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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 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 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 293 37 " Math 132 \n Fall 2006 Exam II" }}{PARA 0 "" 0 "" {TEXT 256 3 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 1 "1" }{TEXT 264 2 ". " }{TEXT 439 13 "Suppose tha t " }{XPPEDIT 440 0 "f(x) = 2^(x^2);" "6#/-%\"fG6#%\"xG)\"\"#*$F'F)" } {TEXT 438 22 ". Calculate D(f)(2)." }{TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 115 "a) 8 ln(2) b) 16 ln(2) c) 32 ln(2 ) d) 64 ln(2) e) 128 ln(2) \nf) 8/ln(2) " } {TEXT 294 10 " g) 1" }{TEXT -1 7 "6/ln(2)" }{TEXT 441 14 " \+ h) " }{TEXT -1 8 "32/ln(2)" }{TEXT 442 12 " i) " }{TEXT -1 8 "64/ln(2)" }{TEXT 443 17 " j) 1" }{TEXT -1 8 "28/ln(2 )" }{TEXT 444 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 640 12 "Solution: d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "A := diff(2^(x^2), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG ,$**\"\"#\"\"\")F'*$)%\"xGF'F(F(F,F(-%#lnG6#F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(x=2, A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#k\"\"\"-%#lnG6#\"\"#F&F&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 335 16 "2. Calculate " }{XPPEDIT 468 0 "Int(log[2](x),x = 1 . . 2);" "6#-%$IntG6$-&%$logG6#\"\"#6#%\"xG/F,;\"\"\"F*" }{TEXT 467 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 346 5 "a) " }{XPPEDIT 470 0 "2-1/ln(2);" "6#,& \"\"#\"\"\"*&F%F%-%#lnG6#F$!\"\"F*" }{TEXT 469 12 " b) " } {XPPEDIT 261 0 "2+1/ln(2);" "6#,&\"\"#\"\"\"*&F%F%-%#lnG6#F$!\"\"F%" } {TEXT 347 13 " c) " }{XPPEDIT 262 0 "2-ln(2);" "6#,&\"\"#\"\" \"-%#lnG6#F$!\"\"" }{TEXT 348 11 " d) " }{XPPEDIT 472 0 "2+ln(2 );" "6#,&\"\"#\"\"\"-%#lnG6#F$F%" }{TEXT 471 10 " e) " } {XPPEDIT 263 0 "2*ln(2)-1;" "6#,&*&\"\"#\"\"\"-%#lnG6#F%F&F&F&!\"\"" } {TEXT 349 9 " \nf) " }{XPPEDIT 264 0 "2*ln(2)+1;" "6#,&*&\"\"#\"\" \"-%#lnG6#F%F&F&F&F&" }{TEXT 350 10 " g) " }{TEXT 357 1 " " } {XPPEDIT 265 0 "1-2/ln(2);" "6#,&\"\"\"F$*&\"\"#F$-%#lnG6#F&!\"\"F*" } {TEXT 351 12 " h) " }{XPPEDIT 266 0 "1+2/ln(2);" "6#,&\"\"\"F$ *&\"\"#F$-%#lnG6#F&!\"\"F$" }{TEXT 352 10 " i) " }{TEXT 355 1 " \+ " }{XPPEDIT 267 0 "log[2](2);" "6#-&%$logG6#\"\"#6#F'" }{TEXT 353 2 " \+ " }{TEXT 356 12 " j) " }{XPPEDIT 268 0 "log[2](2)-1;" "6#,&-& %$logG6#\"\"#6#F(\"\"\"F*!\"\"" }{TEXT 345 1 "\n" }{TEXT 354 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 641 12 "Solution: a" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "int(ln(x)/ln(2), x = 1 .. 2); #log[2](x) = ln(x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"!\"\"*&\"\"#F%-%#lnG6#F(F%F%F%F)F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "testeq(int(log[2](x),x=1..2) = 2 - 1/ln(2)); #Check answer." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%t rueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 262 3 "3. " }{TEXT 296 1 " " } {TEXT 297 15 " Suppose that " }{XPPEDIT 18 0 "f(x) = sqrt(x)*` `^x;" "6#/-%\"fG6#%\"xG*&-%%sqrtG6#F'\"\"\")%\"~GF'F," }{TEXT 445 22 ". Cal culate D(f)(2)." }}{PARA 3 "" 0 "" {TEXT 295 3 "a) " }{XPPEDIT 451 0 "sqrt(2)+1;" "6#,&-%%sqrtG6#\"\"#\"\"\"F(F(" }{TEXT 446 17 " \+ b) " }{XPPEDIT 452 0 "2*sqrt(2)+1;" "6#,&*&\"\"#\"\"\"-%%sqrtG6#F% F&F&F&F&" }{TEXT 447 13 " c) " }{XPPEDIT 453 0 "4*sqrt(2)+1; " "6#,&*&\"\"%\"\"\"-%%sqrtG6#\"\"#F&F&F&F&" }{TEXT 448 12 " d ) " }{XPPEDIT 454 0 "sqrt(2)+ln(2);" "6#,&-%%sqrtG6#\"\"#\"\"\"-%#lnG6 #F'F(" }{TEXT 449 12 " e) " }{XPPEDIT 455 0 "2*sqrt(2)+ln(2); " "6#,&*&\"\"#\"\"\"-%%sqrtG6#F%F&F&-%#lnG6#F%F&" }{TEXT 450 14 " \+ \nf) " }{XPPEDIT 460 0 "4*sqrt(2)+ln(2);" "6#,&*&\"\"%\"\"\"-%%sq rtG6#\"\"#F&F&-%#lnG6#F*F&" }{TEXT 456 8 " g) " }{XPPEDIT 461 0 "l n(2)+1;" "6#,&-%#lnG6#\"\"#\"\"\"F(F(" }{TEXT 457 11 " h) " } {XPPEDIT 462 0 "ln(2)+2;" "6#,&-%#lnG6#\"\"#\"\"\"F'F(" }{TEXT 458 14 " i) " }{XPPEDIT 463 0 "2*ln(2)+1;" "6#,&*&\"\"#\"\"\"-%#lnG 6#F%F&F&F&F&" }{TEXT 459 12 " j) " }{XPPEDIT 465 0 "sqrt(2)*ln (2);" "6#*&-%%sqrtG6#\"\"#\"\"\"-%#lnG6#F'F(" }{TEXT 464 3 " " } {TEXT 466 6 " \n" }{TEXT 642 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 643 13 "Solution: g\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(sqrt(x)^x, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)*$%\"xG#\"\"\"\"\"#F&F(,&-%#lnG6#F%F(F'F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(subs(x=2, %));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#\"\"#\"\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "eqn1 := f(x) = sqrt(x)^x; \n#The n ext steps verify by logarithmic differentiation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%\"fG6#%\"xG)*$F)#\"\"\"\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eqn2 := map(ln, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%#lnG6#-%\"fG6#%\"xG-F'6#)*$F,#\"\"\" \"\"#F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqn3 := map( z - > diff(z,x), eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/*&-%%d iffG6$-%\"fG6#%\"xGF-\"\"\"F*!\"\",&-%#lnG6#*$F-#F.\"\"#F.F5F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn4 := diff(f(x),x) = solve (eqn3, diff(f(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%%d iffG6$-%\"fG6#%\"xGF,,&*&-%#lnG6#*$F,#\"\"\"\"\"#F4F)F4F4*&F3F4F)F4F4 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn5 := subs(eqn1, eqn4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/-%%diffG6$)*$%\"xG#\"\" \"\"\"#F+F+,&*&-%#lnG6#F*F-F)F-F-*&F.!\"\"F)F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "answer := subs(x = 2, rhs(eqn5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG,&*&\"\"#\"\"\"-%#lnG6#*$F'#F(F'F( F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(answer); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#\"\"#\"\"\"F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 257 27 "4. A radioactive substance" } {TEXT 473 1 " " }{TEXT 476 24 "has mass 120g at time " }{XPPEDIT 483 0 "t = 8;" "6#/%\"tG\"\")" }{TEXT 481 26 " and mass 80g at tim e " }{XPPEDIT 484 0 "t = 10;" "6#/%\"tG\"#5" }{TEXT 482 71 ". Express \+ the half-life of this substance in terms of the quantities " } {XPPEDIT 479 0 "alpha = ln(2);" "6#/%&alphaG-%#lnG6#\"\"#" }{TEXT 477 9 " and " }{XPPEDIT 480 0 "beta = ln(3);" "6#/%%betaG-%#lnG6#\"\"$ " }{TEXT 478 1 "." }{TEXT 475 3 " \n\n" }{TEXT 474 4 "a) " }{XPPEDIT 493 0 "beta-alpha;" "6#,&%%betaG\"\"\"%&alphaG!\"\"" }{TEXT 485 15 " \+ b) " }{XPPEDIT 494 0 "2*beta-alpha;" "6#,&*&\"\"#\"\"\"%%bet aGF&F&%&alphaG!\"\"" }{TEXT 486 18 " c) " }{XPPEDIT 495 0 "alpha/(beta-alpha);" "6#*&%&alphaG\"\"\",&%%betaGF%F$!\"\"F(" } {TEXT 487 12 " d) " }{XPPEDIT 496 0 "beta/(beta-alpha);" "6#*& %%betaG\"\"\",&F$F%%&alphaG!\"\"F(" }{TEXT 488 15 " e) " } {TEXT 317 1 " " }{XPPEDIT 497 0 "2*alpha/(beta-alpha);" "6#*(\"\"#\"\" \"%&alphaGF%,&%%betaGF%F&!\"\"F)" }{TEXT 301 19 " " }}{PARA 0 "" 0 "" {TEXT 299 4 "f) " }{XPPEDIT 498 0 "2*beta/(beta-alp ha);" "6#*(\"\"#\"\"\"%%betaGF%,&F&F%%&alphaG!\"\"F)" }{TEXT 489 18 " \+ g) " }{XPPEDIT 499 0 "(beta+alpha)/(beta-alpha);" "6#*&, &%%betaG\"\"\"%&alphaGF&F&,&F%F&F'!\"\"F)" }{TEXT 490 10 " " }{TEXT 500 1 " " }{TEXT 300 5 "h) " }{XPPEDIT 501 0 "(beta-alpha)/be ta;" "6#*&,&%%betaG\"\"\"%&alphaG!\"\"F&F%F(" }{TEXT 491 13 " \+ i) " }{XPPEDIT 502 0 "(beta-alpha)/alpha;" "6#*&,&%%betaG\"\"\"%&alph aG!\"\"F&F'F(" }{TEXT 492 16 " j) " }{XPPEDIT 503 0 "(2*be ta+alpha)/(2*beta-alpha);" "6#*&,&*&\"\"#\"\"\"%%betaGF'F'%&alphaGF'F' ,&*&F&F'F(F'F'F)!\"\"F," }{TEXT 504 3 " " }}{PARA 3 "" 0 "" {TEXT 298 1 " " }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 644 13 "Solution: e\n" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m := t -> m0/2^(t/tau);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&%#m0G\"\"\") \"\"#*&9$F.%$tauG!\"\"F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eqn1 := m(8) = 120;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1 G/*&%#m0G\"\"\")\"\"#,$*&\"\")F(%$tauG!\"\"F(F/\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eqn2 := m(10) = 80;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/*&%#m0G\"\"\")\"\"#,$*&\"#5F(%$tauG!\"\"F (F/\"#!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve(\{eqn1,eq n2\},\{m0,tau\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%$tauG,$*(\"\" #\"\"\"-%#lnG6#F(F)-F+6##F(\"\"$!\"\"F1/%#m0G#\"%:7F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "eqn3 := m0 = solve(eqn1, m0); #Step =by-step calculation of tau" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3 G/%#m0G,$*&\"$?\"\"\"\")\"\"#,$*&\"\")F*%$tauG!\"\"F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn4 := subs(eqn3, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/,$*(\"$?\"\"\"\")\"\"#,$*&\"\")F)%$ tauG!\"\"F)F))F+,$*&\"#5F)F/F0F)F0F)\"#!)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "eqn5 := map(z-> z*2^(10/tau)/2^(8/tau), eqn4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/\"$?\",$*(\"#!)\"\"\")\"\"#,$ *&\"#5F*%$tauG!\"\"F*F*)F,,$*&\"\")F*F0F1F*F1F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "eqn6 := map(simplify, eqn5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/\"$?\",$*&\"#!)\"\"\")\"\"%*&F*F*%$tauG!\" \"F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eqn7 := map(ln, e qn6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/-%#lnG6#\"$?\"-F'6#, $*&\"#!)\"\"\")\"\"%*&F/F/%$tauG!\"\"F/F/" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "eqn8 := expand(eqn7) assuming tau > 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn8G/-%#lnG6#\"$?\",&-F'6#\"#!)\"\"\"*&%$ta uG!\"\"-F'6#\"\"%F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eq n9 := tau = solve(eqn8, tau);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq n9G/%$tauG*&-%#lnG6#\"\"%\"\"\",&-F)6#\"$?\"F,-F)6#\"#!)!\"\"F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn10 := simplify(eqn9);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqn10G/%$tauG,$*(\"\"#\"\"\"-%#lnG6 #F)F*,&F+F*-F,6#\"\"$!\"\"F2F2" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 400 13 "5. The mass " }{XPPEDIT 506 0 "m(t);" "6#-%\"mG6#%\"tG" }{TEXT 505 112 " of a microbe colony feasting in a gourmet nutrient broth do ubles every 2 hours. When the colony's mass is 6 " }{XPPEDIT 508 0 " mu;" "6#%#muG" }{TEXT 507 13 "g, what, in " }{XPPEDIT 510 0 "mu;" "6# %#muG" }{TEXT 509 23 "g/hr, is D(m)(t) ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 403 87 "a) ln(2) b) \+ 2 ln(2) c) 3 ln(2) d) 6 ln(2) e)" } {TEXT 406 1 " " }{TEXT 511 8 "12 ln(2)" }{TEXT 512 1 " " }{TEXT 405 3 " " }{TEXT 404 19 " " }}{PARA 0 "" 0 "" {TEXT 401 42 "f) 1/ln(2) g) 2/ln(2) " }{TEXT -1 1 " " } {TEXT 402 47 "h) 3/ln(2) i) 6/ln(2) j) " }{TEXT 513 8 "12/ln(2)" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 645 13 "Solution: c\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "subs(t=0, diff(6*2^(t/2), t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$\"\"\"-%#lnG6#\"\"#F&F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "m := t -> m0*2^(t/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mGf*6#%\"tG6\"6$%)operatorG%&arr owGF(*&%#m0G\"\"\")\"\"#,$*&#F.F0F.9$F.F.F.F(F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "D(m)(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&#\"\"\"\"\"#F&*(%#m0GF&)F',$*&F'!\"\"%\"tGF&F&F&-%#lnG6#F'F&F&F&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "'D(m)(t)' = 'm(t)'*ln(2)/2 ; \n#Notice that m(t) = m0*2^(t/2) is a factor of D(m)(t)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--%\"DG6#%\"mG6#%\"tG,$*&#\"\"\"\"\"#F.*&-F (F)F.-%#lnG6#F/F.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "an swer := (1/2)*6*ln(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG,$ *&\"\"$\"\"\"-%#lnG6#\"\"#F(F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 377 10 "6. If " }{XPPEDIT 519 0 "u(t);" "6#-%\"uG6#%\"tG" }{TEXT 518 56 " is the unique solution of the initial value problem " } {XPPEDIT 515 0 "diff(u(t),t) = 20-5*u(t),u(0) = 3;" "6$/-%%diffG6$-%\" uG6#%\"tGF*,&\"#?\"\"\"*&\"\"&F--F(6#F*F-!\"\"/-F(6#\"\"!\"\"$" } {TEXT 514 23 ", \n then what is " }{XPPEDIT 517 0 "limit(u(t),t \+ = infinity);" "6#-%&limitG6$-%\"uG6#%\"tG/F)%)infinityG" }{TEXT 516 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 380 1 "\n " }{TEXT 381 4 "a) " }{XPPEDIT 390 0 "1;" "6#\"\"\"" }{TEXT 384 16 " \+ b) " }{XPPEDIT 391 0 "2;" "6#\"\"#" }{TEXT 385 33 " \+ c) 3 d) " }{XPPEDIT 392 0 "4;" "6#\"\"%" }{TEXT 389 16 " e) " }{XPPEDIT 393 0 "5;" "6#\"\"&" }{TEXT 383 2 " \+ " }{TEXT 382 20 " \n" }{TEXT 378 5 "f) " } {XPPEDIT 394 0 "6;" "6#\"\"'" }{TEXT 388 15 " g) " } {XPPEDIT 395 0 "10;" "6#\"#5" }{TEXT 387 10 " " }{TEXT 396 1 " " }{TEXT 379 20 "h) 12 i) " }{XPPEDIT 397 0 "15;" "6#\"#: " }{TEXT 386 15 " j) " }{XPPEDIT 398 0 "20;" "6#\"#?" } {TEXT 399 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 646 13 "Solution: d\n " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "dsolve(\{diff(u(t),t) = 20-5*u(t), u(0) = 3\}, u(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6#%\"tG,&\"\"%\"\"\"- %$expG6#,$*&\"\"&F*F'F*!\"\"F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit(rhs(%), t = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 305 18 "7. Suppose that \+ " }{XPPEDIT 522 0 "f(x) = arcsin(x);" "6#/-%\"fG6#%\"xG-%'arcsinG6#F' " }{TEXT 520 16 ". Calculate " }{XPPEDIT 523 0 "D(f)(sqrt(3)/2);" "6#--%\"DG6#%\"fG6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT 521 3 ". \+ " }{TEXT 538 31 "(The derivative of f(x) at " }{XPPEDIT 258 0 "x \+ = sqrt(3)/2;" "6#/%\"xG*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT 539 3 " )." }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT 304 0 "" }}{PARA 3 "" 0 "" {TEXT 303 20 "a) 1 b) " }{XPPEDIT 531 0 "sqrt(2);" "6 #-%%sqrtG6#\"\"#" }{TEXT 524 15 " c) " }{XPPEDIT 532 0 "1/s qrt(2);" "6#*&\"\"\"F$-%%sqrtG6#\"\"#!\"\"" }{TEXT 525 17 " \+ d) " }{XPPEDIT 533 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT 526 15 " e) " }{XPPEDIT 534 0 "2/sqrt(3);" "6#*&\"\"#\"\"\"-%%sqrtG 6#\"\"$!\"\"" }{TEXT 527 8 " \nf) " }{XPPEDIT 535 0 "1/2;" "6#*&\" \"\"F$\"\"#!\"\"" }{TEXT 528 33 " g) 2 h) " } {XPPEDIT 536 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 529 15 " i) " }{XPPEDIT 537 0 "2*sqrt(3);" "6#*&\"\"#\"\"\" -%%sqrtG6#\"\"$F%" }{TEXT 530 16 " j) 4" }{TEXT 302 1 "\n" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 664 11 "Solution: g" }}{PARA 0 "" 0 " " {TEXT 665 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "D(arcsin)(sqrt(3)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\" \"%#\"\"\"\"\"#" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 320 17 "8. Suppose \+ that " }{XPPEDIT 257 0 "f(x) = -arctan(1/x);" "6#/-%\"fG6#%\"xG,$-%'a rctanG6#*&\"\"\"F-F'!\"\"F." }{TEXT 321 15 ". What is " } {XPPEDIT 324 0 "D(f)(sqrt(3));" "6#--%\"DG6#%\"fG6#-%%sqrtG6#\"\"$" } {TEXT 322 35 "? (The derivative of f(x) at " }{XPPEDIT 325 0 "x \+ = sqrt(3);" "6#/%\"xG-%%sqrtG6#\"\"$" }{TEXT 323 3 " )." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 327 4 "a) \+ " }{XPPEDIT 542 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 541 19 " \+ b) " }{XPPEDIT 336 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" } {TEXT 328 18 " c) " }{XPPEDIT 337 0 "2/3;" "6#*&\"\"#\" \"\"\"\"$!\"\"" }{TEXT 329 19 " d) " }{XPPEDIT 543 0 "1 /4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 540 16 " e) " } {XPPEDIT 338 0 "1/sqrt(3);" "6#*&\"\"\"F$-%%sqrtG6#\"\"$!\"\"" }{TEXT 330 9 " \nf) " }{XPPEDIT 339 0 "sqrt(3)/2;" "6#*&-%%sqrtG6#\"\"$\" \"\"\"\"#!\"\"" }{TEXT 331 16 " g) " }{XPPEDIT 340 0 "sqrt (3);" "6#-%%sqrtG6#\"\"$" }{TEXT 332 14 " h) " }{XPPEDIT 341 0 "2*sqrt(3);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 333 15 " \+ i) " }{XPPEDIT 342 0 "2;" "6#\"\"#" }{TEXT 334 18 " \+ j) " }{XPPEDIT 343 0 "4;" "6#\"\"%" }{TEXT 326 1 "\n" }{TEXT 344 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 662 11 "Solution: d" }}{PARA 0 "" 0 "" {TEXT 663 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "D(x -> -arctan(1/x ))(sqrt(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 260 1 " " }{TEXT 319 1 "9" }{TEXT -1 2 " . " }{TEXT 318 11 "Calculate " }{XPPEDIT 545 0 "Int(x*sin(x),x = 0 .. Pi/2);" "6#-%$IntG6$*&%\"xG\"\"\"-%$sinG6#F'F(/F';\"\"!*&%#PiGF(\"\"# !\"\"" }{TEXT 544 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 358 4 "a) \+ " }{XPPEDIT 553 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 546 18 " \+ b) " }{XPPEDIT 554 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 547 18 " c) " }{XPPEDIT 555 0 "3/4;" "6#*&\"\"$\"\"\"\" \"%!\"\"" }{TEXT 548 42 " d) 1 e) 2 \nf) " }{XPPEDIT 556 0 "pi/4;" "6#*&%#piG\"\"\"\"\"%!\"\"" }{TEXT 549 18 " \+ g) " }{XPPEDIT 557 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" } {TEXT 550 16 " h) " }{XPPEDIT 558 0 "3*Pi/4;" "6#*(\"\"$\" \"\"%#PiGF%\"\"%!\"\"" }{TEXT 551 14 " i) " }{XPPEDIT 559 0 "Pi;" "6#%#PiG" }{TEXT 552 15 " j) " }{XPPEDIT 560 0 "2*Pi; " "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 359 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 660 11 "Solution: d" }}{PARA 0 "" 0 "" {TEXT 661 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "int(x*sin(x),x=0.. Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 315 17 "10. Calculate " }{XPPEDIT 361 0 "int(x^2*exp(-x),x = 0 .. 1);" "6#-%$intG6$*&%\"xG\"\"#-%$expG6# ,$F'!\"\"\"\"\"/F';\"\"!F." }{TEXT 360 2 " ." }}{PARA 0 "" 0 "" {TEXT 314 17 " " }}{PARA 0 "" 0 "" {TEXT 362 5 "a) " } {XPPEDIT 570 0 "1-1/exp(1);" "6#,&\"\"\"F$*&F$F$-%$expG6#F$!\"\"F)" } {TEXT 565 12 " b) " }{XPPEDIT 571 0 "1-2/exp(1);" "6#,&\"\"\"F $*&\"\"#F$-%$expG6#F$!\"\"F*" }{TEXT 566 14 " c) " } {XPPEDIT 572 0 "1-3/exp(1);" "6#,&\"\"\"F$*&\"\"$F$-%$expG6#F$!\"\"F* " }{TEXT 567 14 " d) " }{XPPEDIT 573 0 "1-4/exp(1);" "6#,&\" \"\"F$*&\"\"%F$-%$expG6#F$!\"\"F*" }{TEXT 568 13 " e) " } {XPPEDIT 574 0 "1-5/exp(1);" "6#,&\"\"\"F$*&\"\"&F$-%$expG6#F$!\"\"F* " }{TEXT 569 8 " \nf) " }{XPPEDIT 575 0 "2-1/exp(1);" "6#,&\"\"#\" \"\"*&F%F%-%$expG6#F%!\"\"F*" }{TEXT 564 14 " g) " } {XPPEDIT 576 0 "2-2/exp(1);" "6#,&\"\"#\"\"\"*&F$F%-%$expG6#F%!\"\"F* " }{TEXT 563 13 " h) " }{XPPEDIT 577 0 "2-3/exp(1);" "6#,&\" \"#\"\"\"*&\"\"$F%-%$expG6#F%!\"\"F+" }{TEXT 562 14 " i) " } {XPPEDIT 578 0 "2-4/exp(1);" "6#,&\"\"#\"\"\"*&\"\"%F%-%$expG6#F%!\"\" F+" }{TEXT 561 15 " j) " }{XPPEDIT 579 0 "2-5/exp(1);" "6#, &\"\"#\"\"\"*&\"\"&F%-%$expG6#F%!\"\"F+" }{TEXT 363 1 "\n" }{TEXT 580 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 658 11 "Solution: j" }}{PARA 0 " " 0 "" {TEXT 659 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int(x^2*exp(-x),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"&\"\"\"-%$expG6#!\"\"F&F*\"\"#F&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 19 "11. Calculate " } {XPPEDIT 258 0 "16*Int(x^3*ln(x),x = 1 .. exp(1));" "6#*&\"#;\"\"\"-%$ IntG6$*&%\"xG\"\"$-%#lnG6#F*F%/F*;F%-%$expG6#F%F%" }{TEXT 364 2 " ." } }{PARA 0 "" 0 "" {TEXT 269 4 "a) " }{XPPEDIT 366 1 "exp(4)-1;" "6#,&- %$expG6#\"\"%\"\"\"F(!\"\"" }{TEXT 306 13 " b) " }{XPPEDIT 367 1 "2*exp(4)-1;" "6#,&*&\"\"#\"\"\"-%$expG6#\"\"%F&F&F&!\"\"" } {TEXT 307 13 " c) " }{XPPEDIT 368 1 "3*exp(4)-1;" "6#,&*&\"\" $\"\"\"-%$expG6#\"\"%F&F&F&!\"\"" }{TEXT 308 14 " d) " } {XPPEDIT 369 1 "4*exp(4)-1;" "6#,&*&\"\"%\"\"\"-%$expG6#F%F&F&F&!\"\" " }{TEXT 309 12 " e) " }{XPPEDIT 370 1 "5*exp(4)-1;" "6#,&*&\" \"&\"\"\"-%$expG6#\"\"%F&F&F&!\"\"" }{TEXT 310 11 " \n f) " } {XPPEDIT 371 1 "exp(4)+1;" "6#,&-%$expG6#\"\"%\"\"\"F(F(" }{TEXT 311 13 " g) " }{XPPEDIT 372 1 "2*exp(4)+1;" "6#,&*&\"\"#\"\"\"-%$ expG6#\"\"%F&F&F&F&" }{TEXT 312 13 " h) " }{XPPEDIT 373 1 "3* exp(4)+1;" "6#,&*&\"\"$\"\"\"-%$expG6#\"\"%F&F&F&F&" }{TEXT 313 14 " \+ i) " }{XPPEDIT 374 0 "4*exp(4)+1;" "6#,&*&\"\"%\"\"\"-%$expG6 #F%F&F&F&F&" }{TEXT 365 14 " j) " }{XPPEDIT 375 0 "5*exp(4)+ 1;" "6#,&*&\"\"&\"\"\"-%$expG6#\"\"%F&F&F&F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 656 11 "Solution: h" }}{PARA 0 "" 0 "" {TEXT 657 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "16*int(x^3*ln(x),x = 1 .. exp(1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*&\"\"$\"\"\"-%$expG6#\"\"%F&F&F&F&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 316 17 "12. Calculate " } {XPPEDIT 256 0 "int((3*x+2)/(x^2+x),x = 1 .. 2);" "6#-%$intG6$*&,&*&\" \"$\"\"\"%\"xGF*F*\"\"#F*F*,&*$F+F,F*F+F*!\"\"/F+;F*F," }{TEXT 376 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT -1 14 " \+ b) " }{XPPEDIT 18 0 "ln(3);" "6#-%#lnG6#\"\"$" }{TEXT -1 15 " \+ c) " }{XPPEDIT 18 0 "2*ln(2);" "6#*&\"\"#\"\"\"-%#lnG6#F$F%" } {TEXT -1 11 " d) " }{XPPEDIT 18 0 "2*ln(3);" "6#*&\"\"#\"\"\"-% #lnG6#\"\"$F%" }{TEXT -1 12 " e) " }{XPPEDIT 18 0 "3*ln(2);" " 6#*&\"\"$\"\"\"-%#lnG6#\"\"#F%" }{TEXT -1 20 " \nf) " } {XPPEDIT 18 0 "ln(6);" "6#-%#lnG6#\"\"'" }{TEXT -1 14 " g) \+ " }{XPPEDIT 18 0 "ln(2/3);" "6#-%#lnG6#*&\"\"#\"\"\"\"\"$!\"\"" } {TEXT -1 12 " h) " }{XPPEDIT 18 0 "ln(3/2);" "6#-%#lnG6#*&\"\" $\"\"\"\"\"#!\"\"" }{TEXT -1 14 " i) " }{XPPEDIT 18 0 "ln(9/ 2);" "6#-%#lnG6#*&\"\"*\"\"\"\"\"#!\"\"" }{TEXT -1 14 " j) \+ " }{XPPEDIT 18 0 "ln(12);" "6#-%#lnG6#\"#7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 655 12 "Solutio n: f " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "norm al(1/x+2/(x+1) - 1/(x+1)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,(*& \"\"$\"\"\")%\"xG\"\"#F'F'*&F&F'F)F'F'F'F'F'F)!\"\",&F)F'F'F'!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "testeq(int((3*x+2)/(x^2+x), \+ x= 1..2) = ln(6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT 287 16 "13. Calculate " }{XPPEDIT 408 0 "Int((3*x^2+3*x+1)/x /((x+1)^2),x = 1 .. 2);" "6#-%$IntG6$*(,(*&\"\"$\"\"\"*$%\"xG\"\"#F*F* *&F)F*F,F*F*F*F*F*F,!\"\"*$,&F,F*F*F*F-F//F,;F*F-" }{TEXT 407 4 " . \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 3 "a) \+ " }{XPPEDIT 18 0 "ln(2)-1/3;" "6#,&-%#lnG6#\"\"#\"\"\"*&F(F(\"\"$!\"\" F+" }{TEXT -1 13 " b) " }{XPPEDIT 18 0 "ln(3)-1/2;" "6#,&-%#l nG6#\"\"$\"\"\"*&F(F(\"\"#!\"\"F+" }{TEXT -1 14 " c) " } {XPPEDIT 18 0 "2*ln(2)-1/6;" "6#,&*&\"\"#\"\"\"-%#lnG6#F%F&F&*&F&F&\" \"'!\"\"F," }{TEXT -1 10 " d) " }{XPPEDIT 18 0 "2*ln(3)-1/4;" "6 #,&*&\"\"#\"\"\"-%#lnG6#\"\"$F&F&*&F&F&\"\"%!\"\"F-" }{TEXT -1 11 " \+ e) " }{XPPEDIT 18 0 "3*ln(2)-1/2;" "6#,&*&\"\"$\"\"\"-%#lnG6#\"\" #F&F&*&F&F&F*!\"\"F," }{TEXT -1 20 " \nf) " }{XPPEDIT 18 0 "ln(6)-2/3;" "6#,&-%#lnG6#\"\"'\"\"\"*&\"\"#F(\"\"$!\"\"F," } {TEXT -1 13 " g) " }{XPPEDIT 18 0 "ln(3/2)-1/4;" "6#,&-%#lnG6 #*&\"\"$\"\"\"\"\"#!\"\"F)*&F)F)\"\"%F+F+" }{TEXT -1 11 " h) " }{XPPEDIT 18 0 "ln(3/2)-1/3;" "6#,&-%#lnG6#*&\"\"$\"\"\"\"\"#!\"\"F)*& F)F)F(F+F+" }{TEXT -1 13 " i) " }{XPPEDIT 18 0 "ln(9/2)-1/6; " "6#,&-%#lnG6#*&\"\"*\"\"\"\"\"#!\"\"F)*&F)F)\"\"'F+F+" }{TEXT -1 12 " j) " }{XPPEDIT 18 0 "ln(9/4)-1/3;" "6#,&-%#lnG6#*&\"\"*\"\" \"\"\"%!\"\"F)*&F)F)\"\"$F+F+" }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 671 12 "Solution: i " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A := int((3*x^2+3*x+1)/x/((x +1)^2),x = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,(#\"\"\" \"\"'!\"\"*&\"\"#F'-%#lnG6#\"\"$F'F'-F-6#F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "testeq(A = ln(9/2)-1/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 1 " " }{TEXT 270 17 "14. Calcul ate " }{XPPEDIT 582 0 "Int(sin(x)^2*cos(x)^3,x = 0 .. Pi/2);" "6#-%$ IntG6$*&-%$sinG6#%\"xG\"\"#-%$cosG6#F*\"\"$/F*;\"\"!*&%#PiG\"\"\"F+!\" \"" }{TEXT 581 1 "." }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 409 80 "a) 1/15 b) 2/15 c) 1/5 \+ d) 4/15 e) 1/3 " }{TEXT 412 1 " " }{TEXT 413 18 " \+ " }}{PARA 0 "" 0 "" {TEXT 410 39 "f) 2/5 g) 7/15 \+ " }{TEXT -1 1 " " }{TEXT 411 42 "h) 8/15 i) 3/5 \+ j) 2/3 " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 653 11 "Solution: b" }}{PARA 0 "" 0 "" {TEXT 654 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "int(sin(x)^2*cos(x)^3,x = 0 .. Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 39 "15. From the reduction formula\n " } {XPPEDIT 18 0 "Int(cos(x)^n,x) = sin(x)*cos(x)^(n-1)/n+(n-1)/n*Int(cos (x)^(n-2),x);" "6#/-%$IntG6$)-%$cosG6#%\"xG%\"nGF+,&*(-%$sinG6#F+\"\" \")-F)6#F+,&F,F2F2!\"\"F2F,F7F2*(,&F,F2F2F7F2F,F7-F%6$)-F)6#F+,&F,F2\" \"#F7F+F2F2" }{TEXT -1 77 "\n\n it follows that there are rational num bers A, B, and C for which\n\n " }{XPPEDIT 18 0 "Int(cos(x)^8,x ) = A*sin(x)*cos(x)^7+B*sin(x)*cos(x)^5+C*Int(cos(x)^4,x);" "6#/-%$Int G6$*$-%$cosG6#%\"xG\"\")F+,(*(%\"AG\"\"\"-%$sinG6#F+F0-F)6#F+\"\"(F0*( %\"BGF0-F26#F+F0-F)6#F+\"\"&F0*&%\"CGF0-F%6$*$-F)6#F+\"\"%F+F0F0" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 637 11 "What is \+ C?" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 289 77 "a) 8/15 \+ b) 2/3 c) 7/9 d) 35/48 e) " }{TEXT 414 3 "5/7" }{TEXT 638 3 " " }{TEXT 292 18 " " }} {PARA 0 "" 0 "" {TEXT 290 36 "f) 24/35 g) 16/21 " } {TEXT -1 1 " " }{TEXT 291 38 "h) 9/16 i) 3/4 j)" }{TEXT -1 1 " " }{TEXT 639 3 "3/5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 651 11 "Solution: d" }}{PARA 0 "" 0 "" {TEXT 652 1 " " } {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eqn1 := J(n) = 1/n*sin(x)*cos(x)^(n-1) + (n-1)/n*J(n-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%\"JG6#%\"nG,&*(F)!\"\"-%$sinG6#%\"xG\"\"\")- %$cosGF/,&F)F1F1F,F1F1*(F5F1F)F,-F'6#,&F)F1\"\"#F,F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn2 := subs(n=n-2, eqn1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"JG6#,&%\"nG\"\"\"\"\"#!\" \",&*(F)F--%$sinG6#%\"xGF+)-%$cosGF2,&F*F+\"\"$F-F+F+*(F7F+F)F--F'6#,& F*F+\"\"%F-F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn3 := \+ subs(eqn2, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%\"JG6# %\"nG,&*(F)!\"\"-%$sinG6#%\"xG\"\"\")-%$cosGF/,&F)F1F1F,F1F1*(F5F1F)F, ,&*(,&F)F1\"\"#F,F,F-F1)F3,&F)F1\"\"$F,F1F1*(F " 0 "" {MPLTEXT 1 0 24 "eqn4 := subs(n =8, eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%\"JG6#\"\"),( *&#\"\"\"F)F-*&-%$sinG6#%\"xGF-)-%$cosGF1\"\"(F-F-F-*&#F6\"#[F-*&F/F-) F4\"\"&F-F-F-*&#\"#NF9F--F'6#\"\"%F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "testeq(cos(x)^8 = diff( 1/8*sin(x)*cos(x)^7+7/48*sin( x)*cos(x)^5+35/48*int(cos(x)^4,x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 272 14 "16. Calculate " }{XPPEDIT 599 0 "int(x^2/sqrt(1-x^2),x = 0 .. 1/sqrt(2));" "6#-%$intG6$*&%\"xG\"\"#-%% sqrtG6#,&\"\"\"F-*$F'F(!\"\"F//F';\"\"!*&F-F--F*6#F(F/" }{TEXT 598 1 " ." }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 288 3 "a) \+ " }{XPPEDIT 256 0 "(Pi-2)/8;" "6#*&,&%#PiG\"\"\"\"\"#!\"\"F&\"\")F(" } {TEXT 600 11 " b) " }{XPPEDIT 257 0 "(Pi-2)/4;" "6#*&,&%#PiG\" \"\"\"\"#!\"\"F&\"\"%F(" }{TEXT 601 16 " c) " }{XPPEDIT 258 0 "(Pi-2)/2;" "6#*&,&%#PiG\"\"\"\"\"#!\"\"F&F'F(" }{TEXT 602 14 " \+ d) " }{XPPEDIT 256 0 "(Pi-1)/8;" "6#*&,&%#PiG\"\"\"F&!\"\"F& \"\")F'" }{TEXT 604 11 " e) " }{XPPEDIT 256 0 "(Pi-1)/4;" "6#*& ,&%#PiG\"\"\"F&!\"\"F&\"\"%F'" }{TEXT 605 10 " \nf) " }{XPPEDIT 261 0 "(Pi-1)/2;" "6#*&,&%#PiG\"\"\"F&!\"\"F&\"\"#F'" }{TEXT 603 13 " \+ g) " }{XPPEDIT 256 0 "(4-Pi)/8;" "6#*&,&\"\"%\"\"\"%#PiG!\"\" F&\"\")F(" }{TEXT 606 12 " h) " }{XPPEDIT 256 0 "(4-Pi)/4;" "6 #*&,&\"\"%\"\"\"%#PiG!\"\"F&F%F(" }{TEXT 607 12 " i) " } {XPPEDIT 256 0 "(4-Pi)/2;" "6#*&,&\"\"%\"\"\"%#PiG!\"\"F&\"\"#F(" } {TEXT 608 12 " j) " }{XPPEDIT 265 0 "(8-Pi)/8;" "6#*&,&\"\")\" \"\"%#PiG!\"\"F&F%F(" }{TEXT 609 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 650 14 "Solution: a " }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A := normal(2/x-3/(x^2+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG*(,(*&\"\"#\"\"\")%\"xGF(F)F)F(F)*&\"\"$F)F+F)!\" \"F)F+F.,&*$F*F)F)F)F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(x^2/sqrt(1-x^2),x=0..1/sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"%!\"\"*&\"\")F'%#PiGF%F%" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 274 4 "17. " }{TEXT 286 12 " Calculate " }{XPPEDIT 256 0 "int(1 /sqrt(1+x^2),x = 0 .. 1);" "6#-%$intG6$*&\"\"\"F'-%%sqrtG6#,&F'F'*$%\" xG\"\"#F'!\"\"/F-;\"\"!F'" }{TEXT 610 1 "." }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 611 4 "a) " }{XPPEDIT 630 0 "1/2;" "6#*&\"\"\"F$\"\"#!\" \"" }{TEXT 621 1 " " }{XPPEDIT 631 0 "ln(2);" "6#-%#lnG6#\"\"#" } {TEXT 620 8 " b) " }{XPPEDIT 257 0 "ln(2);" "6#-%#lnG6#\"\"#" } {TEXT 612 23 " c) " }{XPPEDIT 258 0 "2*ln(2);" "6#* &\"\"#\"\"\"-%#lnG6#F$F%" }{TEXT 613 17 " d) " }{XPPEDIT 632 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 622 1 " " }{XPPEDIT 256 0 "sqrt(1+ln(2));" "6#-%%sqrtG6#,&\"\"\"F'-%#lnG6#\"\"#F'" }{TEXT 618 9 " e) " }{XPPEDIT 259 0 "sqrt(1+ln(2));" "6#-%%sqrtG6#,&\"\"\"F' -%#lnG6#\"\"#F'" }{TEXT 614 12 " \nf) " }{XPPEDIT 260 0 "2*ln(3 );" "6#*&\"\"#\"\"\"-%#lnG6#\"\"$F%" }{TEXT 615 10 " g) " } {XPPEDIT 669 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 668 1 " " } {XPPEDIT 261 0 "ln(1+1/sqrt(2));" "6#-%#lnG6#,&\"\"\"F'*&F'F'-%%sqrtG6 #\"\"#!\"\"F'" }{TEXT 616 5 " h) " }{XPPEDIT 262 0 "ln(1+1/sqrt(2)); " "6#-%#lnG6#,&\"\"\"F'*&F'F'-%%sqrtG6#\"\"#!\"\"F'" }{TEXT 619 9 " \+ i) " }{XPPEDIT 667 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 666 1 " " }{XPPEDIT 263 0 "ln(1+sqrt(2));" "6#-%#lnG6#,&\"\"\"F'-%%sqrtG6#\" \"#F'" }{TEXT 617 11 " j) " }{XPPEDIT 264 0 "ln(1+sqrt(2));" "6 #-%#lnG6#,&\"\"\"F'-%%sqrtG6#\"\"#F'" }{TEXT 633 0 "" }}{PAGEBK } {PARA 0 "" 0 "" {TEXT 670 13 "Solution: j\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "te steq(int(1/sqrt(1+x^2),x = 0 .. 1)=ln(sqrt(2)+1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 " " 0 "" {TEXT 276 4 "18. " }{TEXT 583 1 " " }{TEXT 595 56 "There are un ique rational numbers A, B, and C such that " }{XPPEDIT 279 0 "int((2* x^2+2-3*x)/x/(x^2+1),x = 1 .. 2) = A*arctan(2)+B*ln(2)+C*Pi;" "6#/-%$i ntG6$*(,(*&\"\"#\"\"\"*$%\"xGF*F+F+F*F+*&\"\"$F+F-F+!\"\"F+F-F0,&*$F-F *F+F+F+F0/F-;F+F*,(*&%\"AGF+-%'arctanG6#F*F+F+*&%\"BGF+-%#lnG6#F*F+F+* &%\"CGF+%#PiGF+F+" }{TEXT 594 1 "." }{TEXT 596 2 " " }{TEXT 597 16 "W hat is A ?\na) " }{XPPEDIT 256 0 "-5;" "6#,$\"\"&!\"\"" }{TEXT 585 11 " b) " }{XPPEDIT 262 0 "-4;" "6#,$\"\"%!\"\"" }{TEXT 586 13 " c) " }{XPPEDIT 270 0 "-3;" "6#,$\"\"$!\"\"" }{TEXT 587 12 " d) " }{XPPEDIT 271 0 "-2;" "6#,$\"\"#!\"\"" }{TEXT 588 12 " \+ e) " }{XPPEDIT 272 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT 589 10 " \+ \nf) " }{XPPEDIT 273 0 "5;" "6#\"\"&" }{TEXT 590 16 " g) \+ " }{XPPEDIT 274 0 "4;" "6#\"\"%" }{TEXT 591 14 " h) " } {XPPEDIT 275 0 "3;" "6#\"\"$" }{TEXT 592 15 " i) " } {XPPEDIT 276 0 "2;" "6#\"\"#" }{TEXT 593 16 " j) " } {XPPEDIT 629 0 "1;" "6#\"\"\"" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 584 0 "" }{TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 647 13 "Solut ion: c\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int((2*x^2+2-3*x)/x/(x^2+1),x=1..2) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"$\"\"\"-%'arctanG6#\"\"#F &!\"\"*&F*F&-%#lnGF)F&F&*(F%F&\"\"%F+%#PiGF&F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "convert((2*x^2+2-3*x)/x/(x^2+1), parfrac, x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"$\"\"\",&*$)%\"xG\"\"#F&F& F&F&!\"\"F,*&F+F&F*F,F&" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 277 55 "1 9. The region in the first quadrant bounded above by " }{XPPEDIT 18 0 "y = 3-x^2;" "6#/%\"yG,&\"\"$\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT 635 17 " and below by " }{XPPEDIT 18 0 "y = 2*x;" "6#/%\"yG*&\"\"#\"\"\"% \"xGF'" }{TEXT 636 88 " is rotated about the x-axis. What is the vol ume of the resulting solid of revolution?" }}{PARA 260 "" 0 "" {TEXT 634 5 "a) " }{XPPEDIT 18 0 "88*Pi/15;" "6#*(\"#))\"\"\"%#PiGF%\"#:! \"\"" }{TEXT 423 12 " b) " }{XPPEDIT 265 0 "92*Pi/15;" "6#*(\" ##*\"\"\"%#PiGF%\"#:!\"\"" }{TEXT 415 14 " c) " }{XPPEDIT 266 0 "33*Pi/5;" "6#*(\"#L\"\"\"%#PiGF%\"\"&!\"\"" }{TEXT 416 12 " \+ d) " }{XPPEDIT 256 0 "20*Pi/3;" "6#*(\"#?\"\"\"%#PiGF%\"\"$!\"\" " }{TEXT 421 13 " e) " }{XPPEDIT 267 0 "36*Pi/5;" "6#*(\"#O\" \"\"%#PiGF%\"\"&!\"\"" }{TEXT 417 12 " \nf) " }{XPPEDIT 268 0 " 112*Pi/15;" "6#*(\"$7\"\"\"\"%#PiGF%\"#:!\"\"" }{TEXT 418 13 " \+ g) " }{XPPEDIT 273 0 "40*Pi/3;" "6#*(\"#S\"\"\"%#PiGF%\"\"$!\"\"" } {TEXT 419 12 " h) " }{XPPEDIT 274 0 "48*Pi/5;" "6#*(\"#[\"\"\" %#PiGF%\"\"&!\"\"" }{TEXT 422 15 " i) " }{XPPEDIT 275 0 "14 8*Pi/15;" "6#*(\"$[\"\"\"\"%#PiGF%\"#:!\"\"" }{TEXT 420 11 " j) \+ " }{XPPEDIT 276 0 "10*Pi;" "6#*&\"#5\"\"\"%#PiGF%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 648 13 "Solution: a\n " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "Pi*int((3-x^2)^2-(2*x)^2,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#))\"\"\"\"#:!\"\"%#PiGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 280 57 "20. The region above the x-axis and under the graph of " } {XPPEDIT 626 0 "y = sin(x)/x;" "6#/%\"yG*&-%$sinG6#%\"xG\"\"\"F)!\"\" " }{TEXT 623 6 " , " }{XPPEDIT 627 0 "0 < x;" "6#2\"\"!%\"xG" } {TEXT 624 1 " " }{XPPEDIT 628 0 "` ` <= Pi/2;" "6#1%\"~G*&%#PiG\"\"\" \"\"#!\"\"" }{TEXT 625 89 " is rotated about the y-axis. What is the volume of the resulting solid of revolution?\n" }}{PARA 260 "" 0 "" {TEXT 279 5 "a) " }{XPPEDIT 429 1 "Pi/6;" "6#*&%#PiG\"\"\"\"\"'!\"\" " }{TEXT 424 13 " b) " }{XPPEDIT 430 1 "Pi/4;" "6#*&%#PiG\"\" \"\"\"%!\"\"" }{TEXT 281 16 " c) " }{XPPEDIT 431 1 "Pi/3; " "6#*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 282 13 " d) " }{XPPEDIT 432 1 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 283 12 " e) \+ " }{XPPEDIT 433 0 "3*Pi/4;" "6#*(\"\"$\"\"\"%#PiGF%\"\"%!\"\"" }{TEXT 425 10 " \nf) " }{XPPEDIT 434 1 "Pi;" "6#%#PiG" }{TEXT 426 13 " \+ g) " }{XPPEDIT 18 0 "3*Pi/2;" "6#*(\"\"$\"\"\"%#PiGF%\"\"#!\" \"" }{TEXT 284 13 " h) " }{XPPEDIT 435 0 "2*Pi;" "6#*&\"\"#\" \"\"%#PiGF%" }{TEXT 285 11 " i) " }{XPPEDIT 436 0 "3*Pi;" "6#*& \"\"$\"\"\"%#PiGF%" }{TEXT 427 13 " j) " }{XPPEDIT 437 1 "4*P i;" "6#*&\"\"%\"\"\"%#PiGF%" }{TEXT 428 2 " " }{TEXT 278 1 "\n" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 649 13 "Solution: h\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int(2*Pi*x*sin(x)/x,x=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"%#PiGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "13 7 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }