THE ESSENTIALS OF CALCULUS ANSWERS TO SELECTED EXERCISES

Transcription

1 Assignmen - page. m.. f THE ESSENTIALS OF CALCULUS ANSWERS TO SELECTED EXERCISES m.... no collinear 8...,,.,.8 or.,..78,.7 or.7,.8., 8.87 or., 8.88.,,, 7..7 Assignmen - page 7. b. d... Do:, Ra: 8. no a funcion (circle). a.. c. 8 e... c.,. a. b. c.., or.8. Do: all reals, Ra: Do:.,. or.,. Ra:,. or,. Assignmen - page, d.. -in., no -in. 8. -in.,, -in.. no -in.,.,,,,, -in.,, 7.,,, -ais smm.. k 8,. origin. -ais

2 . -in. -in., -ais smm.,, 8... or.7.,. Assignmen - page.. DNE.. or DNE. or DNE.., 7., 8. b. d.. b. or DNE g. i.. d. e.. c. or DNE e.,,,,,. b. DNE. c. f. ever ineger a. b. c. DNE. d.,,, Assignmen - page Problem 8. a. b.. c. d. = -.,.,. a. 8. a.do: all reals Ra: b..,. b. -in:, -in: (, ) e. c. -ais smmer d. even c. 7. f

3 . a. Do:, Ra:. a. Do: all reals Ra: b. -in.,, -in., b. -in:, -in: (, -) c. no smmer d. neiher c. -ais smmer d. even e. e.... Ra: all reals. a. Do:.8,.8, Ra:., b. -in: (,) -in: (, ) b. -in..8,, -in.,.. a. Do: c. origin smmer d. odd c. -ais smm. d. even e. e.. a. Do: all reals 7.,.7,.,.7 or., or., -.7 Ra: 7. b. -in:. 88,, (.88, ). odd. Neiher. - -in: (, -). -. DNE. - c. no smmer d. neiher. DNE e.... =. =.. -. DNE =

4 Assignmen - page..... or DNE b.. or.7. DNE. (removable), (nonrem.). (removable). (rem.). ever even ineger (nonrem.). ever inegral muliple of (nonremovable). a =. 7. c = 8. c =. c =. 7, b a Assignmen - page. (even). (boh odd) DNE... DNE.. a. b or DNE.. 7. (nonremovable), (removable). (nonremovable). (nonremovable). or 7. Do:, Ra: 7..,.. DNE. (removable), (nonremovable). nonremovable disconinui a ever inegral muliple of Assignmen - page. Do: all reals. Do:,. Do:,,, EB: Horiz. Asmp. Hole: (, ) (even) HA: VA: = (odd). -in:, (odd). Do:. -in:, (even) VA: (odd) -in, -in:, -in:, EB: like -in:, EB: HA, (odd), Sar P: Smm: origin. Do:. Do:, f fred red VA: (odd) VA: (odd) Hole:,,, (odd) -in:, EB: like, (odd) Holes: -in:, (even), -in:, EB: HA DNE.. or DNE. 7.

5 Assignmen - page. f. f. f.. f. f 7. f 8. f Assignmen - page 7. d. f f d..,,,,,..,,,..,.88 or.,.8,., 7.88 or., or f f is undefined,. f. g. f. f 7. f 8. h. f.. f. g. 7. f,... f. f. f 8... ARC = 8. f = (sharp urn),. (ver. asmp.), = (hole). = (ver. ang.). = (sharp urn) Assignmen - page DNE.. or DNE or DNE or DNE DNE.... DNE or DNE..,... DNE a, b f ais. neiher..

6 .... Do: VA: (odd) red -in:, (odd) Hole:, -in:, EB: HA f f. f a. AROC = b.. 8. a. Yes, c = -, b. No, c. Yes, c =.. g.,. a. alwas coninuous b. f c. alwas differeniable,,. a. = b. f c. =,,. a. alwas coninuous b. f c. =, 7. a. = -, b. = -,, 8. a. D: -,,, Holes: (,.), (,) odd VA: = - odd, = even EB: = f red -in (-,) even -in: (,-) 8. b. D: VA: = EB:, -in: (, -/) 8. c. D: (-,-), (, ) VA: =, = - EB:, -in: (,) Assignmen - page. b. m d. A 7 m e. f. V. m h.. m. a. b. f d. V f. b..7,.7. c. slowing down (he slope of S is decreasing). a. disp. = cm. b. V cm d. =, min e. TD = 8 cm f. Vavg cm. g. avg. speed =. a. 7 c. f f V min V 8 cm min min

7 V f 7. a.. or. b... f 8. 8 f. f.. f 8..., 7. f 8. AROC = Assignmen - page 8. f 8. f.. f, 8. f.. 8..,. AROC. a. g f f dv ds moving lef on,. =, h 7. f.. TD =. m 7. f 7.7 or 7.8 V. b. 7 f d. V f f. jus before = f, Assignmen - page f.... g ,8,, 8.. a.. f 7. b. Avg. Speed =. e. g. beween 8 and m 8. or. f is a cubic, f is a parabola, and f is a line. disp. =., TD =. or.7. a. f c. g or.77.. m m h V 8 f Assignmen - page... d. ; a d d d,,

8 d. ; a,, d 7. ; d d 8. ;,...,,,. 8,. m m or.78. V, V., dv dr Assignmen - page d d d d. 8.. a. b.. d da cm d d da in dv in. a. b.. 7. a. d d d min da f da in 7. b. c. increasing, no 8.. d 8 d min..8 f. =. a. AROC = b. f. a. b. c. es. a. b. c. es cm dc cm dr d da f d 8 d f 8 min Assignmen - page. f. f... f g. f 7., 8. f 8. f (using he limi definiion). f (using he alernae form)..... a. b. c.. a. v b. a c. =,. d. e., f. v g. disp. = h. TD = 7. a. 8.7 or 8.7 b c.

9 7 8. a. AROC = b. f.. a. es b. no, sharp urn. a. es b. no, verical angen. a. es b. no, sharp urn. a. no b. no, jump. a. es b. es. a. no b. no.,,, Assignmen - page..,,..,8 8.. ma. f =,. min. f =. ma. f = 8, min. f =. ma. f =, min. f =. ma. f =, min. f =. ma. f =, min. f =. ma. value =,. min. value = 7. no ma. value, min. value =. c =.. MVT does no appl. f is no differeniable a = (sharp urn) c 7. c = 8. c.,.8 or.. c =.. b. Assignmen - page. a.,, b., c., d.. a.,, b.,,, c., d.. a.,, b., c. = d. 7. problem. a. b., c., d. none. a., b. c. none d.. a.,, b. c. d.. a., b. none c. none d. none. a. none b.,,,,, c. none d. none. a.,,, b. none c. none d. none 8. es. no.,. a. never. a, b. ma. f = 7, min. f = 7. c. Assignmen - page 7. a., b. c.. a., b.,,, c. none. a.,,,, 8, b. c.. a. 8. b. c.,,, 7. a. none b.,,, c. none 8. rel. ma. (,) rel. min. (,). rel. ma., rel. min. (,) d d cm,,,,,,,,8,,,,,,,8,,,,, c

10 8. rel. ma. a, rel. min. a.. concave down ( f.. ). ma. f = 8, min. f = 7. c = 8. a, b, c, d.a. b. - c. relaive maimum Assignmen - page 7. rel. ma. rel. min. PI. no rel. erema, PI. rel. min., PI,,,.. rel. min., P.I.,, E.B. horiz. asmp. =. Do:, -in.,, rel. ma.,. = (nonrem.), = (rem.). = (nonrem.).. 8.a.,,,. 8,. B. (A), (G),,,,, 7. Do:, rel. ma.,, rel. min.,, V. A. = (odd), E. B. like,,,, Assignmen - page f f. f These f graphs could be shifed vericall.. V.A.:,, H.A.:, Rel. Ma.:,, no Rel. Min.,,,. Rel. Min.:, 7, no Rel. Ma., P.I.:. Rel. Ma.:,, Rel. Min.:,. 7. abs. min. f = 8, abs. ma. f =. a, b, c c

11 Assignmen - page 7. -in.:,,,,. Do.:,,, -in.:,, -in.:, Rel. Ma.: -in.:, odd,,,, Rel. Ma.: 8,8, Rel. Min.: 8, 8, P.I.:, 8,8,,. Do.:,,. -in.:,,, all odd,,, V. A.: (odd), -in.:, Rel. Ma.:,, -in.:, (odd), Rel. Min.:,, P.I.: Hole:,, H. A.: 8, 8.. Min.:,, Ma.:, 7. Abs. Ma. f =, Abs. Min. f = 8. Abs. Ma. g =, Abs. Min. g =. MVT does no appl. (V.A. a = ). c.7. MVT does no appl. (Sharp urn a = ). a. 7 b. -. a =, b = -. a. b. DNE c. d.. f. f 7. f

12 8.. or Eiher or boh branches of his graph could be shifed vericall.... This graph could be shifed vericall a. b., v cm 7. c. a d. a e. cm f... g. cm 7. h. T. D. =. or. i.. j. Assignmen - page 8. The numbers are boh.. The recangle s lengh and widh are boh cm.. The recangle s lengh and widh are boh f.. = f., = f.. = f., = f.. = in. 7.,.. 7 mph. f.,, l.,,, Assignmen - page 8. The firs number is and he ond is.. The firs number is. and he ond is... The firs number is and he ond is.. = m, = 8 m. =. f or.7 f. V =. f 7. 8, lengh = cm, widh = cm, heigh = cm. Rel. Ma.:,,. no Rel. Min., P. I.:,,,. Rel. Ma.:, Rel. Min.:,,. Concave Down:,,,. v cm,..,. dv d in min, g

13 Assignmen - page 87. a. b...7 c. f... b.... f a. A sq. in. 8. f or.8. a.. b..7. Boh numbers are.. 8,. V. A.: (odd), Rel. Ma.:,, no Rel. Min., H. A.: = 8. V.A.: = (even), -in.:, (odd), -in.:, Rel. Min.:,, H.A: =. f Assignmen - page. C.... C C C C 7. C. C. C. C. f.. a. C b. 7. g. a. v c.,. c. s 7.. a. v,,. c. s e...,. Do.:, V.A.: (odd), no hole, -in.:, (even), -in.:,, E.B.: H.A. =. f Assignmen - page 7. C... C C. v C 8. C.. or. C C u u uc or u C d, c. or. 7 s C C

14 . C. C.. C a. a b. e. a all imes f c.7 Tangen and an lines have he same slope.. a. b., C 7,,, C f f Assignmen - page a. b.. a a. c. 7. a. b or.8 a..8 or. b..8 or.. c. f. 8 g a. Do.: all reals, V.A.: none,. a. H.A.: =, -in.: (,) odd, -in.: (,), smm.: origin. b. rel. min. a, rel. ma. a c. P.I. a, 8 Assignmen -7 page., a. s. b. mile c. mile. d., hours 7. a 8. f speed f,,,,7. onds..,,,. onds

15 . Do.:. -in: V. A.: = (odd) -in: -in.:, rel. ma:, no -in. rel. min: E. B.: like Rel. Min.: no Rel. Ma. P. I.:,,,,, and,, P. I.:,.., and, 7. ma. poin min. poin 8. c =.877. MVT does no appl. (sharp urn a = ). b 8. unis per.. abs. ma. f =, abs. min. f =. local maimum a =. 7 fee. + C. C 7. C or or 7. Assignmen - page a. A 8: AM waer is flowing from he reservoir a he rae of.. b. gal e. gal. b. approimael c. Approimael, babies were born in he Uah ci during,,, and. ker. c... hi s mi mi. c. v., v e. miles f. a. d. f e. f f.. f 7.. a.. b. c.. c. 7 f.., and,,, 7 7 d d 8 a sin D mile hr. f v hr gal hr

16 Assignmen - page. a. b.. c =. or d d or cm 8 d Assignmen - page = , This is no accurae because onl a small par of he area under he curve is covered b he hree recangles.. sin()... a. d C 8. C.,,. a. reverse, forward, a res, b. Si minues afer he sar of he es drive he car s veloci was fee per minue. d. f/min/min f. h., f i., f. The firs number is and he ond is., Assignmen - page....7 or

17 . c = a. 8 b. 7. b a. = =. b., square ards. a. f. b. Assignmen - page. a. V d b. V d c.. a. V d. a. V d. a =. or.. f.. 8 C. f C h C.. Assignmen - page (Be sure o show skeches wih radii labeled!). a. d b.. b. d. a. d b.. a. d b. d c. 8. or 8. f 8 8 V d V d 7. a. d b. 8. a. 8 d g g C. Do.:, red, no V.A., Holes:,, no -in., -in.:,, E.B.: H.A. a, -ais sm c.7 7 d. b. d. b. d c. d d d

18 7 8. b.. a.. or. b or 8.8. b. 7. b. -in.:,,, d. range:.. c.78 or.7, c = 8. 8 d Assignmen -7 page. v v vc. C.. C.. 7. cos cos 8. a. a b. 8. c. d. e. f.. a... or.7 b.. c. disp. =.7 or.8 d. T. D. =.77 or.78 or.78.. D. D. appro. miles from he railhead avg hi ker s mile. D d represens he oal number of hikers on he rail a a disance beween and miles from he railhead.. larger. smaller 7. The are equal. 8. A = d. A =. V = 8. a. V d b. V d. V d.8 or C C v a avg hi ker s mile..7 or.8 D avg D d. 7. g g g

19 7 8. = an even ineger (nonremovable)., (nonrem.). = (nonrem.). (rem.), (nonrem.)... a, b angen: normal:. Assignmen 7- page e = = (,) e 8 a, b,,,,,, (,-) = - (,).. f. A 7. c 8. (,) avg e e. Mehod : Mehod : 7. f 8... C. m m. a. c. e.. m f.. m i.. j.. 7. f or 7.. V d

20 8 Assignmen 7- page. a. b. e e. f e. h ln 7. e. e e e... e e e. Rel. Min. a,, P.I. a,.. e ln u 7. e 8. e C. e C.. e C. a.. e e e g 7. a.. d..7 and 7. or or.7. V ln d.. or. or. a., b. A = Assignmen 7- page. a. g c. f, g d.. e. The are reciprocals.. a. g. a. c. f, g. a. no possible b. c ,.,.. 8 ln. f. e. C. e e 8. e e. A. a. b.. e. a g. cm h.. cm..7 or g.8 or.87.. e e e e 7 e b b a C f, g 8 cm 7 g, d d C cm v e e e

21 Assignmen 7- page 7. false. false. false. false. false. rue 7. false 8. rue. log. 7.. p 7. = 8 8. e.... = (,) (-,) = = (,). f ln 7. f e 8. ln a ln b ln c. ln ab lnc. log. ln ac ln.,.. is eraneous b 8..8 or.8..7 or.77. monoonic increasing.. f. dh d f min Assignmen 7- page.. f. g 7. h ln ln 8.. f.. f e ln ln. ln. ln. d. 7. e 8e 8. c. d. Do.:, Rel. Min.:,, no P.I... e C. Ae k a. = Assignmen 7- page e.. ln... ln 7. ln C 8. ln C. ln C ln ln V e e e C k ln C

22 . u C. C. ln 8 ln C. f e. g ln log p. qr. e. f.. ln a. Do.:, Ra.: all reals. a.. or. b.. c..7 or.8 d..,. or.. b. A e d c. V e d d.. e. V e d. a, b, c Assignmen 7-7 page 8. a. Do: all reals, poins,,,e, HA: b. Do:, poins,, e,, VA: g ln ln V e d,e ln e,. HA:. ln, VA:.. Tangen:, Normal:., ,... g e e. ln... ln ln d c 7. log 8.. c. d ln. no possible wih curren echniques. ln.. ln C. ln C. ln C 7. e 7 C e e e e e e c ln c

23 . a. e. b. e c. e e. d. f e cm e Assignmen 8- page. C. e C or e C. ln ln C or ln ln C... e 7. k e 8... f e e.. ln e C ln...7 or or.8. a..7 c. ln 7. d. TD. or b. Assignmen 8- page..7. e or e or e. a. $.,. a. A $7. c.. or. rs. baceria..7 or.7 rs., rs 7..8 or. min e 8.. a.. b.. or.7. ln. ln. ln u C. C 7. C 8. d. e d ln ln C e f e ln a.7 A A a b $. ln Assignmen 8- page. a. c. e.. b. i,ii.. d.. c. e for C or C,

24 . a,b. c.. c. 7. c. 8. c. e. a,c. b.. a (sharp urns). (hole), (VA). (verical angen). a. c.. Do:,, red, VA: (odd), Hole:,, f f 8 f -in:, (odd), no -in., HA:, -ais smm. a Assignmen 8- page 7... or DNE e... f C. C. e ln.....% ln ln ln C ln C., people or 8. g 8. a. c. e.. e..8 or... Assignmen 8- page a. b. d. f.. a. c.. a. c.. a. c.. a. c.. a. 7 c. 7. a. c.. 8. a. c a. c.. a. Quadran. a. or c.. a.,

25 .,,,. a. f, f or.. ln.. ln ln ln 8 ln. sin. Assignmen 8- page 8... e. B. D. a. b. c. d. e. 7,8.. a. R,,e b. 8. rs. c. 7. rs (earl in 7). Rd gives he oal amoun of coal (in ons) produced b he mine from he beginning of hrough.. a.,8, ons b.,, ons.,7, ons DNE.,,, e 7. sin 8. sin or... or.. a.. or. b a. ma a = -, min a =, PI a =. b. ma a = -, min a =, PI a =, a. ln( e ) C b.,. abs ma =, abs min = Lengh, widh 7 (ln ) C c. ln C Assignmen - page 8. a.. b..7 or.8 d.. or..... in or.7, a. b. c. d. (upward). a. b. none c. (righ) d.. a. b. c. (lef ) d..., 7., f e.8... DNE or c. or. ln. 8. a. sin (ohers are possible) 7,,,

26 7 7. sin, an. a. sin, an,. b. sin, an, c. sin, cos, an is undef cos, an.,.,.,, Assignmen - page 8. sin. f cos. csc. f. cscco an co an 7. f. h cos sin cos sincos. f. P sin cos sin. f ln. g an. sin cos h e sin cos Tangen: Normal: g cos sin or sin.,. sin sin... 7., 8. a. b.. sin, cos, an. sin, cos, an.. amp., per., ph. sh. (righ).. disc. a,,, an. b. d d radians

27 Assignmen - page... ln ln. an C.. sin C 7. sin C 8. C. ln sin C an. ln sin ln cos C. e C... or cos d e e e e an an an an.. f e an. h. g csc.. Assignmen - page or cos. sin 7. e. g. h. e. amp. =, per. =, ph. sh. = (righ) an C cos. d sin cos cos sin C arccos. sin ln. sin cos sin. a, b. b. c...8 or.. b. e. b.. d..87 or.877 ears e. 7 rabbis g. ears Assignmen - page... ln. arcan C an + C cos ln sin + C ln C N sin. arcan C. arcanc

28 . v e arcan C. C e.. f. e arcsin. an C. ln sin e C 7. co C Assignmen - page. C. C. C.. C 7. an C 8. ln cos C or. ln C. sin C.. sin + C.. cos sin cossin 8. g. v h ln an.. v...,. 7.. or.7 8.a. sin C or sin C b.. a. b. c. d. e.. Assignmen -7 page 7. ln C. an C... arcan C. e e e C 7 7. ln e C or ln e C 8. ln C.. ln e e C. an + C. arcan C ln arcan C ln an C e e e C ln C e sin 7 ln cos C arcan ln C

29 7.. f ln arcan arcan. g lnarcsin arcsin. cos e e 7.I. 7. II. maches wih c maches wih b 7. III. 7. IV. 8.a. I III 8.b. IV maches wih d maches wih a d d II d d d d d d. C. a. k.8 b. or baceria c., baceria. a. 8 f. b f. b. 7 f. a. b. hours f hr