"\"-%% sqrtG6#\"\"$F%" }{TEXT 601 16 " i) " }{XPPEDIT 600 1 "3*sq rt(3);" "6#*&\"\"$\"\"\"-%%sqrtG6#F$F%" }{TEXT 599 15 " j) \+ " }{XPPEDIT 608 1 "4*sqrt(6)" "6#*&\"\"%\"\"\"-%%sqrtG6#\"\"'F%" } {TEXT 607 2 " " }{TEXT 698 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 597 12 "Solution : c" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "r := t -> [t^3/3+3*t,4*t,t^2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"tG6\"6$%)operatorG%&arrowGF(7%,&*$9$\"\"$# \"\"\"F0F/F0,$F/\"\"%*$F/\"\"#F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "velocity(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&* $%\"tG\"\"#\"\"\"\"\"$F(\"\"%,$F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "speed(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$*$,&\" \"&\"\"\"*$%\"tG\"\"#F'F*#F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "int(magnitude(velocity(t)),t=0..1); # exact" }}{PARA 11 "" 1 " " {XPPMATH 20 "6##\"#;\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 4 "11. \+ " }{TEXT 356 75 "A particle is moving along a space curve in such a wa y that its speed is " }{XPPEDIT 357 1 "t+t^2;" "6#,&%\"tG\"\"\"*$F$ \"\"#F%" }{TEXT 358 144 " at time t. The curvature of the particle' s path is 3/4 at time t = 1. What is the magnitude of its accele ration vector at that time? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 359 5 "a) " } {XPPEDIT 19 1 "3*sqrt(2);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 361 17 " b) " }{XPPEDIT 19 1 "4*sqrt(2);" "6#*&\"\"%\"\" \"-%%sqrtG6#\"\"#F%" }{TEXT 362 13 " c) " }{XPPEDIT 19 1 "5*s qrt(2);" "6#*&\"\"&\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 363 17 " \+ d) " }{XPPEDIT 19 1 "6*sqrt(2);" "6#*&\"\"'\"\"\"-%%sqrtG6#\"\"#F% " }{TEXT 360 14 " e) " }{XPPEDIT 19 1 "7*sqrt(2);" "6#*&\"\" (\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 364 12 " \n\nf) " }{XPPEDIT 19 1 "2*sqrt(3);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 365 15 " \+ g) " }{XPPEDIT 19 1 "3*sqrt(3);" "6#*&\"\"$\"\"\"-%%sqrtG6#F$F% " }{TEXT 366 16 " h) " }{XPPEDIT 19 1 "4*sqrt(10);" "6#*& \"\"%\"\"\"-%%sqrtG6#\"#5F%" }{TEXT 367 14 " i) " }{XPPEDIT 19 1 "5*sqrt(3);" "6#*&\"\"&\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 368 15 " \+ j) " }{XPPEDIT 19 1 "6*sqrt(3);" "6#*&\"\"'\"\"\"-%%sqrtG6# \"\"$F%" }{TEXT 369 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 370 12 "Solution: a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v := t + t^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG,&%\"tG\" \"\"*$F&\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a_T := d iff(v,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_TG,&\"\"\"F&%\"tG\" \"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a_T := subs(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#>%$a_NG*&%&kappaG\"\"\",&%\"tGF'*$F)\"\"#F'F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a_N := subs(\{t=1,kappa=3/4 \}, 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#>%$accG,$*$\"\"##\"\"\"F'\"\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 609 23 "12. For the function" }{TEXT 485 3 " " }{XPPEDIT 257 1 "f(x,y) = 3*x^2+y^3+x*y^2;" "6#/-%\"fG6$% \"xG%\"yG,(*&\"\"$\"\"\"*$F'\"\"#F,F,*$F(F+F,*&F'F,*$F(F.F,F," }{TEXT 610 3 " , " }{TEXT 611 11 " calculate" }{TEXT 486 7 " \n\n " } {TEXT 612 5 " " }{XPPEDIT 259 1 "Diff(f(x,y),x,x)*Diff(f(x,y),y,y) -(Diff(f(x,y),x,y))^2" "6#,&*&-%%DiffG6%-%\"fG6$%\"xG%\"yGF+F+\"\"\"-F &6%-F)6$F+F,F,F,F-F-*$-F&6%-F)6$F+F,F+F,\"\"#!\"\"" }{TEXT 260 2 " \n " }{TEXT 583 1 "\n" }{TEXT 480 27 "at the point ( 4 , 1 ) . " } {TEXT 481 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 148 "a) 86 b) 68 c) 44 d) 2 1 e) 80 \nf) - 72 g) -20 h) - 48 i ) - 68 j) - 84\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 9 "Solution:" }{TEXT 512 2 " " }{TEXT 511 1 "e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f := (x,y) -> 3*x^2 + y^3 + x*y^2: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "A := diff(f(x,y),x,x)*d iff(f(x,y),y,y)-(diff(f(x,y),x,y))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,(%\"yG\"#O%\"xG\"#7*$F&\"\"#!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(\{x=4,y=1\},A );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#!)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 42 "13. A moving particl e starts at position " }{TEXT 620 1 "r" }{TEXT 621 3 "(0)" }{TEXT 622 1 " " }{TEXT 623 1 "=" }{TEXT 624 5 " i . " }{TEXT -1 34 " Its ini tial velocity is given by " }{TEXT 625 2 " v" }{TEXT 613 3 "(0)" } {TEXT 626 12 " = i - j + k" }{TEXT -1 24 ". \nIts acceleration is " } {TEXT 616 2 " a" }{TEXT 614 3 "(t)" }{TEXT 617 3 " = " }{TEXT 619 2 "4 t" }{TEXT 618 5 " i + " }{TEXT 615 19 "6t j + k . If " }{TEXT 630 1 "r" }{TEXT 627 6 " ( 3 )" }{TEXT 629 4 " = " }{TEXT 631 1 "b" } {TEXT 632 7 " i + " }{TEXT 628 2 "c " }{TEXT 635 1 "j" }{TEXT 636 5 " + d " }{TEXT 633 1 "k" }{TEXT 634 21 " , what is b + c ?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 175 "a) 2 0 b) 22 c) 24 d) 26 e) \+ 29 \nf) 32 g) 36 h) 38 \+ i) 42 j) 46" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 352 13 "Solution: j " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "v1 := t -> 1 + int(4*u, u = 0 .. t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1Gf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&\"\"\"F --%$intG6$,$%\"uG\"\"%/F2;\"\"!9$F-F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 6 "v1(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$ *$%\"tG\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "r1 := t - > 1 + int(v1(s), s = 0 .. t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1 Gf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&\"\"\"F--%$intG6$-%#v1G6#%\"sG/F 4;\"\"!9$F-F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "r1(t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$%\"tGF$*$F%\"\"$#\"\"#F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "v2 := t -> -1 + int(6*u , u = 0 .. t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2Gf*6#%\"tG6\"6$ %)operatorG%&arrowGF(,&!\"\"\"\"\"-%$intG6$,$%\"uG\"\"'/F3;\"\"!9$F.F( F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "v2(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&!\"\"\"\"\"*$%\"tG\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "r2 := t -> 0 + int(v2(s), s = 0 .. \+ t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"tG6\"6$%)operatorG %&arrowGF(-%$intG6$-%#v2G6#%\"sG/F2;\"\"!9$F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "r2(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& %\"tG!\"\"*$F$\"\"$\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "r1(3) + r2(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#Y" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 263 13 "14. The plane" }{TEXT 701 1 " " }{TEXT 482 3 " " }{XPPEDIT 291 1 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 290 2 " " }{TEXT 292 24 " inte rsects the graph of" }{TEXT 293 2 " " }{XPPEDIT 294 1 "z = x*y^3-2*x; " "6#/%\"zG,&*&%\"xG\"\"\"*$%\"yG\"\"$F(F(*&\"\"#F(F'F(!\"\"" }{TEXT 289 2 " " }{TEXT 295 64 " in a curve. \n The tangent line to this curve at the point " }{XPPEDIT 297 1 "``(1, 2, 6);" "6#-%!G6%\"\"\" \"\"#\"\"'" }{TEXT 296 1 " " }{TEXT 298 30 " passes through the \n po int " }{XPPEDIT 299 1 "``(1,4,c);" "6#-%!G6%\"\"\"\"\"%%\"cG" }{TEXT 300 12 ". What is " }{TEXT 302 1 " " }{XPPEDIT 303 1 "c" "6#%\"cG" } {TEXT 301 5 "? " }}{PARA 258 "" 0 "" {TEXT 304 5 "\na) " } {XPPEDIT 305 1 "8;" "6#\"\")" }{TEXT 306 12 " b) " }{XPPEDIT 307 1 "12" "6#\"#7" }{TEXT 308 12 " c) " }{XPPEDIT 309 1 "16" "6#\"#;" }{TEXT 310 11 " d) " }{XPPEDIT 311 1 "18" "6#\"#=" } {TEXT 312 12 " e) " }{XPPEDIT 313 1 "22" "6#\"#A" }{TEXT 314 12 " \nf) " }{XPPEDIT 315 1 "26" "6#\"#E" }{TEXT 316 12 " \+ g) " }{XPPEDIT 317 1 "30" "6#\"#I" }{TEXT 318 12 " h) " } {XPPEDIT 319 1 "32" "6#\"#K" }{TEXT 320 11 " i) " }{XPPEDIT 321 1 "36" "6#\"#O" }{TEXT 322 13 " j) " }{XPPEDIT 324 1 "40 " "6#\"#S" }{TEXT -1 1 " " }{TEXT 323 3 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 354 12 "Solution: g" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := (x,y) -> x*y^3-2*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&9$\"\"\"9%\"\"$F0F/!\"#F )F)6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "m := subs(\{x=1,y =2\}, diff(f(x,y),y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "r := t -> [1,2+t,6+m*t]; \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"tG6\"6$%)operatorG%&a rrowGF(7%\"\"\",&\"\"#F-9$F-,&\"\"'F-*&%\"mGF-F0F-F-F(F(6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "t_0 := solve( r(t)[2]=4,t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t_0G\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(t=t_0,r(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"\"\"%\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 449 9 "15. Let " }{TEXT 662 1 "L" } {TEXT 663 80 " be the vertical line that is defined by the Cartesian \+ equations x = 1 , y = " }{TEXT 664 1 "-" }{TEXT 665 41 " 1. The ta ngent plane to the graph of " }{TEXT 447 2 " " }{XPPEDIT 448 1 "z=3* x^2+2*y+3" "6#/%\"zG,(*&\"\"$\"\"\"*$%\"xG\"\"#F(F(*&F+F(%\"yGF(F(F'F( " }{TEXT 446 55 " at the point ( 1 , - 2 , 2 ) is intersected by \+ " }{TEXT 666 1 "L" }{TEXT 667 29 " at the point ( 1 , -1 , " } {TEXT 579 1 "c" }{TEXT 580 17 " ). \nWhat is " }{TEXT 577 1 "c" } {TEXT 578 4 " ? \n" }}{PARA 0 "" 0 "" {TEXT 504 4 "a) " }{XPPEDIT 257 1 "-4" "6#,$\"\"%!\"\"" }{TEXT -1 6 " " }{TEXT 505 4 "b) " } {XPPEDIT 259 1 "-3" "6#,$\"\"$!\"\"" }{TEXT 260 9 " c)" }{TEXT 506 2 " " }{XPPEDIT 261 1 "-2" "6#,$\"\"#!\"\"" }{TEXT 262 9 " d ) " }{XPPEDIT 263 1 "-1" "6#,$\"\"\"!\"\"" }{TEXT 264 9 " e)" } {TEXT 265 2 " " }{XPPEDIT 638 1 "0" "6#\"\"!" }{TEXT 266 1 " " } {TEXT 503 6 " \n" }{TEXT 507 2 "f)" }{TEXT 508 1 " " }{XPPEDIT 267 1 "1" "6#\"\"\"" }{TEXT 268 12 " " }{TEXT 509 2 "g)" } {TEXT 510 3 " " }{XPPEDIT 269 1 "2" "6#\"\"#" }{TEXT 270 7 " \+ " }{TEXT 483 3 " h)" }{TEXT 484 2 " " }{XPPEDIT 271 1 "3" "6#\"\"$" } {TEXT 272 11 " i) " }{XPPEDIT 273 1 "4" "6#\"\"%" }{TEXT 274 10 " 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 -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 13 "Solution: i \n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f := (x,y) -> 3*x^2+2*y+3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(*$ 9$\"\"#\"\"$9%F0F1\"\"\"F)F)6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "testeq( f(1,-2) = 2); #Verification of the asserted point" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "tp_eqn := subs(\{x=1,y=-2\}, diff(f(x,y),x))*(x-1) + subs(\{x=1,y=-2\}, diff(f(x,y),y))*(y-(-2))-(z-2) = 0;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'tp_eqnG/,(%\"xG\"\"'%\"yG\"\"#%\"zG!\"\"\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "c = solve( subs(\{x=1, y = -1\},tp_eqn) , z); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG\"\" %" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 4 "16. " }{TEXT 565 31 "The vector-valued function\n " } {TEXT 563 10 " " }{XPPEDIT 564 1 "r(u,v) = (3*v)*i+u^2*j+(u+2 *v)*k;" "6#/-%\"rG6$%\"uG%\"vG,(*(\"\"$\"\"\"F(F,%\"iGF,F,*&F'\"\"#%\" jGF,F,*&,&F'F,*&F/F,F(F,F,F,%\"kGF,F," }{TEXT 560 33 " \n \nof the t wo real variables " }{TEXT 568 1 "u" }{TEXT 569 6 " and " }{TEXT 570 1 "v" }{TEXT 571 1 " " }{TEXT 562 1 " " }{TEXT 561 103 "defines a \+ surface. The plane that is tangent to this surface at the \npoint ( 6, 1, 5 ) has equation " }{TEXT 566 17 "Ax + y + Cz = D" }{TEXT 567 13 ". What is " }{TEXT 575 1 "D" }{TEXT 576 3 "? \n" }}{PARA 256 "" 0 "" {TEXT 573 4 "a) " }{XPPEDIT 540 1 "-4" "6#,$\"\"%!\"\"" } {TEXT 541 13 " b) " }{XPPEDIT 542 1 "-3" "6#,$\"\"$!\"\"" } {TEXT 543 12 " c) " }{XPPEDIT 544 1 "-2" "6#,$\"\"#!\"\"" } {TEXT 545 9 " d) " }{XPPEDIT 546 1 "-1" "6#,$\"\"\"!\"\"" }{TEXT 547 10 " e) " }{XPPEDIT 548 1 "0" "6#\"\"!" }{TEXT 549 9 " \+ \nf)" }{TEXT 574 3 " " }{XPPEDIT 550 1 "1" "6#\"\"\"" }{TEXT 551 17 " g) " }{XPPEDIT 552 1 "2" "6#\"\"#" }{TEXT 553 8 " \+ " }{TEXT 572 5 " h) " }{XPPEDIT 554 1 "3" "6#\"\"$" }{TEXT 555 12 " i) " }{XPPEDIT 556 1 "4" "6#\"\"%" }{TEXT 557 14 " j ) " }{XPPEDIT 559 1 "5" "6#\"\"&" }{TEXT 702 1 " " }{TEXT 558 2 " " }{TEXT 703 1 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 14 "Solution: d \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "r := (u,v) -> [3*v , u^2 , u+2*v];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6$%\"uG%\"v G6\"6$%)operatorG%&arrowGF)7%,$*&\"\"$\"\"\"9%F1F1*$)9$\"\"#F1,&F5F1*& F6F1F2F1F1F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "r(u,v); P0 := r(1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&\"\"$\"\"\"% \"vGF'F'*$)%\"uG\"\"#F',&F+F'*&F,F'F(F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P0G7%\"\"'\"\"\"\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "v1 := map( w -> diff(w,u), r(u,v) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G7%\"\"!,$*&\"\"#\"\"\"%\"uGF*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "v2 := map(w->diff(w,v), r(u, v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G7%\"\"$\"\"!\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "N1 :=convert(linalg[crosspro d](v1,v2),list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N1G7%,$*&\"\"% \"\"\"%\"uGF)F)\"\"$,$*&\"\"'F)F*F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "N := evalm(subs(\{u=1,v=2\},N1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"NG-%'vectorG6#7%\"\"%\"\"$!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "eqn := expand(N[1]*('x'-P0[1]) + N[2]*('y '-P0[2]) + N[3]*('z'-P0[3])) = 0 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$eqnG/,**&\"\"%\"\"\"%\"xGF)F)\"\"$F)*&F+F)%\"yGF)F)*&\"\"'F)%\"zGF) !\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(z->z/3, % );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**(\"\"%\"\"\"\"\"$!\"\"%\"xG F'F'F'F'%\"yGF'*&\"\"#F'%\"zGF'F)\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 325 8 "17. Let" }{TEXT 329 2 " " } {XPPEDIT 330 1 "f(x,y) = 6*sqrt(x+2*y^2);" "6#/-%\"fG6$%\"xG%\"yG*&\" \"'\"\"\"-%%sqrtG6#,&F'F+*&\"\"#F+*$F(F1F+F+F+" }{TEXT 326 42 ". Wha t is the linear approximation of " }{TEXT 331 1 " " }{XPPEDIT 332 1 "f(4/3,7/3);" "6#-%\"fG6$*&\"\"%\"\"\"\"\"$!\"\"*&\"\"(F(F)F*" }{TEXT 327 1 " " }{TEXT 333 7 " when " }{XPPEDIT 334 1 "``(1,2);" "6#-%!G6$ \"\"\"\"\"#" }{TEXT 328 29 " is used as the base point? " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 175 "a) 58/3 b) 59/3 c) 61/3 \+ d) 62/3 e) 64/3 \nf) 65/3 g) 67/3 \+ h) 20 i) 21 j) 22" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 355 12 "Solution: i" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f := (x,y) -> 6*sqrt(x+2*y^2); a : = 1: b := 2:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6 \"6$%)operatorG%&arrowGF),$-%%sqrtG6#,&9$\"\"\"*$9%\"\"#F6\"\"'F)F)6\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A := simplify( subs(\{u =a,v=b\}, diff(f(u,v),u)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "B := simplify( su bs(\{u=a,v=b\}, diff(f(u,v),v)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"BG\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "L := (x,y) - > f(a,b) + A*(x-a) + B*(y-b); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" LGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(-%\"fG6$%\"aG%\"bG\"\"\"*& %\"AGF3,&9$F3F1!\"\"F3F3*&%\"BGF3,&9%F3F2F8F3F3F)F)6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(L(4/3,7/3)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 255 "### WARNING: incomplete string; use \" to end the string (note t hat the ditto operator is now % instead of \")\nf(4/3,7/3); evalf(\"); #to compare approximation with actual value\nperCentError := evalf( \+ abs(L(4/3,7/3)-f(4/3,7/3)) / f(4/3,7/3) )*100*per_cent;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*$\"$5\"#\"\"\"\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'p%-pe rCentErrorG,$%)per_centG$\"+q)=d8\"!#5" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 335 10 "18. Let " }{XPPEDIT 336 1 "z=x^3*y^4 " "6#/%\"zG*&%\"x G\"\"$%\"yG\"\"%" }{TEXT 337 5 ", " }{XPPEDIT 338 1 "x=2*t^2+t-1" " 6#/%\"xG,(*&\"\"#\"\"\"*$%\"tGF'F(F(F*F(F(!\"\"" }{TEXT 339 9 ", and \+ " }{XPPEDIT 340 1 "y = t^2+2*t-2" "6#/%\"yG,(*$%\"tG\"\"#\"\"\"*&F(F )F'F)F)F(!\"\"" }{TEXT 341 13 ". Calculate" }{TEXT 344 1 " " } {XPPEDIT 345 1 "dz/dt" "6#*&%#dzG\"\"\"%#dtG!\"\"" }{TEXT 342 1 " " } {TEXT 346 8 " when " }{XPPEDIT 347 1 "t = 1;" "6#/%\"tG\"\"\"" } {TEXT 343 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 183 "a) 164 b) 168 c) 172 d) 176 e) 180 \+ \nf) 184 g) 188 h) 192 i) \+ 196 j) 200" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 353 12 "Solution: g" }} {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\"\" $%\"yG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "mess := subs (\{x=2*t^2+t-1,y=t^2+2*t-2\}, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%messG*&,(*$%\"tG\"\"#F)F(\"\"\"!\"\"F*\"\"$,(F'F*F(F)!\"#F*\"\"%" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "messier := diff(mess,t);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(messierG,&*(,(*$%\"tG\"\"#F*F)\"\" \"!\"\"F+F*,(F(F+F)F*!\"#F+\"\"%,&F)F/F+F+F+\"\"$*(F'F1F-F1,&F)F*F*F+F +F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(subs(t=1, m essier));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$)=" }}}{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\"\"$%\"yG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "B := diff(z,x)*Diff(x,t)+diff(z,y)*Diff(y,t); \n# Comment: The capital D in \"Diff\" is used to delay the differentiatio n until after the substitutions " }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"BG,&*(%\"xG\"\"#%\"yG\"\"%-%%DiffG6$F'%\"tG\"\"\"\"\"$*(F'F0F)F0-F ,6$F)F.F/F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "B := subs(\{ x=2*t^2+t-1,y=t^2+2*t-2\}, B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" BG,&*(,(*$%\"tG\"\"#F*F)\"\"\"!\"\"F+F*,(F(F+F)F*!\"#F+\"\"%-%%DiffG6$ F'F)F+\"\"$*(F'F3F-F3-F16$F-F)F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "B := value(B);\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#>%\"B G,&*(,(*$%\"tG\"\"#F*F)\"\"\"!\"\"F+F*,(F(F+F)F*!\"#F+\"\"%,&F)F/F+F+F +\"\"$*(F'F1F-F1,&F)F*F*F+F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(subs(t=1, B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$ )=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 487 20 "19. Suppose that " }{XPPEDIT 533 1 "F(x,y,z) = x*y^2*z^3;" "6#/-%\"FG6%%\"xG%\"yG%\"zG*(F'\"\"\"*$F(\"\"#F+F)\"\"$" }{TEXT 532 4 ", " }{XPPEDIT 536 1 "f(s,t) = s+t;" "6#/-%\"fG6$%\"sG%\"tG,&F'\"\" \"F(F*" }{TEXT 534 5 ", " }{XPPEDIT 537 1 "g(s,t) = 2*s-t;" "6#/-% \"gG6$%\"sG%\"tG,&*&\"\"#\"\"\"F'F,F,F(!\"\"" }{TEXT 535 4 ", " } {TEXT 581 4 "and " }{TEXT 582 3 " " }{XPPEDIT 538 1 "h(s,t) = s+3*t; " "6#/-%\"hG6$%\"sG%\"tG,&F'\"\"\"*&\"\"$F*F(F*F*" }{TEXT 539 11 " . \+ \nDefine " }{XPPEDIT 669 1 "G(s,t) = F(f(s,t),g(s,t),h(s,t));" "6#/-% \"GG6$%\"sG%\"tG-%\"FG6%-%\"fG6$F'F(-%\"gG6$F'F(-%\"hG6$F'F(" }{TEXT 668 38 ". \nCalculate the partial derivative " }{XPPEDIT 671 1 "diff (G(s,t),s);" "6#-%%diffG6$-%\"GG6$%\"sG%\"tGF)" }{TEXT 670 1 " " } {TEXT 672 25 " at the point ( 1 , 0 )." }{TEXT -1 0 "" }}{PARA 260 " " 0 "" {TEXT -1 163 "a) 8 b) 9 c) 10 \+ d) 12 e) 14 \nf) 15 g) 16 \+ h) 18 i) 20 j) 24" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 637 13 "Solution: j " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "w := x*y^2*z^3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG*(%\"xG\"\"\"%\"yG\"\"#%\"zG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "A := diff(w,x)*diff(s+t,s) + diff(w ,y)*diff(2*s-t,s) + diff(w,z)*diff(s+3*t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,(*&%\"yG\"\"#%\"zG\"\"$\"\"\"*(%\"xGF+F'F+F)F*\" \"%*(F-F+F'F(F)F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "B := subs(\{x=s+t, y = 2*s-t, z = s+3*t\}, A );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,(*&,&%\"sG\"\"#%\"tG!\"\"F),&F(\"\"\"F*\"\"$F.F- *(,&F(F-F*F-F-F'F-F,F.\"\"%*(F0F-F'F)F,F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(\{s=1,t=0\},B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "C := diff( s ubs(\{x=s+t,y = 2*s-t,z = s+3*t\}, w), s); #Alternative (longer) calcu lation, without Chain Rule" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG,( *&,&%\"sG\"\"#%\"tG!\"\"F),&F(\"\"\"F*\"\"$F.F-*(,&F(F-F*F-F-F'F-F,F. \"\"%*(F0F-F'F)F,F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "su bs(\{s=1,t=0\},C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 488 4 "20. " } {TEXT -1 1 " " }{TEXT 517 27 "The graph of the equation " }{XPPEDIT 518 1 "x^3*z^2-y^2+z^7 +2 = 0" "6#/,**&%\"xG\"\"$%\"zG\"\"#\"\"\"*$%\" yGF)!\"\"*$F(\"\"(F*F)F*\"\"!" }{TEXT 516 41 " is a surface that con tains the point " }{XPPEDIT 514 1 "`(`*1,2,1*`)`;" "6%*&%\"(G\"\"\"F% F%\"\"#*&F%F%%\")GF%" }{TEXT 513 45 " . \nThe part of the surface near the point " }{XPPEDIT 519 1 "`(`*1,2,1*`)`;" "6%*&%\"(G\"\"\"F%F%\" \"#*&F%F%%\")GF%" }{TEXT 520 22 " implicitly defines " }{XPPEDIT 522 1 "z" "6#%\"zG" }{TEXT 521 19 " as a function of " }{TEXT 530 1 " " }{XPPEDIT 528 1 "x" "6#%\"xG" }{TEXT 523 5 " and " }{XPPEDIT 529 1 "y" "6#%\"yG" }{TEXT 524 60 " near the \npoint (1,2). Calculate the partial derivative " }{TEXT 531 1 " " }{XPPEDIT 525 1 "z[x](1,2)" "6# -&%\"zG6#%\"xG6$\"\"\"\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 515 150 "a) -1/3 b) 1/3 \+ c) -1/2 d) 1/2 e) -1 \nf) 1 g) \+ -2 h) 2 i) -3 j) 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 489 11 "Solution: " }{TEXT 526 1 " " }{TEXT 527 1 "a" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "F := (x,y,z) -> x^3*z^2 - y^2 + z^7 + 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arro wGF*,**&9$\"\"$9&\"\"#\"\"\"*$9%F3!\"\"*$F2\"\"(F4F3F4F*F*6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "P0 := [1,2,1];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#P0G7%\"\"\"\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "F(P0[1],P0[2],P0[3]);#Check given point is on surf ace!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "subs(\{x=P0[1],y=P0[2],z=P0[3]\},-diff(F(x,y,z), x)/diff(F(x,y,z),z));" }}{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 "" }}}} {MARK "9 0 11" 4 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }