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648 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Nor mal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 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 "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title " -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 } 3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 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 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 292 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 436 13 "Suppose tha t " }{XPPEDIT 437 0 "f(x) = 2^(-1/x);" "6#/-%\"fG6#%\"xG)\"\"#,$*&\"\" \"F,F'!\"\"F-" }{TEXT 435 22 ". Calculate D(f)(2)." }{TEXT -1 0 "" } }{PARA 261 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 18 0 "sqrt(2)*ln(2)/8;" "6#*(-%%sqrtG6#\"\"#\"\"\"-%#lnG6#F'F(\"\")!\"\"" }{TEXT -1 10 " \+ b) " }{XPPEDIT 18 0 "sqrt(2)*ln(2)/4;" "6#*(-%%sqrtG6#\"\"#\"\"\"-%#l nG6#F'F(\"\"%!\"\"" }{TEXT -1 12 " c) " }{XPPEDIT 18 0 "sqrt(2 )*ln(2)/2;" "6#*(-%%sqrtG6#\"\"#\"\"\"-%#lnG6#F'F(F'!\"\"" }{TEXT -1 9 " d) " }{XPPEDIT 18 0 "sqrt(2)*ln(2);" "6#*&-%%sqrtG6#\"\"#\"\" \"-%#lnG6#F'F(" }{TEXT -1 9 " e) " }{XPPEDIT 18 0 "2*sqrt(2)*ln(2 );" "6#*(\"\"#\"\"\"-%%sqrtG6#F$F%-%#lnG6#F$F%" }{TEXT -1 7 " \nf) \+ " }{XPPEDIT 18 0 "sqrt(2)/ln(2)/8;" "6#*(-%%sqrtG6#\"\"#\"\"\"-%#lnG6# F'!\"\"\"\")F," }{TEXT -1 5 " " }{TEXT 293 8 " g) " }{XPPEDIT 18 0 "sqrt(2)/ln(2)/4;" "6#*(-%%sqrtG6#\"\"#\"\"\"-%#lnG6#F'!\"\"\"\"% F," }{TEXT -1 1 " " }{TEXT 438 13 " h) " }{XPPEDIT 18 0 "sqrt (2)/ln(2)/2;" "6#*(-%%sqrtG6#\"\"#\"\"\"-%#lnG6#F'!\"\"F'F," }{TEXT -1 1 " " }{TEXT 636 13 " i) " }{XPPEDIT 18 0 "sqrt(2)/ln(2); " "6#*&-%%sqrtG6#\"\"#\"\"\"-%#lnG6#F'!\"\"" }{TEXT -1 1 " " }{TEXT 439 16 " j) " }{XPPEDIT 18 0 "2*sqrt(2)/ln(2);" "6#*(\"\"# \"\"\"-%%sqrtG6#F$F%-%#lnG6#F$!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PAGEBK }{PARA 0 "" 0 "" {TEXT 610 12 "Solution: a" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A := diff(2^(-1/x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG*()\"\"#,$*&\"\"\"F*%\"xG!\"\"F,F*F+!\"#-%#lnG6#F 'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(x=2, A);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\")F&*&\"\"##F&F)-%#lnG6# F)F&F&F&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 334 16 "2. Calculate " }{XPPEDIT 463 0 "Int(log[10](x),x = 1 .. 10);" "6#-%$IntG6$-&%$logG6# \"#56#%\"xG/F,;\"\"\"F*" }{TEXT 462 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 345 5 "a) " }{XPPEDIT 465 0 "10-1/ln(10);" "6#,&\"#5\"\"\"*&F%F%-%#lnG6#F$!\" \"F*" }{TEXT 464 12 " b) " }{XPPEDIT 261 0 "10+1/ln(10);" "6#, &\"#5\"\"\"*&F%F%-%#lnG6#F$!\"\"F%" }{TEXT 346 13 " c) " } {XPPEDIT 262 0 "10-ln(10);" "6#,&\"#5\"\"\"-%#lnG6#F$!\"\"" }{TEXT 347 11 " d) " }{XPPEDIT 467 0 "10+ln(10);" "6#,&\"#5\"\"\"-%#ln G6#F$F%" }{TEXT 466 10 " e) " }{XPPEDIT 263 0 "10*ln(10)-1;" "6# ,&*&\"#5\"\"\"-%#lnG6#F%F&F&F&!\"\"" }{TEXT 348 9 " \nf) " } {XPPEDIT 264 0 "10*ln(10)+1;" "6#,&*&\"#5\"\"\"-%#lnG6#F%F&F&F&F&" } {TEXT 349 10 " g) " }{TEXT 356 1 " " }{XPPEDIT 265 0 "10-9/ln(10 );" "6#,&\"#5\"\"\"*&\"\"*F%-%#lnG6#F$!\"\"F+" }{TEXT 350 12 " \+ h) " }{XPPEDIT 266 0 "1+10/ln(10);" "6#,&\"\"\"F$*&\"#5F$-%#lnG6#F&! \"\"F$" }{TEXT 351 10 " i) " }{TEXT 354 1 " " }{XPPEDIT 267 0 "l og[10](10);" "6#-&%$logG6#\"#56#F'" }{TEXT 352 2 " " }{TEXT 355 12 " \+ j) " }{XPPEDIT 268 0 "log[10](10)-1;" "6#,&-&%$logG6#\"#56#F( \"\"\"F*!\"\"" }{TEXT 344 1 "\n" }{TEXT 353 0 "" }}{PAGEBK }{PARA 0 " " 0 "" {TEXT 611 12 "Solution: g" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "int(log[10]( x),x=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(\"\"*!\"\"*&\"#5\" \"\"-%#lnG6#\"\"#F)F)*&F(F)-F+6#\"\"&F)F)F),&F*F)F/F)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "testeq(int(log[10](x),x=1..10) = 10 -9/ln(10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 262 3 "3. " }{TEXT 295 1 " " }{TEXT 296 15 " Suppose that " }{XPPEDIT 18 0 "f(x) = x^sqrt(x);" "6#/-%\"fG6#%\"xG)F'-%%sqr tG6#F'" }{TEXT 440 22 ". Calculate D(f)(4)." }}{PARA 3 "" 0 "" {TEXT 294 3 "a) " }{XPPEDIT 446 0 "sqrt(2)+1;" "6#,&-%%sqrtG6#\"\"#\" \"\"F(F(" }{TEXT 441 17 " b) " }{XPPEDIT 447 0 "2*sqrt(2) +1;" "6#,&*&\"\"#\"\"\"-%%sqrtG6#F%F&F&F&F&" }{TEXT 442 13 " c ) " }{XPPEDIT 448 0 "4*sqrt(2)+1;" "6#,&*&\"\"%\"\"\"-%%sqrtG6#\"\"#F &F&F&F&" }{TEXT 443 12 " d) " }{XPPEDIT 449 0 "sqrt(2)+ln(2); " "6#,&-%%sqrtG6#\"\"#\"\"\"-%#lnG6#F'F(" }{TEXT 444 14 " e) \+ " }{XPPEDIT 450 0 "2*sqrt(2)+ln(2);" "6#,&*&\"\"#\"\"\"-%%sqrtG6#F%F& F&-%#lnG6#F%F&" }{TEXT 445 14 " \nf) " }{XPPEDIT 455 0 "4*sqr t(2)+ln(2);" "6#,&*&\"\"%\"\"\"-%%sqrtG6#\"\"#F&F&-%#lnG6#F*F&" } {TEXT 451 8 " g) " }{XPPEDIT 456 0 "ln(2)+1;" "6#,&-%#lnG6#\"\"#\" \"\"F(F(" }{TEXT 452 11 " h) " }{XPPEDIT 457 0 "ln(2)+2;" "6#,& -%#lnG6#\"\"#\"\"\"F'F(" }{TEXT 453 14 " i) " }{XPPEDIT 458 0 "8*(ln(2)+1);" "6#*&\"\")\"\"\",&-%#lnG6#\"\"#F%F%F%F%" }{TEXT 454 10 " j) " }{XPPEDIT 460 0 "sqrt(2)*ln(2);" "6#*&-%%sqrtG6#\"\"# \"\"\"-%#lnG6#F'F(" }{TEXT 459 3 " " }{TEXT 461 6 " \n" }{TEXT 612 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 613 13 "Sol ution: i\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(x^sqrt(x ), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG*$F%#\"\"\"\"\"#F(,& *&F'F(*&F%#!\"\"F)-%#lnG6#F%F(F(F(*&F(F(*$F%#F(F)F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(subs(x=4, %));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\")\"\"\"-%#lnG6#\"\"#F&F&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 468 1 " " }{TEXT 471 134 "is measured. Only 1/3 of the substan ce remains 3 hours later. Express the half-life of this substance in \+ terms of the quantities " }{XPPEDIT 474 0 "alpha = ln(2);" "6#/%&alp haG-%#lnG6#\"\"#" }{TEXT 472 9 " and " }{XPPEDIT 475 0 "beta = ln( 3);" "6#/%%betaG-%#lnG6#\"\"$" }{TEXT 473 1 "." }{TEXT 470 3 " \n\n" } {TEXT 469 4 "a) " }{XPPEDIT 483 0 "beta-alpha;" "6#,&%%betaG\"\"\"%&a lphaG!\"\"" }{TEXT 476 12 " b) " }{XPPEDIT 484 0 "exp(alpha)*b eta-alpha;" "6#,&*&-%$expG6#%&alphaG\"\"\"%%betaGF)F)F(!\"\"" }{TEXT 477 12 " c) " }{XPPEDIT 485 0 "alpha/(beta-alpha);" "6#*&%&alp haG\"\"\",&%%betaGF%F$!\"\"F(" }{TEXT 478 9 " d) " }{XPPEDIT 486 0 "beta/(beta-alpha);" "6#*&%%betaG\"\"\",&F$F%%&alphaG!\"\"F(" } {TEXT 479 13 " e) " }{TEXT 316 1 " " }{XPPEDIT 487 0 "exp(alp ha)*alpha/(beta-alpha);" "6#*(-%$expG6#%&alphaG\"\"\"F'F(,&%%betaGF(F' !\"\"F+" }{TEXT 300 19 " " }}{PARA 0 "" 0 "" {TEXT 298 4 "f) " }{XPPEDIT 488 0 "exp(alpha)*beta/alpha;" "6#*(-%$expG6#%& alphaG\"\"\"%%betaGF(F'!\"\"" }{TEXT 480 18 " g) " } {XPPEDIT 489 0 "beta/alpha;" "6#*&%%betaG\"\"\"%&alphaG!\"\"" }{TEXT 481 15 " " }{TEXT 490 1 " " }{TEXT 299 5 "h) " } {XPPEDIT 491 0 "alpha/beta;" "6#*&%&alphaG\"\"\"%%betaG!\"\"" }{TEXT 482 17 " i) " }{XPPEDIT 638 0 "exp(alpha)*alpha/beta;" "6 #*(-%$expG6#%&alphaG\"\"\"F'F(%%betaG!\"\"" }{TEXT 637 16 " \+ j) " }{XPPEDIT 492 0 "exp(beta)*alpha/beta;" "6#*(-%$expG6#%%betaG\" \"\"%&alphaGF(F'!\"\"" }{TEXT 493 3 " " }}{PARA 3 "" 0 "" {TEXT 297 1 " " }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 614 13 "Solution: j\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.%$tauG!\"\"F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eq n := m(3) = m(0)/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/*&%#m0G \"\"\")\"\"#,$*&\"\"$F(%$tauG!\"\"F(F/,$*&F-F/F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "tau = solve(eqn, tau);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%$tauG,$*(\"\"$\"\"\"-%#lnG6#\"\"#F(-F*6#F'!\"\"F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 397 99 "5. The mass of an Eek. oli \+ colony is growing exponentially and is increasing at the rate of 20 " }{XPPEDIT 258 0 "mu;" "6#%#muG" }{TEXT 647 35 "g/hr when the colon y's mass is 60 " }{XPPEDIT 495 0 "mu;" "6#%#muG" }{TEXT 494 4 "g. " } {TEXT 496 37 " How many hours is the half-life? " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 400 87 "a) ln(2) b ) 2 ln(2) c) 3 ln(2) d) 6 ln(2) e)" } {TEXT 403 1 " " }{TEXT 498 8 "12 ln(2)" }{TEXT 499 1 " " }{TEXT 402 3 " " }{TEXT 401 19 " " }}{PARA 0 "" 0 "" {TEXT 398 42 "f) 1/ln(2) g) 2/ln(2) " }{TEXT -1 1 " " } {TEXT 399 47 "h) 3/ln(2) i) 6/ln(2) j) " }{TEXT 500 8 "12/ln(2)" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 615 13 "Solution: c\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m := t -> m0*2^(t/tau);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mGf*6#%\"tG6\"6$%)operatorG%&arrow GF(*&%#m0G\"\"\")\"\"#*&9$F.%$tauG!\"\"F.F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "D(m)(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**% #m0G\"\"\")\"\"#*&%\"tGF%%$tauG!\"\"F%F*F+-%#lnG6#F'F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "eqn := 'D(m)(t)' = 'm(t)'*ln(2)/tau ; \n#Notice that m(t) = m0*2^(t/tau) is a factor of D(m)(t)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/--%\"DG6#%\"mG6#%\"tG*(-F*F+\"\"\"-% #lnG6#\"\"#F/%$tauG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqn2 := 20*mu*g/hr = 60*mu*g*ln(2)/tau;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,$**\"#?\"\"\"%#muGF)%\"gGF)%#hrG!\"\"F),$*,\"#gF)F*F)F +F)-%#lnG6#\"\"#F)%$tauGF-F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "answer := tau = solve(eqn2, tau);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG/%$tauG,$*(\"\"$\"\"\"-%#lnG6#\"\"#F*%#hrGF*F*" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 376 374 "6. An intravenous drip deliver s a drug to a patient so that the drug is absorbed into the patient's \+ bloodstream at the constant rate of 60 mg/hr. Drip, drip, drip, drip, \+ drip, and so on. The drug is eliminated from the bloodstream \nat a r ate proportional to the mass of the drug in the bloodstream. If the m ass is measured in mg, then the proportionality constant is 5/hr" } {TEXT 648 312 ". (Once you write out the differential equation that th e mass satisfies, you will see that the proportionality constant bears the units of 1/time. Its numerical value depends on the units chosen. ) Long term, what, approximately, is the nearly constant mass of the d rug maintained in the patient's \nbloodstream?\n\n" }{TEXT -1 0 "" } {TEXT 378 4 "a) " }{XPPEDIT 387 0 "1;" "6#\"\"\"" }{TEXT 381 16 " \+ b) " }{XPPEDIT 388 0 "2;" "6#\"\"#" }{TEXT 382 33 " \+ c) 3 d) " }{XPPEDIT 389 0 "4;" "6#\"\"%" }{TEXT 386 16 " e) " }{XPPEDIT 390 0 "5;" "6#\"\"&" }{TEXT 380 2 " " } {TEXT 379 25 " \nf) " }{XPPEDIT 391 0 "6;" "6#\"\" '" }{TEXT 385 15 " g) " }{XPPEDIT 392 0 "10;" "6#\"#5" } {TEXT 384 10 " " }{TEXT 393 1 " " }{TEXT 377 20 "h) 12 \+ i) " }{XPPEDIT 394 0 "15;" "6#\"#:" }{TEXT 383 15 " j) \+ " }{XPPEDIT 395 0 "20;" "6#\"#?" }{TEXT 396 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 616 13 "Solution: h\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dsolve(\{dif f(u(t),t) = 60-5*u(t), u(0) = u0\},u(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6#%\"tG,&\"#7\"\"\"*&-%$expG6#,$*&\"\"&F*F'F*!\" \"F*,&F)F2%#u0GF*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "li mit(rhs(%), t = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" } }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 304 18 "7. Suppose that " } {XPPEDIT 503 0 "f(x) = arcsin(sqrt(x));" "6#/-%\"fG6#%\"xG-%'arcsinG6# -%%sqrtG6#F'" }{TEXT 501 16 ". Calculate " }{XPPEDIT 504 0 "D(f)(1 /2);" "6#--%\"DG6#%\"fG6#*&\"\"\"F*\"\"#!\"\"" }{TEXT 502 3 ". " } {TEXT 519 31 "(The derivative of f(x) at " }{XPPEDIT 258 0 "x = 1/ 2;" "6#/%\"xG*&\"\"\"F&\"\"#!\"\"" }{TEXT 520 3 " )." }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT 303 0 "" }}{PARA 3 "" 0 "" {TEXT 302 20 "a) 1 b) " }{XPPEDIT 512 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" } {TEXT 505 15 " c) " }{XPPEDIT 513 0 "1/sqrt(2);" "6#*&\"\" \"F$-%%sqrtG6#\"\"#!\"\"" }{TEXT 506 17 " d) " }{XPPEDIT 514 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT 507 15 " e) " } {XPPEDIT 515 0 "2/sqrt(3);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$!\"\"" } {TEXT 508 8 " \nf) " }{XPPEDIT 516 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\" " }{TEXT 509 33 " g) 2 h) " }{XPPEDIT 517 0 "2 *sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 510 15 " i ) " }{XPPEDIT 518 0 "2*sqrt(3);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$F%" } {TEXT 511 16 " j) 4" }{TEXT 301 1 "\n" }}{PAGEBK }{PARA 0 " " 0 "" {TEXT 634 11 "Solution: a" }}{PARA 0 "" 0 "" {TEXT 635 1 " " } {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "D(x -> arcsi n(sqrt(x)) )(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 319 17 "8. Suppose that " }{XPPEDIT 257 0 "f(x) = arctan(x);" "6#/-%\"fG6#%\"xG-%'arctanG6#F'" }{TEXT 320 15 ". What is " }{XPPEDIT 323 0 "D(f)(1/sqrt(3));" "6#--%\"DG6#% \"fG6#*&\"\"\"F*-%%sqrtG6#\"\"$!\"\"" }{TEXT 321 35 "? (The derivati ve of f(x) at " }{XPPEDIT 324 0 "x = 1/sqrt(3);" "6#/%\"xG*&\"\"\" F&-%%sqrtG6#\"\"$!\"\"" }{TEXT 322 3 " )." }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 326 4 "a) " }{XPPEDIT 523 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 522 19 " b ) " }{XPPEDIT 335 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 327 18 " \+ c) " }{XPPEDIT 336 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" } {TEXT 328 19 " d) " }{XPPEDIT 524 0 "1/4;" "6#*&\"\"\"F $\"\"%!\"\"" }{TEXT 521 16 " e) " }{XPPEDIT 337 0 "1/sqrt( 3);" "6#*&\"\"\"F$-%%sqrtG6#\"\"$!\"\"" }{TEXT 329 9 " \nf) " } {XPPEDIT 338 0 "sqrt(3)/2;" "6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" } {TEXT 330 16 " g) " }{XPPEDIT 339 0 "sqrt(3);" "6#-%%sqrtG 6#\"\"$" }{TEXT 331 14 " h) " }{XPPEDIT 340 0 "2*sqrt(3);" " 6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 332 15 " i) " } {XPPEDIT 341 0 "3/4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 333 18 " \+ j) " }{XPPEDIT 342 0 "4/3;" "6#*&\"\"%\"\"\"\"\"$!\"\"" } {TEXT 325 1 "\n" }{TEXT 343 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 632 11 "Solution: i" }}{PARA 0 "" 0 "" {TEXT 633 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "D(arctan)(1/sqrt(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$ \"\"%" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 1 " " }{TEXT 318 1 "9" } {TEXT -1 2 ". " }{TEXT 317 11 "Calculate " }{XPPEDIT 526 0 "Int(x*exp (x),x = 0 .. 1);" "6#-%$IntG6$*&%\"xG\"\"\"-%$expG6#F'F(/F';\"\"!F(" } {TEXT 525 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 357 4 "a) " } {XPPEDIT 534 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 527 15 " \+ b) " }{XPPEDIT 535 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 528 18 " c) " }{XPPEDIT 536 0 "3/4;" "6#*&\"\"$\"\"\"\"\"%! \"\"" }{TEXT 529 46 " d) 1 e) 2 \nf) " }{XPPEDIT 537 0 "exp(1);" "6#-%$expG6#\"\"\"" }{TEXT 530 17 " \+ g) " }{XPPEDIT 538 0 "exp(1)-1;" "6#,&-%$expG6#\"\"\"F'F'!\"\"" } {TEXT 531 11 " h) " }{XPPEDIT 539 0 "exp(1)-2;" "6#,&-%$expG6# \"\"\"F'\"\"#!\"\"" }{TEXT 532 13 " i) " }{XPPEDIT 540 0 "2*e xp(1)-1;" "6#,&*&\"\"#\"\"\"-%$expG6#F&F&F&F&!\"\"" }{TEXT 533 9 " \+ j) " }{XPPEDIT 541 0 "2*exp(1)+1;" "6#,&*&\"\"#\"\"\"-%$expG6#F&F&F& F&F&" }{TEXT 358 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 630 11 "Solution: \+ d" }}{PARA 0 "" 0 "" {TEXT 631 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "int(x*exp(x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " } {TEXT 314 17 "10. Calculate " }{XPPEDIT 360 0 "int(x^2*sin(x),x = 0 .. Pi);" "6#-%$intG6$*&%\"xG\"\"#-%$sinG6#F'\"\"\"/F';\"\"!%#PiG" } {TEXT 359 2 " ." }}{PARA 0 "" 0 "" {TEXT 313 17 " " }} {PARA 0 "" 0 "" {TEXT 361 5 "a) " }{XPPEDIT 551 0 "Pi^2+1/2;" "6#,&* $%#PiG\"\"#\"\"\"*&F'F'F&!\"\"F'" }{TEXT 546 10 " b) " } {XPPEDIT 552 0 "Pi^2+1;" "6#,&*$%#PiG\"\"#\"\"\"F'F'" }{TEXT 547 13 " \+ c) " }{XPPEDIT 553 0 "Pi^2+2;" "6#,&*$%#PiG\"\"#\"\"\"F&F'" } {TEXT 548 12 " d) " }{XPPEDIT 554 0 "Pi^2+4;" "6#,&*$%#PiG\"\" #\"\"\"\"\"%F'" }{TEXT 549 11 " e) " }{XPPEDIT 555 0 "Pi^2+5;" "6#,&*$%#PiG\"\"#\"\"\"\"\"&F'" }{TEXT 550 8 " \nf) " }{XPPEDIT 556 0 "Pi^2-1/2;" "6#,&*$%#PiG\"\"#\"\"\"*&F'F'F&!\"\"F)" }{TEXT 545 12 " g) " }{XPPEDIT 557 0 "Pi^2-1;" "6#,&*$%#PiG\"\"#\"\"\"F'! \"\"" }{TEXT 544 12 " h) " }{XPPEDIT 558 0 "Pi^2-2;" "6#,&*$%# PiG\"\"#\"\"\"F&!\"\"" }{TEXT 543 12 " i) " }{XPPEDIT 559 0 "P i^2-4;" "6#,&*$%#PiG\"\"#\"\"\"\"\"%!\"\"" }{TEXT 542 13 " j) \+ " }{XPPEDIT 560 0 "Pi^2-5;" "6#,&*$%#PiG\"\"#\"\"\"\"\"&!\"\"" } {TEXT 362 1 "\n" }{TEXT 561 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 628 11 "Solution: i" }}{PARA 0 "" 0 "" {TEXT 629 1 " " }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int(x^2*sin(x),x = 0 .. Pi); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%#PiG\"\"#\"\"\"F(\"\"%!\"\" " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 19 "11. Calc ulate " }{XPPEDIT 258 0 "36*Int(x^5*ln(x),x = 1 .. exp(1));" "6#*& \"#O\"\"\"-%$IntG6$*&%\"xG\"\"&-%#lnG6#F*F%/F*;F%-%$expG6#F%F%" } {TEXT 363 2 " ." }}{PARA 0 "" 0 "" {TEXT 269 4 "a) " }{XPPEDIT 365 1 "exp(6)-1;" "6#,&-%$expG6#\"\"'\"\"\"F(!\"\"" }{TEXT 305 13 " \+ b) " }{XPPEDIT 366 1 "2*exp(6)-1;" "6#,&*&\"\"#\"\"\"-%$expG6#\"\"'F& F&F&!\"\"" }{TEXT 306 13 " c) " }{XPPEDIT 367 1 "3*exp(6)-1; " "6#,&*&\"\"$\"\"\"-%$expG6#\"\"'F&F&F&!\"\"" }{TEXT 307 14 " \+ d) " }{XPPEDIT 368 1 "4*exp(6)-1;" "6#,&*&\"\"%\"\"\"-%$expG6#\"\"' F&F&F&!\"\"" }{TEXT 308 12 " e) " }{XPPEDIT 369 1 "5*exp(6)-1; " "6#,&*&\"\"&\"\"\"-%$expG6#\"\"'F&F&F&!\"\"" }{TEXT 309 11 " \n \+ f) " }{XPPEDIT 370 1 "exp(6)+1;" "6#,&-%$expG6#\"\"'\"\"\"F(F(" } {TEXT 310 13 " g) " }{XPPEDIT 371 1 "2*exp(6)+1;" "6#,&*&\"\" #\"\"\"-%$expG6#\"\"'F&F&F&F&" }{TEXT 311 13 " h) " } {XPPEDIT 372 1 "3*exp(6)+1;" "6#,&*&\"\"$\"\"\"-%$expG6#\"\"'F&F&F&F& " }{TEXT 312 14 " i) " }{XPPEDIT 373 0 "4*exp(6)+1;" "6#,&*& \"\"%\"\"\"-%$expG6#\"\"'F&F&F&F&" }{TEXT 364 14 " j) " } {XPPEDIT 374 0 "5*exp(6)+1;" "6#,&*&\"\"&\"\"\"-%$expG6#\"\"'F&F&F&F& " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 626 11 "Solution: j" }}{PARA 0 "" 0 "" {TEXT 627 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "36*int(x^5*ln(x),x = 1 .. \+ exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"&\"\"\"-%$expG6#\" \"'F&F&F&F&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 315 17 "12. Calculate " }{XPPEDIT 256 0 "int((x+3)/(x^2-1),x = 2 .. 3);" " 6#-%$intG6$*&,&%\"xG\"\"\"\"\"$F)F),&*$F(\"\"#F)F)!\"\"F./F(;F-F*" } {TEXT 375 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT -1 14 " b) " }{XPPEDIT 18 0 "ln(3);" "6#-%#lnG6#\"\"$" }{TEXT -1 15 " c) " }{XPPEDIT 18 0 "2*ln(2);" "6#*&\"\"#\"\"\"-%#l nG6#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#* &\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 14 " i) " }{XPPEDIT 18 0 " ln(9/2);" "6#-%#lnG6#*&\"\"*\"\"\"\"\"#!\"\"" }{TEXT -1 14 " \+ j) " }{XPPEDIT 18 0 "ln(12);" "6#-%#lnG6#\"#7" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 625 12 "Solution: b " }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int((x+3)/(x^2-1),x = 2 .. 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%#lnG6#\"\"$" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 287 16 "13. Calcula te " }{XPPEDIT 405 0 "Int((5*x^2-x-1)/(x^2)/(x-1),x = 2 .. 3);" "6#- %$IntG6$*(,(*&\"\"&\"\"\"*$%\"xG\"\"#F*F*F,!\"\"F*F.F**$F,F-F.,&F,F*F* F.F./F,;F-\"\"$" }{TEXT 404 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 261 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 18 0 "ln(12)-1/3;" "6#,& -%#lnG6#\"#7\"\"\"*&F(F(\"\"$!\"\"F+" }{TEXT -1 13 " b) " } {XPPEDIT 18 0 "ln(6)-1/2;" "6#,&-%#lnG6#\"\"'\"\"\"*&F(F(\"\"#!\"\"F+ " }{TEXT -1 14 " c) " }{XPPEDIT 18 0 "ln(18)+1/6;" "6#,&-%#l nG6#\"#=\"\"\"*&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(\"\"$!\"\"F," }{TEXT -1 13 " g) " }{XPPEDIT 18 0 "2*ln(4)- 1/4;" "6#,&*&\"\"#\"\"\"-%#lnG6#\"\"%F&F&*&F&F&F*!\"\"F," }{TEXT -1 12 " h) " }{XPPEDIT 18 0 "ln(24)+1/3;" "6#,&-%#lnG6#\"#C\"\"\" *&F(F(\"\"$!\"\"F(" }{TEXT -1 13 " i) " }{XPPEDIT 18 0 "ln(12 )-1/6;" "6#,&-%#lnG6#\"#7\"\"\"*&F(F(\"\"'!\"\"F+" }{TEXT -1 12 " \+ j) " }{XPPEDIT 18 0 "ln(8)+1/3;" "6#,&-%#lnG6#\"\")\"\"\"*&F(F(\" \"$!\"\"F(" }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 646 11 "Solutio n: c" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "int ((5*x^2-x-1)/(x^2)/(x-1),x = 2 .. 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\"\"\"\"'F%-%#lnG6#\"\"#F%*&F*F%-F(6#\"\"$F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "testeq(A = ln(9/2)-1/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 1 " " }{TEXT 270 17 "14. Calculate " }{XPPEDIT 563 0 "Int(sin(x )^3,x = 0 .. Pi/2);" "6#-%$IntG6$*$-%$sinG6#%\"xG\"\"$/F*;\"\"!*&%#PiG \"\"\"\"\"#!\"\"" }{TEXT 562 1 "." }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 406 80 "a) 1/15 b) 2/15 c) 1/5 d) 4/15 e) 1/3 " }{TEXT 409 1 " " }{TEXT 410 18 " " }}{PARA 0 "" 0 "" {TEXT 407 39 "f) 2/5 \+ g) 7/15 " }{TEXT -1 1 " " }{TEXT 408 42 "h) 8/15 \+ i) 3/5 j) 2/3 " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 623 11 "Solution: j" }}{PARA 0 "" 0 "" {TEXT 624 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int(sin(x)^3,x = 0 .. Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"$" }}}}{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#/-%$IntG 6$)-%$cosG6#%\"xG%\"nGF+,&*(-%$sinG6#F+\"\"\")-F)6#F+,&F,F2F2!\"\"F2F, F7F2*(,&F,F2F2F7F2F,F7-F%6$)-F)6#F+,&F,F2\"\"#F7F+F2F2" }{TEXT -1 77 " \n\n it follows that there are rational numbers A, B, and C such that \n\n " }{XPPEDIT 18 0 "Int(cos(x)^4,x) = A*cos(x)^3*sin(x)+B*co s(x)*sin(x)+C*x+D;" "6#/-%$IntG6$*$-%$cosG6#%\"xG\"\"%F+,**(%\"AG\"\" \"*$-F)6#F+\"\"$F0-%$sinG6#F+F0F0*(%\"BGF0-F)6#F+F0-F66#F+F0F0*&%\"CGF 0F+F0F0%\"DGF0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 607 36 "where D is a constant. What is C?" }{TEXT -1 1 "\n" } }{PARA 0 "" 0 "" {TEXT 288 75 "a) 8/15 b) 2/3 \+ c) 7/9 d) 3/8 e) " }{TEXT 411 3 "5/7" }{TEXT 608 3 " " }{TEXT 291 18 " " }}{PARA 0 "" 0 "" {TEXT 289 36 "f) 24/35 g) 16/21 " }{TEXT -1 1 " " }{TEXT 290 34 "h) 9/16 i) 3/4 j)" }{TEXT -1 1 " " }{TEXT 609 3 " 3/5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 621 11 "Solution: d" }}{PARA 0 "" 0 "" {TEXT 622 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "eqn1 := int(cos(x)^4,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G,(*&#\"\"\"\"\"%F(*&)-%$cosG6#%\"xG\"\"$F(-%$si nGF.F(F(F(*&#F0\"\")F(*&F,F(F1F(F(F(*(F0F(F5!\"\"F/F(F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 14 "16. Calculate " }{XPPEDIT 580 0 "int(x^3 /sqrt(1-x^2),x = 0 .. 1);" "6#-%$intG6$*&%\"xG\"\"$-%%sqrtG6#,&\"\"\"F -*$F'\"\"#!\"\"F0/F';\"\"!F-" }{TEXT 579 1 "." }}{PARA 0 "" 0 "" {TEXT 640 71 "a) 1/3 b) 2/3 c) 3/4 d) \+ 3/5 e) 4" }{TEXT 644 2 "/5" }{TEXT 645 3 " " }{TEXT 643 18 " \+ " }}{PARA 0 "" 0 "" {TEXT 641 36 "f) 1/4 \+ g) 1/2 " }{TEXT -1 1 " " }{TEXT 642 37 "h) 1/8 i) 3 /8 j) 5/8" }{TEXT -1 2 " " }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 620 14 "Solution: b " }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int(x^3/sqrt(1-x^2),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"$" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 274 4 "1 7. " }{TEXT 286 12 " Calculate " }{XPPEDIT 256 0 "int(1/((1+x^2)^(3/2 )),x = 0 .. 1);" "6#-%$intG6$*&\"\"\"F'),&F'F'*$%\"xG\"\"#F'*&\"\"$F'F ,!\"\"F//F+;\"\"!F'" }{TEXT 581 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 582 5 "a) " }{XPPEDIT 600 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 592 1 " " }{XPPEDIT 601 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 591 14 " b) " }{XPPEDIT 257 0 "ln(2);" "6#-%#lnG6#\"\"#" } {TEXT 583 16 " c) " }{XPPEDIT 258 0 "2*ln(2);" "6#*&\"\"# \"\"\"-%#lnG6#F$F%" }{TEXT 584 12 " d) " }{XPPEDIT 602 0 "1/2; " "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 593 1 " " }{XPPEDIT 256 0 "ln(3);" " 6#-%#lnG6#\"\"$" }{TEXT 589 14 " e) " }{XPPEDIT 259 0 "ln(3) ;" "6#-%#lnG6#\"\"$" }{TEXT 585 12 " \nf) " }{XPPEDIT 260 0 "2* ln(3);" "6#*&\"\"#\"\"\"-%#lnG6#\"\"$F%" }{TEXT 586 17 " g) " }{XPPEDIT 261 0 "3*ln(3);" "6#*&\"\"$\"\"\"-%#lnG6#F$F%" }{TEXT 587 12 " h) " }{XPPEDIT 262 0 "1;" "6#\"\"\"" }{TEXT 590 23 " \+ i) " }{XPPEDIT 263 0 "2;" "6#\"\"#" }{TEXT 588 25 " j) " }{XPPEDIT 264 0 "1/sqrt(2);" "6#*&\"\"\"F$- %%sqrtG6#\"\"#!\"\"" }{TEXT 603 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 639 13 "Solution: j\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "in t(1/(1+x^2)^(3/2),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*& \"\"#!\"\"F%#\"\"\"F%F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 276 4 "18. \+ " }{TEXT 564 1 " " }{TEXT 576 52 "There are unique rational numbers A \+ and B such that " }{XPPEDIT 279 0 "int((4*x^2-x+4)/x/(x^2+1),x = 1/sqr t(3) .. 1) = A*ln(3)+B*Pi;" "6#/-%$intG6$*(,(*&\"\"%\"\"\"*$%\"xG\"\"# F+F+F-!\"\"F*F+F+F-F/,&*$F-F.F+F+F+F//F-;*&F+F+-%%sqrtG6#\"\"$F/F+,&*& %\"AGF+-%#lnG6#F8F+F+*&%\"BGF+%#PiGF+F+" }{TEXT 575 1 "." }{TEXT 577 2 " \n" }{TEXT 578 17 "What is A ?\n\na) " }{XPPEDIT 256 0 "-5;" "6#, $\"\"&!\"\"" }{TEXT 566 11 " b) " }{XPPEDIT 262 0 "-4;" "6#,$\" \"%!\"\"" }{TEXT 567 13 " c) " }{XPPEDIT 270 0 "-3;" "6#,$\" \"$!\"\"" }{TEXT 568 12 " d) " }{XPPEDIT 271 0 "-2;" "6#,$\"\" #!\"\"" }{TEXT 569 12 " e) " }{XPPEDIT 272 0 "-1;" "6#,$\"\"\" !\"\"" }{TEXT 570 10 " \nf) " }{XPPEDIT 273 0 "5;" "6#\"\"&" } {TEXT 571 16 " g) " }{XPPEDIT 274 0 "4;" "6#\"\"%" }{TEXT 572 14 " h) " }{XPPEDIT 275 0 "3;" "6#\"\"$" }{TEXT 573 15 " i) " }{XPPEDIT 276 0 "2;" "6#\"\"#" }{TEXT 574 16 " \+ j) " }{XPPEDIT 599 0 "1;" "6#\"\"\"" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 565 0 "" }{TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 617 13 "Solution: i\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "int(-1/(x^2+1)+4/x, x = \+ 1/sqrt(3)..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#7!\"\"%#PiG\" \"\"F&*&\"\"#F(-%#lnG6#\"\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int((4*x^2-x+4)/x/(x^2+1),x=1/sqrt(3)..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#7!\"\"%#PiG\"\"\"F&*&\"\"#F(-%#lnG6# \"\"$F(F(" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 277 55 "19. The region \+ in the first quadrant bounded above by " }{XPPEDIT 18 0 "y = 2-x;" "6 #/%\"yG,&\"\"#\"\"\"%\"xG!\"\"" }{TEXT 605 17 " and below by " } {XPPEDIT 18 0 "y = sqrt(x);" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT 606 88 " is rotated about the x-axis. What is the volume of the resulting so lid of revolution?" }}{PARA 260 "" 0 "" {TEXT 604 5 "a) " }{XPPEDIT 18 0 "12*Pi/5;" "6#*(\"#7\"\"\"%#PiGF%\"\"&!\"\"" }{TEXT 420 12 " \+ b) " }{XPPEDIT 265 0 "31*Pi/15;" "6#*(\"#J\"\"\"%#PiGF%\"#:!\"\"" }{TEXT 412 13 " c) " }{XPPEDIT 266 0 "15*Pi/8;" "6#*(\"#:\"\" \"%#PiGF%\"\")!\"\"" }{TEXT 413 12 " d) " }{XPPEDIT 256 0 "8*P i/3;" "6#*(\"\")\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 418 13 " e) \+ " }{XPPEDIT 267 0 "16*Pi/5;" "6#*(\"#;\"\"\"%#PiGF%\"\"&!\"\"" }{TEXT 414 12 " \nf) " }{XPPEDIT 268 0 "11*Pi/6;" "6#*(\"#6\"\"\"%#PiG F%\"\"'!\"\"" }{TEXT 415 14 " g) " }{XPPEDIT 273 0 "5*Pi/3; " "6#*(\"\"&\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 416 14 " h) " } {XPPEDIT 274 0 "9*Pi/5;" "6#*(\"\"*\"\"\"%#PiGF%\"\"&!\"\"" }{TEXT 419 16 " i) " }{XPPEDIT 275 0 "32*Pi/15;" "6#*(\"#K\"\"\"% #PiGF%\"#:!\"\"" }{TEXT 417 12 " j) " }{XPPEDIT 276 0 "21/8*Pi ;" "6#*(\"#@\"\"\"\"\")!\"\"%#PiGF%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PAGEBK }{PARA 0 "" 0 "" {TEXT 618 13 "Solution: f\n" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Pi*int((2-x)^2-(sqrt(x))^2,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#6\"\"\"\"\"'!\"\"%#PiGF&F&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 280 57 "20. The region above the x-axis and under the gr aph of " }{XPPEDIT 596 0 "y = sin(Pi*x^2);" "6#/%\"yG-%$sinG6#*&%#Pi G\"\"\"*$%\"xG\"\"#F*" }{TEXT 594 6 " , " }{XPPEDIT 597 0 "0 <= x; " "6#1\"\"!%\"xG" }{XPPEDIT 598 0 "` ` <= 1;" "6#1%\"~G\"\"\"" }{TEXT 595 89 " is rotated about the y-axis. What is the volume of the resu lting solid of revolution?\n" }}{PARA 260 "" 0 "" {TEXT 279 5 "a) " }{XPPEDIT 426 1 "1;" "6#\"\"\"" }{TEXT 421 15 " b) " } {XPPEDIT 427 1 "2;" "6#\"\"#" }{TEXT 281 15 " c) " } {XPPEDIT 428 1 "3;" "6#\"\"$" }{TEXT 282 14 " d) " } {XPPEDIT 429 1 "4;" "6#\"\"%" }{TEXT 283 12 " e) " }{XPPEDIT 430 0 "5;" "6#\"\"&" }{TEXT 422 10 " \nf) " }{XPPEDIT 431 1 "Pi; " "6#%#PiG" }{TEXT 423 12 " g) " }{XPPEDIT 18 0 "3*Pi/2;" "6#* (\"\"$\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT 284 12 " h) " }{XPPEDIT 432 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 285 9 " i) " } {XPPEDIT 433 0 "3*Pi;" "6#*&\"\"$\"\"\"%#PiGF%" }{TEXT 424 11 " \+ j) " }{XPPEDIT 434 1 "4*Pi;" "6#*&\"\"%\"\"\"%#PiGF%" }{TEXT 425 2 " \+ " }{TEXT 278 1 "\n" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 619 13 "Solution : b\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int(2*Pi*x*sin(Pi*x^2),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}}}{MARK "5 0 5" 16 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }