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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 "Times" 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 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 "" 3 263 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 "" 3 266 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 "" 3 267 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 "" 258 268 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 "" 258 269 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 "" 0 272 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 "" 3 273 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 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 " Math 2331 Practice Exam \+ 2" }}{PARA 0 "" 0 "" {TEXT 260 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 1 "1" }{TEXT 316 2 ". " }{TEXT 294 28 "The vector valued function " }{TEXT 383 2 " " }{XPPEDIT 315 1 "r(t)=`<`*cos(t),sin(t),z(t)*`>`" "6%/-%\"rG6#%\"tG*&%\"GF*" }{TEXT 373 61 " is a par ameterization of the intersection of the cylinder" }{TEXT 384 2 " " } {XPPEDIT 381 1 "x^2+y^2=1" "/,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'F'" } {TEXT 379 16 " and the plane " }{TEXT 385 2 " " }{XPPEDIT 382 1 "x+y +z=2" "/,(%\"xG\"\"\"%\"yGF%%\"zGF%\"\"#" }{TEXT 380 13 ". What is \+ " }{TEXT 387 1 " " }{XPPEDIT 388 1 "z*`'`(Pi/3)" "*&%\"zG\"\"\"-%\"'G6 #*&%#PiGF$\"\"$!\"\"F$" }{TEXT 386 3 "?\n " }}{PARA 268 "" 0 "" {TEXT -1 6 "a) " }{XPPEDIT 19 1 "sqrt(3)/2" "*&-%%sqrtG6#\"\"$\"\"\"\"\"# !\"\"" }{TEXT -1 5 "\nb) " }{XPPEDIT 19 1 "(sqrt(3)+1)/2" "*&,&-%%sqr tG6#\"\"$\"\"\"F(F(F(\"\"#!\"\"" }{TEXT -1 14 " \nc) " } {XPPEDIT 19 1 "(sqrt(3)-1)/2" "*&,&-%%sqrtG6#\"\"$\"\"\"F(!\"\"F(\"\"# F)" }{TEXT -1 12 " \nd) " }{XPPEDIT 19 1 "(sqrt(3)-2)/2" "*&,&- %%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F(F)F*" }{TEXT -1 7 " \ne) " } {XPPEDIT 19 1 "(1-sqrt(3))/2" "*&,&\"\"\"F$-%%sqrtG6#\"\"$!\"\"F$\"\"# F)" }{TEXT -1 6 " \nf) " }{XPPEDIT 19 1 "(sqrt(2)-1)/2" "*&,&-%%sqrtG 6#\"\"#\"\"\"F(!\"\"F(F'F)" }{TEXT -1 1 " " }}{PARA 269 "" 0 "" {TEXT 295 3 "g) " }{XPPEDIT 19 1 "(sqrt(2)+1)/2" "*&,&-%%sqrtG6#\"\"#\"\"\"F (F(F(F'!\"\"" }{TEXT 389 11 " \nh) " }{XPPEDIT 19 1 "sqrt(2)/2" "*&-%%sqrtG6#\"\"#\"\"\"F&!\"\"" }{TEXT 390 17 " \ni) " } {XPPEDIT 19 1 "0" "\"\"!" }{TEXT 391 7 " \nj) " }{XPPEDIT 19 1 "1" " \"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 263 "" 0 "" {TEXT 276 4 "2. " }{TEXT -1 2 "If" } {TEXT 395 2 " " }{TEXT 398 1 " " }{XPPEDIT 396 1 "r(t)=`<`*t^t,t^3,(t ^2*ln(t))*`>`" "6%/-%\"rG6#%\"tG*&%\"GF*" }{TEXT 397 37 " then what is the dot product o f " }{XPPEDIT 19 1 "limit( (r(2+Delta*t)-r(2))/(Delta*t),Delta*t=0)" "-%&limitG6$*&,&-%\"rG6#,&\"\"#\"\"\"*&%&DeltaGF,%\"tGF,F,F,-F(6#F+!\" \"F,*&F.F,F/F,F2/*&F.F,F/F,\"\"!" }{TEXT -1 7 " with " }{TEXT 399 1 " j" }{TEXT -1 3 " ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 296 4 "a) " }{XPPEDIT 297 1 "0" "\"\"!" }{TEXT 298 10 " \+ b) " }{XPPEDIT 299 1 "1" "\"\"\"" }{TEXT 300 10 " c) " } {XPPEDIT 301 1 "2" "\"\"#" }{TEXT 302 8 " d) " }{XPPEDIT 303 1 "3 " "\"\"$" }{TEXT 304 10 " e) " }{XPPEDIT 305 1 "4" "\"\"%" } {TEXT 306 11 " f) " }{XPPEDIT 307 1 "6" "\"\"'" }{TEXT 308 11 " g) " }{XPPEDIT 309 1 "9" "\"\"*" }{TEXT 310 9 " h) " } {XPPEDIT 311 1 "12" "\"#7" }{TEXT 312 9 " i) " }{XPPEDIT 313 1 "1 5" "\"#:" }{TEXT 314 10 " j) " }{XPPEDIT 401 1 "18" "\"#=" } {TEXT -1 1 " " }{TEXT 400 3 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 40 "3. The position vecto r of a particle is " }{TEXT 319 2 " " }{XPPEDIT 322 1 "r(t)=`<`*8*t,2 *t^3,sin(Pi*t)*ln(t)*`>`" "6%/-%\"rG6#%\"tG*(%\"GF*" } {TEXT 317 1 " " }{TEXT 320 22 ". What is its speed at" }{TEXT 321 2 " \+ " }{XPPEDIT 323 1 "t=1" "/%\"tG\"\"\"" }{TEXT 318 1 " " }{TEXT 324 2 "? " }{TEXT 325 2 " " }}{PARA 264 "" 0 "" {TEXT 275 4 "a) " } {XPPEDIT 19 1 "6" "\"\"'" }{TEXT 412 14 " b) " }{XPPEDIT 19 1 "2*sqrt(10" "*&\"\"#\"\"\"-%%sqrtG6#\"#5F$" }{TEXT 411 13 " \+ c) " }{XPPEDIT 19 1 "7" "\"\"(" }{TEXT 410 14 " d) " } {XPPEDIT 19 1 "3*sqrt(6)" "*&\"\"$\"\"\"-%%sqrtG6#\"\"'F$" }{TEXT 409 17 " e) " }{XPPEDIT 19 1 "8" "\"\")" }{TEXT 408 15 " \+ \nf) " }{XPPEDIT 19 1 "6*sqrt(2)" "*&\"\"'\"\"\"-%%sqrtG6#\"\"#F $" }{TEXT 407 11 " g) " }{XPPEDIT 19 1 "9" "\"\"*" }{TEXT 406 15 " h) " }{XPPEDIT 19 1 "sqrt(86)" "-%%sqrtG6#\"#')" } {TEXT 405 13 " i) " }{XPPEDIT 19 1 "10" "\"#5" }{TEXT 404 17 " j) " }{XPPEDIT 19 1 "sqrt(120)" "-%%sqrtG6#\"$?\"" } {TEXT 403 1 " " }{TEXT 326 7 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 263 1 " " }{TEXT 331 36 "4. A space curve is parameterized by" }{TEXT 332 3 " " }{XPPEDIT 330 1 "r(t) = `<`*t,t^2,2*t*`>`" "6%/-%\"rG6#%\" tG*&%\"GF*" }{TEXT 327 1 " " } {TEXT 329 88 ". What is the second component of its unit tangent vecto r at the point corresponding to " }{TEXT 453 1 " " }{XPPEDIT 454 1 "t= 1" "/%\"tG\"\"\"" }{TEXT 452 1 " " }{TEXT 328 1 "?" }}{PARA 0 "" 0 "" {TEXT 268 4 "a) " }{XPPEDIT 256 0 "0" "\"\"!" }{TEXT 257 11 " \+ " }{TEXT 333 3 "b) " }{TEXT 334 1 " " }{XPPEDIT 277 0 "1/6" "*&\"\" \"F#\"\"'!\"\"" }{TEXT 278 14 " c) " }{XPPEDIT 279 0 "1/3" " *&\"\"\"F#\"\"$!\"\"" }{TEXT 280 7 " " }{TEXT 335 5 " d) " } {XPPEDIT 281 0 "1/sqrt(3)" "*&\"\"\"F#-%%sqrtG6#\"\"$!\"\"" }{TEXT 282 5 " " }{TEXT 336 6 " e)" }{TEXT 337 2 " " }{XPPEDIT 374 0 "sqrt(3)/2" "*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT 375 3 " " } {TEXT 285 2 " " }}{PARA 262 "" 0 "" {TEXT 261 4 "f) " }{TEXT 258 2 " " }{XPPEDIT 283 0 "2/3" "*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 284 18 " \+ g) " }{XPPEDIT 286 0 "5/6" "*&\"\"&\"\"\"\"\"'!\"\"" } {TEXT 287 15 " h) " }{XPPEDIT 288 0 "1/12" "*&\"\"\"F#\"#7! \"\"" }{TEXT 289 14 " i) " }{XPPEDIT 290 0 "sqrt(2)/sqrt(3) " "*&-%%sqrtG6#\"\"#\"\"\"-F$6#\"\"$!\"\"" }{TEXT 259 16 " \+ j) " }{XPPEDIT 292 0 "sqrt(2)/12" "*&-%%sqrtG6#\"\"#\"\"\"\"#7!\"\"" }{TEXT 293 1 " " }{TEXT 291 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 269 1 " " }{TEXT 415 2 "5." }{TEXT -1 1 " " }{TEXT 338 124 "Referring to the curve of the preceding problem at the same point, what is the first component of its principal unit nor mal?" }}{PARA 261 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 19 1 "-1/sqrt(5) " ",$*&\"\"\"F$-%%sqrtG6#\"\"&!\"\"F)" }{TEXT -1 11 " b) " } {XPPEDIT 19 1 "-2/3/sqrt(5)" ",$*(\"\"#\"\"\"\"\"$!\"\"-%%sqrtG6#\"\"& F'F'" }{TEXT -1 10 " c) " }{XPPEDIT 19 1 "-1/3" ",$*&\"\"\"F$\" \"$!\"\"F&" }{TEXT -1 11 " d) " }{XPPEDIT 19 1 "-2/9" ",$*&\"\" #\"\"\"\"\"*!\"\"F'" }{TEXT -1 16 " e) " }{XPPEDIT 19 1 "- sqrt(5)/3" ",$*&-%%sqrtG6#\"\"&\"\"\"\"\"$!\"\"F*" }}{PARA 265 "" 0 " " {TEXT -1 5 "f) " }{XPPEDIT 19 1 "sqrt(5)/3" "*&-%%sqrtG6#\"\"&\"\" \"\"\"$!\"\"" }{TEXT -1 14 " g) " }{XPPEDIT 19 1 "2/9" "*&\" \"#\"\"\"\"\"*!\"\"" }{TEXT -1 20 " h) " }{XPPEDIT 19 1 "1/3" "*&\"\"\"F#\"\"$!\"\"" }{TEXT -1 16 " i) " } {XPPEDIT 19 1 "2/3/sqrt(5)" "*(\"\"#\"\"\"\"\"$!\"\"-%%sqrtG6#\"\"&F& " }{TEXT -1 14 " j) " }{XPPEDIT 19 1 "1/sqrt(5)" "*&\"\"\"F# -%%sqrtG6#\"\"&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 265 81 "6. Referring to the curve of problem 4 at the same point, what is the curvature ?" }}{PARA 0 "" 0 "" {TEXT 266 8 "\na) " }{XPPEDIT 19 1 "sqrt(5)/3" "*&-%%sqrtG6#\"\"&\"\"\"\"\"$!\"\"" }{TEXT -1 5 " " }{TEXT 417 11 " b) " }{XPPEDIT 19 1 "2*sqrt(5)/3" "*(\"\"#\"\" \"-%%sqrtG6#\"\"&F$\"\"$!\"\"" }{TEXT -1 5 " " }{TEXT 418 14 " \+ c) " }{XPPEDIT 19 1 "sqrt(5)/27" "*&-%%sqrtG6#\"\"&\"\"\"\"#F! \"\"" }{TEXT -1 5 " " }{TEXT 419 13 " d) " }{XPPEDIT 19 1 "2*sqrt(5)/27" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$\"#F!\"\"" }{TEXT -1 5 " " }{TEXT 420 13 " e) " }{XPPEDIT 19 1 "sqrt(5)/81" "* &-%%sqrtG6#\"\"&\"\"\"\"#\")!\"\"" }{TEXT -1 5 " " }{TEXT 421 11 " \n\n f) " }{XPPEDIT 19 1 "2*sqrt(5)/81" "*(\"\"#\"\"\"-%%sqrtG6# \"\"&F$\"#\")!\"\"" }{TEXT 422 15 " g) " }{XPPEDIT 19 1 "2* sqrt(5)/5" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }{TEXT 423 18 " \+ h) " }{XPPEDIT 19 1 "3*sqrt(5)/5" "*(\"\"$\"\"\"-%%sqrtG6# \"\"&F$F(!\"\"" }{TEXT 424 14 " i) " }{XPPEDIT 19 1 "6*sqrt( 5)/5" "*(\"\"'\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }{TEXT 425 16 " \+ j) " }{XPPEDIT 19 1 "12*sqrt(5)/5" "*(\"#7\"\"\"-%%sqrtG6#\"\"&F$F (!\"\"" }{TEXT 426 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 267 61 "7. What is the length of the cu rve that is parameterized by " }{TEXT 340 1 " " }{XPPEDIT 344 1 "r(t)= `<`*cos(t^2),sin(t^2),t^2*`>`" "6%/-%\"rG6#%\"tG*&%\"GF*" }{TEXT 345 1 " " }{TEXT 341 5 " for " }{XPPEDIT 19 1 "t" "I\"tG6\"" }{TEXT 427 18 " in the interva l " }{TEXT 346 2 " " }{XPPEDIT 342 1 "[1,2]" "7$\"\"\"\"\"#" }{TEXT 343 3 " ?" }{TEXT 339 1 "\n" }}{PARA 0 "" 0 "" {TEXT 347 5 "a) " } {XPPEDIT 19 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 429 16 " b) " }{XPPEDIT 19 1 "2*sqrt(2)" "*&\"\"#\"\"\"-%%sqrtG6#F#F$" }{TEXT 430 14 " c) " }{XPPEDIT 19 1 "3*sqrt(2)" "*&\"\"$\"\"\"-%%sq rtG6#\"\"#F$" }{TEXT 431 15 " d) " }{XPPEDIT 19 1 "sqrt(2)/ 2" "*&-%%sqrtG6#\"\"#\"\"\"F&!\"\"" }{TEXT 432 14 " e) " } {XPPEDIT 19 1 "3*sqrt(2)/2" "*(\"\"$\"\"\"-%%sqrtG6#\"\"#F$F(!\"\"" } {TEXT 433 8 " \n\n f) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#\"\"$" } {TEXT 434 15 " g) " }{XPPEDIT 19 1 "2*sqrt(3)" "*&\"\"#\"\" \"-%%sqrtG6#\"\"$F$" }{TEXT 435 16 " h) " }{XPPEDIT 19 1 " 3*sqrt(3)" "*&\"\"$\"\"\"-%%sqrtG6#F#F$" }{TEXT 436 16 " i) " }{XPPEDIT 19 1 "sqrt(3)/2" "*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" } {TEXT 437 15 " j) " }{XPPEDIT 19 1 "3*sqrt(3)/2" "*(\"\"$\" \"\"-%%sqrtG6#F#F$\"\"#!\"\"" }{TEXT 438 2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 352 6 "8. If " }{TEXT 350 1 " " }{XPPEDIT 348 1 "r(t)=`<`*cos(t),sin(t),t*`>` " "6%/-%\"rG6#%\"tG*&%\"GF*" }{TEXT 349 1 " " }{TEXT 351 79 " \+ is the position vector of a particle then what is its tangential comp onent, " }{XPPEDIT 19 1 "a[T]" "&%\"aG6#%\"TG" }{TEXT 451 22 ", of a cceleration at " }{TEXT 377 1 " " }{XPPEDIT 376 1 "t=Pi/2" "/%\"tG*&%# PiG\"\"\"\"\"#!\"\"" }{TEXT 378 3 " ?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 353 81 "a) 0 b) 1 c) 2 d) 3 e) " } {XPPEDIT 19 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 440 8 " \n\nf) " } {XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#\"\"$" }{TEXT 441 12 " g) \+ " }{XPPEDIT 19 1 "2*sqrt(2)" "*&\"\"#\"\"\"-%%sqrtG6#F#F$" }{TEXT 442 13 " h) " }{XPPEDIT 19 1 "2*sqrt(3)" "*&\"\"#\"\"\"-%%sqrtG6# \"\"$F$" }{TEXT 443 13 " i) " }{XPPEDIT 19 1 "sqrt(2)/2" "*&- %%sqrtG6#\"\"#\"\"\"F&!\"\"" }{TEXT 444 14 " j) " }{XPPEDIT 19 1 "sqrt(3)/2" "*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT 445 5 " \+ " }}{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 270 61 "9. At an instant of time the ac celeration of a particle is " }{TEXT 456 1 " " }{XPPEDIT 455 1 "`<`*8 ,-1,4*`>` " "6%*&%\"GF%" }}{PARA 258 "" 0 "" {TEXT 354 46 "and its tangential component of acceleration , " }{XPPEDIT 19 1 "a[T]" "&%\"aG6#%\"TG" }{TEXT 460 8 " , is " } {XPPEDIT 19 1 "sqrt(65)" "-%%sqrtG6#\"#l" }{TEXT 470 50 ".\n What is i ts normal component of acceleration, " }{XPPEDIT 19 1 "a[N]" "&%\"aG6 #%\"NG" }{TEXT 461 16 " , at that time?" }}{PARA 3 "" 0 "" {TEXT 447 69 "a) 1 b) 2 c) 3 d) 4 e ) " }{TEXT 448 22 " 5 " }}{PARA 0 "" 0 "" {TEXT 449 6 "f) " }{XPPEDIT 19 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 462 18 " g) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#\"\"$" } {TEXT 464 12 " " }{TEXT -1 1 " " }{TEXT 450 6 "h) " } {XPPEDIT 19 1 "sqrt(5)" "-%%sqrtG6#\"\"&" }{TEXT 466 20 " i ) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#\"\"$" }{TEXT 468 15 " \+ j) " }{TEXT -1 1 " " }{TEXT 469 23 " Not enough information" }{TEXT -1 2 ". " }}{PARA 3 "" 0 "" {TEXT 446 1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 271 9 " 10. Let" }{TEXT 474 2 " " }{XPPEDIT 475 1 "f(x,y)=(x^2-y^2)/(x^4+y^ 2)" "/-%\"fG6$%\"xG%\"yG*&,&*$F&\"\"#\"\"\"*$F'F+!\"\"F,,&*$F&\"\"%F,* $F'F+F,F." }{TEXT 473 49 " for all points other than the origin. L et " }{TEXT 477 1 " " }{XPPEDIT 478 1 "r(t)=`<`*3*t,t*`>`" "6$/-%\"r G6#%\"tG*(%\"GF*" }{TEXT 476 2 ". " } {TEXT 472 3 " \n\n" }{TEXT 480 13 "Calculate " }{XPPEDIT 481 1 "lim it(f(r(t)),t=0)" "-%&limitG6$-%\"fG6#-%\"rG6#%\"tG/F+\"\"!" }{TEXT 479 2 ".\n" }}{PARA 0 "" 0 "" {TEXT 487 4 "a) " }{TEXT 484 1 "0" } {TEXT -1 1 " " }{TEXT 523 6 " " }{TEXT 496 4 "b ) " }{TEXT 497 2 " " }{TEXT 520 1 "1" }{TEXT 521 7 " " }{TEXT 507 2 "c " }{TEXT 508 6 ") 2 " }{TEXT 498 20 " d) 3 e)" }{TEXT 500 2 " " }{TEXT 522 1 "4" }}{PARA 270 "" 0 "" {TEXT 486 4 "f) " }{TEXT 502 46 " 5 g) 6 h) 7 i) " }{TEXT 485 31 " 8 \+ j) Does not exist " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 1 " " }{TEXT 360 5 "11. " }{TEXT -1 1 " " } {TEXT 512 6 "If " }{XPPEDIT 513 1 "f(x,y)=2*x*y - 3*x^2/y" "/-%\"fG 6$%\"xG%\"yG,&*(\"\"#\"\"\"F&F+F'F+F+*(\"\"$F+*$F&F*F+F'!\"\"F/" } {TEXT 514 18 " then calculate " }{XPPEDIT 515 1 "Diff(f(x,y),y)" "-% %DiffG6$-%\"fG6$%\"xG%\"yGF)" }{TEXT 516 7 " at " }{XPPEDIT 517 1 " `(`*2,1*`)`" "6$*&%\"(G\"\"\"\"\"#F%*&F%F%%\")GF%" }{TEXT 518 4 ". \n " }{TEXT 519 1 "\n" }{TEXT 362 129 "a) 4 b) 5 c) \+ 12 d) 16 e) 20 \nf) 24 g) 28 h) 32 \+ i) 36 j) 40" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 273 "" 0 "" {TEXT -1 11 "12. \+ If " }{XPPEDIT 19 1 "x^2 + 4*z^4=6*y+ x*z" "/,&*$%\"xG\"\"#\"\"\"*& \"\"%F'*$%\"zGF)F'F',&*&\"\"'F'%\"yGF'F'*&F%F'F+F'F'" }{TEXT -1 20 " \+ then calculate " }{XPPEDIT 19 1 "diff( z ,x)" "-%%diffG6$%\"zG%\"xG " }{TEXT -1 10 " at " }{XPPEDIT 19 1 "`(`*2,1,1*`)`" "6%*&%\"(G \"\"\"\"\"#F%F%*&F%F%%\")GF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 184 "a) -5/14 b) -2/7 \+ c) -3/14 d) -1/7 e) 0 \n f) 1/7 g) 3/14 h) 2/7 i) 5/14 \+ j) 5/7 " }}{PARA 266 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 369 15 "13. The plane " }{XPPEDIT 530 1 "x=2" "/%\"xG \"\"#" }{TEXT 529 2 " " }{TEXT 531 24 " intersects the graph of" } {TEXT 532 2 " " }{XPPEDIT 533 1 "z=5 - x + y^2" "/%\"zG,(\"\"&\"\"\"% \"xG!\"\"*$%\"yG\"\"#F&" }{TEXT 528 2 " " }{TEXT 534 50 " in a curve. The tangent line to this curve at " }{XPPEDIT 536 1 "`(`*2,3,12*`) `" "6%*&%\"(G\"\"\"\"\"#F%\"\"$*&\"#7F%%\")GF%" }{TEXT 535 1 " " } {TEXT 537 27 " passes through the point " }{XPPEDIT 538 1 "`(`*2,b,24 *`)`" "6%*&%\"(G\"\"\"\"\"#F%%\"bG*&\"#CF%%\")GF%" }{TEXT 539 12 ". \+ What is " }{TEXT 541 1 " " }{XPPEDIT 542 1 "b" "I\"bG6\"" }{TEXT 540 5 "? " }}{PARA 258 "" 0 "" {TEXT 543 5 "\na) " }{XPPEDIT 544 1 "0 " "\"\"!" }{TEXT 545 10 " b) " }{XPPEDIT 546 1 "1" "\"\"\"" } {TEXT 547 10 " c) " }{XPPEDIT 548 1 "2" "\"\"#" }{TEXT 549 8 " \+ d) " }{XPPEDIT 550 1 "3" "\"\"$" }{TEXT 551 10 " e) " } {XPPEDIT 552 1 "4" "\"\"%" }{TEXT 553 11 " f) " }{XPPEDIT 554 1 "5" "\"\"&" }{TEXT 555 11 " g) " }{XPPEDIT 556 1 "6" "\"\"'" }{TEXT 557 9 " h) " }{XPPEDIT 558 1 "7" "\"\"(" }{TEXT 559 9 " \+ i) " }{XPPEDIT 560 1 "8" "\"\")" }{TEXT 561 10 " j) " } {XPPEDIT 563 1 "9" "\"\"*" }{TEXT -1 1 " " }{TEXT 562 3 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 267 "" 0 "" {TEXT -1 9 "14. If " }{XPPEDIT 565 1 "`<`*8,-1,4*`>` " "6%*&%\"GF%" }{TEXT -1 57 " is perpendicular to the tangent plane to the graph of " }{XPPEDIT 19 1 "f" "I\"fG6\"" }{TEXT -1 5 " at " }{XPPEDIT 19 1 "P " "I\"PG6\"" }{TEXT -1 16 " then what is " }{XPPEDIT 19 1 "f[x](P" " -&%\"fG6#%\"xG6#%\"PG" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 260 "" 0 "" {TEXT -1 162 "a) -5 b) -4 c) -3 d) -2 e) -1 \nf) 5 g ) 4 h) 3 i) 2 j) 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 273 10 "15. Let " }{XPPEDIT 568 1 "z=y*x^3 " "/%\"zG*&%\"yG\"\"\"*$%\"xG\"\"$F&" }{TEXT 569 3 ", \+ " }{XPPEDIT 570 1 "x=2*t+ s^2" "/%\"xG,&*&\"\"#\"\"\"%\"tGF'F'*$%\"sG F&F'" }{TEXT 571 7 ", and " }{XPPEDIT 572 1 "y=3*t-s" "/%\"yG,&*&\"\" $\"\"\"%\"tGF'F'%\"sG!\"\"" }{TEXT 573 13 ". Calculate" }{TEXT 577 1 " " }{XPPEDIT 578 1 "diff(z,s)" "-%%diffG6$%\"zG%\"sG" }{TEXT 574 1 " " }{TEXT 579 8 " when " }{XPPEDIT 580 1 "t=1" "/%\"tG\"\"\"" } {TEXT 575 8 " and " }{XPPEDIT 581 1 "s=2" "/%\"sG\"\"#" }{TEXT 576 7 " . " }}{PARA 3 "" 0 "" {TEXT 274 147 "a) 12 b) \+ 24 \nc) 48 d) 96 \ne) 144 \+ f) 188 \ng) 216 h) 232" }{TEXT 371 7 " " }{TEXT 372 32 " \ni) 256 j) 280 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 392 10 " Solutions" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 393 2 "1)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "z := t -> 2 - cos(t) - sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"zG:6#%\"tG6\"6$%)operatorG%&arrowGF(,(\"\"#\"\"\"-%$cosG6#9$!\"\"- %$sinGF1F3F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D(z)(Pi/3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$\"\"$#\"\"\"\"\"#F&#!\"\"F(F '" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 394 2 "2) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(t^3, t);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*$%\"tG\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "subs(t=2, \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 402 2 " 3)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "magnitude := r -> sqrt(r[1]^ 2+r[2]^2+r[3]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*magnitudeG:6#% \"rG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,(*$&9$6#\"\"\"\"\"#F4*$&F26#F 5F5F4*$&F26#\"\"$F5F4F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "x := t -> 8*t: y := t -> 2*t^3: z := t -> sin(Pi*t)*ln(t):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "velocity := map(u->diff(u,t) , [x(t),y(t),z(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7% \"\"),$*$%\"tG\"\"#\"\"',&*(-%$cosG6#*&%#PiG\"\"\"F)F3F3F2F3-%#lnG6#F) F3F3*&-%$sinGF0F3F)!\"\"F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(t=1, \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\")\"\"',&*( -%$cosG6#%#PiG\"\"\"F+F,-%#lnG6#F,F,F,-%$sinGF*F," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 13 "magnitude(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 413 2 "4)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x := t -> t: y := t -> t^2: z := t -> 2*t:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "velocity := map(u->diff(u,t) , [x(t),y(t),z(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7% \"\"\",$%\"tG\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "uni tTangent := map( u -> u/magnitude(velocity) , velocity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%,unitTangentG7%*$,&\"\"&\"\"\"*$%\"tG\"\"#\"\" %#!\"\"F,,$*&F+F)F'F.F,,$F&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(t=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*$\"\"*#\" \"\"\"\"##F(F&,$F%#F)F&F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"\"\"\"$#\" \"#F&F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 414 2 "5)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "unitTangent;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$,&\"\"&\"\"\"*$%\"tG\"\"#\"\"%#!\"\"F*,$ *&F)F'F%F,F*,$F$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "N := \+ map(u->diff(u,t), unitTangent);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"NG7%,$*&,&\"\"&\"\"\"*$%\"tG\"\"#\"\"%#!\"$F-F,F*!\"%,&*$F(#!\"\"F-F -*&F,F-F(F/!\"),$F'F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "N \+ := subs(t = 1, N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG7%,$*$\"\" *#\"\"\"\"\"##!\"%\"#\"),$F'#\"#5F.,$F'#!\")F." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "principalUnitNormal := map( u -> u/magnitude( N), N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4principalUnitNormalG7%,$ *&\"\"*#\"\"\"\"\"#\"\"&F)#!\"#\"#X,$F'#F*F(,$F'#!\"%F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify( \"[1] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"&#\"\"\"\"\"##!\"#\"#:" }}}{PARA 0 "" 0 " " {TEXT -1 2 "\n " }{TEXT 416 2 "6)" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "velocity;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\" \"\",$%\"tG\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "accel eration := map(u->diff(u,t), velocity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accelerationG7%\"\"!\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning , new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ma gnitude(crossprod(velocity,acceleration))/magnitude(velocity)^3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"&#\"\"\"\"\"#,&F%F'*$%\"tGF(\" \"%#!\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(t=1,\" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"*#\"\"\"\"\"#\"\"&F&#F( \"#\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"&#\"\"\"\"\"##F(\"#F" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 428 2 "7)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "x := t -> cos(t^2): y := t \+ -> sin(t^2): z := t -> t^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "velocity := map(u->diff(u,t), [x(t),y(t),z(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7%,$*&-%$sinG6#*$%\"tG\"\"#\"\"\"F,F.! \"#,$*&-%$cosGF*F.F,F.F-,$F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "arcLength := Int(magnitude(velocity),t=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*arcLengthG-%$IntG6$,$*$,(*&-%$sinG6#*$%\"tG\"\" #F1F0F1\"\"\"*&-%$cosGF.F1F0F1F2F/F2#F2F1F1/F0;F2F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "value(arcLength);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"##\"\"\"F%\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 439 2 "8)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "x := t -> cos(t): y := t -> sin(t): z := t -> t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "velocity := map(u->diff(u,t), [x(t) ,y(t),z(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7%,$-%$si nG6#%\"tG!\"\"-%$cosGF)\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "speed := magnitude(velocity);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&speedG*$,(*$-%$sinG6#%\"tG\"\"#\"\"\"*$-%$cosGF*F,F-F-F-#F-F," }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "spedd := simplify(speed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&speddG*$\"\"##\"\"\"F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "tangentialComponentAccelerat ion := diff(speed,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%@tangential ComponentAccelerationG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 526 2 "9)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "acceleration := [8, -1, 4]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accelerationG7%\"\")!\"\"\"\"% " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eqn := a[T]^2 + a[N]^2 \+ = magnitude(acceleration)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqn G/,&*$&%\"aG6#%\"TG\"\"#\"\"\"*$&F)6#%\"NGF,F-\"#\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(a[T] = sqrt(65), \" );" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&\"#l\"\"\"*$&%\"aG6#%\"NG\"\"#F&\"#\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(\",a[N]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 605 3 "10)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "subs (\{x = 3*t, y = t\}, (x^2-y^2)/(x^4+y^2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"tG\"\"#,&*$F%\"\"%\"#\")*$F%F&\"\"\"!\"\"\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "limit(\",t=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 604 3 "11)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "diff(2*x*y-3*x^2/y ,y );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"#*&F$F%%\"yG!\"#\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(\{x=2,y=1\},\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 525 3 "12) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "eqn := x^2+4*z(x,y)^4 = 6*y+x*z(x,y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&*$%\"xG\"\"#\"\"\"*$-%\"zG6$ F(%\"yG\"\"%F0,&F/\"\"'*&F(F*F,F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "map(u->diff(u,x), eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"#*&-%\"zG6$F%%\"yG\"\"$-%%diffG6$F(F%\"\"\"\"#;,&F(F 0*&F%F0F-F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(\",d iff(z(x,y),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG!\"#-%\"zG 6$F%%\"yG\"\"\"F+,&*$F'\"\"$\"#;F%!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "subs(\{x=2,y=1\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&!\"%\"\"\"-%\"zG6$\"\"#F&F&F&,&*$F'\"\"$\"#;!\"#F&!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(z(2,1)=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"$\"#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 583 3 "14)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " eqn := [8, -1, 4] = map(u -> c*u, [f[x](P), f[y](P), -1 ]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/7%\"\")!\"\"\"\"%7%*&%\"cG\"\" \"-&%\"fG6#%\"xG6#%\"PGF-*&F,F--&F06#%\"yGF3F-,$F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "c = solve(lhs(eqn)[3] = rhs(eqn)[3] , c); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG!\"%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "subs(\", eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/7%\"\")!\"\"\"\"%7%,$-&%\"fG6#%\"xG6#%\"PG!\"%,$-&F,6#%\"yGF/F1 F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f[x](P) = solve(lhs( \")[1] = rhs(\")[1] , f[x](P));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-& %\"fG6#%\"xG6#%\"PG!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 607 3 "15)" }}{PARA 0 "" 0 "" {TEXT 584 1 " " }{XPPEDIT 585 1 "z=y*x^3 " "/%\"zG*&%\"yG\"\"\"* $%\"xG\"\"$F&" }{TEXT 586 3 ", " }{XPPEDIT 587 1 "x=2*t+ s^2" "/%\"xG ,&*&\"\"#\"\"\"%\"tGF'F'*$%\"sGF&F'" }{TEXT 588 7 ", and " }{XPPEDIT 589 1 "y=3*t-s" "/%\"yG,&*&\"\"$\"\"\"%\"tGF'F'%\"sG!\"\"" }{TEXT 590 13 ". Calculate" }{TEXT 594 1 " " }{XPPEDIT 595 1 "diff(z,s)" "-%%di ffG6$%\"zG%\"sG" }{TEXT 591 1 " " }{TEXT 596 8 " when " }{XPPEDIT 597 1 "t=1" "/%\"tG\"\"\"" }{TEXT 592 8 " and " }{XPPEDIT 598 1 "s= 2" "/%\"sG\"\"#" }{TEXT 593 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "diff(y*x^3,x)*diff(2*t+s^2,s ) + diff(y*x^3,y)*diff(3*t-s,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& *(%\"yG\"\"\"%\"xG\"\"#%\"sGF&\"\"'*$F'\"\"$!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 24 "dz_by_ds := subs(s=2,\");" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%)dz_by_dsG,&*&%\"yG\"\"\"%\"xG\"\"#\"#7*$F)\"\"$!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "x = subs(\{t=1,s=2\}, 2*t+s^2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dz_by_ds := subs(\", dz_by_ds);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)dz_by_dsG,&%\"yG\"$K%!$;#\"\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y = subs(\{t=1,s=2\}, 3*t-s );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dz_by_ds := subs(\", dz_by_ds);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)dz_by_dsG\"$;#" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "16 105 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 }