CHAPTER 5 Logarithmic, Exponential, and Other Transcendental Functions

Transcription

1 CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. The Natural Logarithmic Function: Dierentiation.... Section 5. The Natural Logarithmic Function: Integration Section 5. Inverse Functions Section 5. Eponential Functions: Dierentiation and Integration.. Section 5.5 Bases Other than e and Applications Section 5.6 Inverse Trigonometric Functions: Dierentiation Section 5.7 Inverse Trigonometric Functions: Integration Section 5.8 Hperbolic Functions Review Eercises Problem Solving

2 CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. The Natural Logarithmic Function: Dierentiation. Simpson s Rule: n t dt Note:.5 t dt t dt. (a) ln dt.867 t 5. (a) ln dt. t ln Vertical shit units upward Matches 9. ln Horizontal shit unit to the right Matches (a). ln. ln 5. ln Domain: > Domain: > Domain: > 5 7. (a) ln 6 ln ln.797 ln ln ln.55 (c) ln 8 ln ln.9 (d) ln ln ln ln ln ln. ln z ln ln ln z. ln a lna lna 5. ln ln ln ln ln ln

3 Section 5. The Natural Logarithmic Function: Dierentiation 5 7. ln zz ln z lnz 9. ln ln ln ln z lnz. ln ln ln ln ln. ln 9 ln ln 9 ln ln 5. (a) = g 9 ln ln ln ln ln g since >. 7. lim ln 9. lim ln ln.86. ln ln. ln ln 5. g ln ln g Slope at, is. Slope at, is. Tangent line Tangent line 7. ln 9. d d ln ln ln ln ln d d 5. ln ln ln 5. gt ln t t gt t t t ln t t ln t t 55. lnln 57. d d d ln d ln ln ln ln ln ln ln d d 59. ln ln ln

4 ( 6 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions d d ln ln sin d cos cot d sin cos 67. lncos ln cos ln cos d sin d cos sin cos tan sin cos sin ln sin d d ln sin ln sin cos sin cos sin cos sin sin 69. ln t dt ln ln Second solution: ln t dt t t ln ln ln ln ln 7. (a) ln,, d d 6 When, d 5. d Tangent line: 5 (, ) (a) ln sin ln sin,, ln π (, ln Tangent line: sin cos sin cos sin sin ln ln

5 Section 5. The Natural Logarithmic Function: Dierentiation (a) ln,, 77. ln Tangent line: ln d d d d d d d d (, ) 79. At, : ln,, 8. Tangent line: ln 8. ln 85. ln Domain: > Domain: > (, ) when. > > Relative minimum: ln ln when e. e, e Relative minimum:, ( e, e ) 87. ln Domain: < <, > ln ln ln ln ln when e. ln ln ln ln when e. Relative minimum: ln e, e Point o inlection: e, e (, e e) ( e, e/) 9

6 8 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 89. ln, 9. Find such that ln.,, P, P ln n n n n n ln n n P, P P, P P, P P, P The values o, P, P, and their irst derivatives agree at. The values o the second derivatives o and P agree at. n n n Approimate root:.567 P P ln ln ln d d d d ln ln ln ln d d d d ln ln ln ln d d d d 99. Answers will var. See Theorems 5. and 5..

7 Section 5. The Natural Logarithmic Function: Dierentiation 9. g ln, > g (a) Yes. I the graph o g is increasing, then g >. Since >, ou know that g and thus, >. Thereore, the graph o is increasing. No. Let (positive and concave up). g ln is not concave up.. False ln ln 5 ln5 ln t ln, (a) 5 (c) t67. ears T 67. $8,78. t68.5 ears < T 68.5 $8,6. (d) dt d ln ln When 67., dtd.65. When 68.5, dtd.585. (e) There are two obvious beneits to paing a higher monthl pament:. The term is lower. The total amount paid is lower. 7. (a) 5 T p.96 p.955 p (c) T.75 deglbin T7.97 deglbin lim Tp p 9. ln ln ln (a) (c) d lim d d d When 5, dd. When 9, dd 99.

8 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions. ln > or >. Since ln is increasing on its entire domain,, it is a strictl monotonic unction and thereore, is one-to-one. Section 5. The Natural Logarithmic Function: Integration. 5 d 5 d 5 ln C. u, du d d ln C 5. u, du d 7. d d u, du d d d ln C ln C ln C 9.. d d. ln C 6 d d 5. u 9, du d 9 d 9 d 5 ln 9 C d 5 d 7. d 6 ln C d ln C 9. u ln, du d ln d ln C 5 ln C. u, du d. d d C C d d d d d d ln C

9 Section 5. The Natural Logarithmic Function: Integration 5. u, d where C C. du d u du d u u u lnu C du u du ln C ln C 7. u, du d u du d d u du u u where C C 7. 6u 9 u u 6u 9 ln u C u u 8 ln u C du u 6 9 u du 8 ln C 6 8 ln C 9. cos sin d lnsin C. u sin, du cos d csc d csc d ln csc cot C sin t dt ln sin t C 5. sec tan sec d ln sec C. cos t 7. d 9. s tan d (, ) d tan d ln C, : ln C C ln ln cos C, : ln cos C C (, ) s ln cos

10 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions., > C C C ln C C C ln. d d, (a) (, ), 5. (a) 5 (, ) 8 d ln C ln C C ln Hence, ln ln ln. d d,, ln C C C ln d 5 ln 5 ln u ln, e ln du d d ln e 7 5. d d 5. ln ln cos sin d ln sin sin ln sin d ln C ln C where C C.

11 Section 5. The Natural Logarithmic Function: Integration 57. d ln C 59. csc sin d ln csc cot cos ln.7 Note: In Eercises 6 6, ou can use the Second Fundamental Theorem o Calculus or integrate the unction. 6. F t dt F 6. F t dt F (b Second Fundamental Theorem o Calculus) Alternate Solution: F F t dt ln t ln A d ln ln 69. A tan d ln cos ln ln ln A.5; Matches (d) A d d 7. ln 8 ln 5 8 ln.5 square units sec d 6 ln sec tan ln 5. sec 6 6 ln d sec tan 6 6 ln 6

12 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 75., b a 5, n Trapezoid: Simpson: Calculator: Eact: ln d ln, b a 6, n Trapezoid: Simpson: Calculator: ln d Power Rule 8. Substitution: u and 8. Log Rule ln cos C ln cos C ln sec C ln sec tan sec tan sec tan C ln sec tan ln C Average value d Pt.5t dt.5 sec tan C ln sec tan C 8, ln.5t C P, ln.5 C C Pt, ln.5t ln.5t P ln d 89. ln sec tan sec tan C Average value e e ln e e e.9 e.5t dt , 5 d ln 5 $68.7 ln d

13 Section 5. The Natural Logarithmic Function: Integration (a) 8 e d e ln C e ln e C k 8 8 Let k and graph., 8 (c) In part (a): 8 In part : Using a graphing utilit the graphs intersect at.,.. The slopes are.95 and..95, respectivel. 97. False ln ln ln 99. True d ln C ln ln C ln C, C. (a) (c) intersects :.5 5 A d ln ln Hence, or < m <, the graphs o and m enclose a inite region..5 () = + m m m m m m m = m m, intersection point m A mm ln m mm ln m m ln m m m d, < m < m m m [m lnm

14 6 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. Inverse Functions. (a) 5 g 5 g g g g (a) g g g g g 5. (a) g, g g g 8 6 g (a) g = g g g 9. Matches (c). Matches (a). 6 One-to-one; has an inverse 8 5. sin Not one-to-one; does not have an inverse 7. hs s One-to-one; has an inverse π π θ 7

15 Section 5. Inverse Functions 7 9. ln. One-to-one; has an inverse 5 g 5. One-to-one; has an inverse ln, > > or > is increasing on,. Thereore, is strictl monotonic and has an inverse when,, is not strictl monotonic on,. Thereore, does not have an inverse. < or all is decreasing on,. Thereore, is strictl monotonic and has an inverse ,,, 7. 9.,, 6 The graphs o and across the line. are relections o each other The graphs o and across the line. are relections o each other

16 8 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions , < < (, ) (, ) (, ) (, ) The graphs o and across the line. are relections o each other 5. (a) Let be the number o pounds o the commodit costing.5 per pound. Since there are 5 pounds total, the amount o the second commodit is 5. The total cost is , 5. (c) Domain o inverse is We ind the inverse o the original unction: Inverse: represents cost and represents pounds. (d) I 7 in the inverse unction, pounds. 7. on, 9. > on, is increasing on,. Thereore, is strictl monotonic and has an inverse. on, 8 < on, is decreasing on,. Thereore, is strictl monotonic and has an inverse. 5. cos on, sin < on, is decreasing on,. Thereore, is strictl monotonic and has an inverse. 5. a, b, c ± 6,, Domain o : all Range o : < < on, ± 6 i i The graphs o and the line. are relections o each other across

17 Section 5. Inverse Functions (a), 6 5 (c) Yes, is one-to-one and has an inverse. The inverse relation is an inverse unction. 57. (a), 6 g g (c) g is not one-to-one and does not have an inverse. The inverse relation is not an inverse unction , Domain: is one-to-one; has an inverse > or >, 6., 6. is one-to-one or. is one-to-one; has an inverse,, (Answer is not unique.) 65. is one-to-one or., (Answer is not unique.) 67. Yes, the volume is an increasing unction, and hence one-to-one. The inverse unction gives the time t corresponding to the volume V. 69. No, Ct is not one-to-one because long distance costs are 7. step unctions. A call lasting. minutes costs the same as one lasting. minutes., a 5 7. sin, 6 a 75. cos 6 cos6, 6 a 6 6

18 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 77. (a) Domain Domain, Range Range, (c) (d),, 8, 8, 79. (a) Domain,, Domain, Range,, Range, (c) (d) , 5, 5,, d d d d d d At,, d d. Alternate Solution: Let 7. Then and. Hence, d d. In Eercises 8 85, use the ollowing. and g 8 8 and g 8. g g In Eercises 87 89, use the ollowing. and g 5 and g g g 89. g 5 g g 5 5 Hence, g. Note: g g

19 Section 5. Inverse Functions 9. Answers will var. See page and Eample. 9. is not one-to-one because man dierent -values ield the same -value. Eample: Not continuous at n, where n is an integer. 95. k is one-to-one. Since, k k k. 97. Let and g be one-to-one unctions. (a) Let g g g g g g (Because is one-to-one.) (Because g is one-to-one.) Thus, g is one-to-one. Let g, then g. Also: g g g g g Thus, g g and g g. 99. Suppose g and h are both inverses o. Then the graph o contains the point a, b i and onl i the graphs o g and h contain the point b, a. Since the graphs o g and h are the same, g h. Thereore, the inverse o is unique.. False. Let.. True 5. Not true. Let,, <. is one-to-one, but not strictl monotonic. 7. dt, t (a) Hence, i then., The graph o is smmetric about the line.

20 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. Eponential Functions: Dierentiation and Integration. e ln. e 5. 9 e e ln.85 e e e 5 ln 5 ln ln. ln. ln e 7.89 e e e e.89 e e e 7. e Smmetric with respect to the -ais Horizontal asmptote: 9. (a) 7 g (c) 7 q 5 7 Horizontal shit units to the right h A relection in the -ais and a vertical shrink 8 Vertical shit units upward and a relection in the -ais. Ce a. C e a Horizontal asmptote: Matches (c) Vertical shit C units Relection in both the - and -aes Matches (a) 5. e 7. g ln ln 6 g 6 e 9. g ln 6 g 6 g As, the graph o approaches the graph o g..5 lim e.5

21 Section 5. Eponential Functions: Dierentiation and Integration.,,,, e e >,,,,. (a) e e e e Tangent line Tangent line 5. e 7. e e 9. d e d gt e t e t gt e t e t e t e t. ln e. d d e e e e e e d d e e e e e e e e 5. e sin cos 7. d d e cos sin sin cos e e cos e cos F ln cos e t dt F cose ln cos 9. e,, 5. e, Tangent line lne,,, Tangent line 5. e e e,, e 55. e e e e e e e e e e Tangent line e ln,, e e ln e ln e e e e Tangent line 57. e 59. e d d e d d d d e e d e d e e e,, e e e e At, : e e Tangent line: e e

22 ( Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 6. e e e 7 6e 7 6e 6e 6 5e 6. e cos sin e sin cos e cos sin e cos sin e sin cos e cos sin e sin cos e cos sin e cos sin e cos sin Thereore,. 65. e e e e e e > Relative minimum:, when. 6 (, ) 67. g e g g e Relative maimum: Points o inlection:, e,,.99 e,,.8 (, e,.,,. (, ( π e.5, π ( ( e.5 π 69. e e e e when,. e e e when ±. Relative minimum:, Relative maimum:, e ± ± e ± Points o inlection:.,.8,.586,.9 (, ) (, e ) 5 ( ±, (6 ± )e ( ± ) )

23 Section 5. Eponential Functions: Dierentiation and Integration 5 7. gt te t gt te t (, + e) 5 (, ) g t te t 6 6 Relative maimum:, e,.78 Point o inlection:, 7. A baseheight e da d e e e when. A e (, e ) 75. Laeb ae b ae b b L, a >, b >, L > aeb L a b eb ae b al e b b al b eb aeb a b eb ae b ae b al e b b al b a eb b eb ae b i b ln a ae b al b eb ae b ae b b ln b ln a a L ae L b ln ab aa L Thereore, the -coordinate o the inlection point is L. 77. e e Let,, e be the point on the graph where the tangent line passes through the origin. Equating slopes, e e 8 () = e, e, e. (, e ( () = (e) Point:, e Tangent line: e e e

24 6 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 79. V 5,e.686t, t (a), dv dt 99e.686t (c), When t, dv dt When t 5, dv dt 8. h 5 5 P, ln P (a) ln P ah b P e ahb e b e ah P Ce ah, C e b (c).99h 9.8 is the regression line or data h, ln P., (d) For our data, a.99 and C e9.8, P,957.7e.99h dp, e.99h dh For 6.56e.99h dp h 5, dh For h 8, dp.6. dh 8. e, 7 e, P P e, 6 6 P, P P, P P 8 P 8 P, P P, P The values o, P, P and their irst derivatives agree at. The values o the second derivatives o and agree at. P 85. Let u 5, du 5 d. e 5 5 d e 5 C 87. e d e d e C

25 Section 5. Eponential Functions: Dierentiation and Integration Let u e, du e d. e e d e e d ln e C ln e e C lne C 9. Let u e, du e d. e e d e e d 9. Let u e e, du e e d. e e e e d ln e e C e C e d 5e e d e d 97. e tane d tane e d 5 e e C ln cose C 99. Let u, du d.. e d e d e e e e e d e e e d. Let u, du d. 5. d e d e e e e e e sin cos d e e esin e sin e sin cos d 7. Let u a, du a d. Assume a. 9. e e d e e C e a d ae a a d a ea C C e e d e e C C C e e. (a) 5 d d e,, 6 (, ) 5 e d e d 8 e C, : e C C C 5 e 5

26 8 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions. 5 e d e 5 e e d e 6 e e d, n e.9t8 dt 8 Midpoint Rule: Graphing utilit: % Trapezoidal Rule: 9.87 Simpson s Rule: Graphing utilit: 9.77 e t dt dt. e. Domain is, and range is,. et t e e or is continuous, increasing, one-to-one, and concave upwards on its entire domain. lim e and lim e. 5. Yes. Ce, C a constant. 7. e e e n n n n n n e n e n We approimate the root o to be ln ea e b ln ea ln e b a b ln e ab a b Thereore, ln ea and since is one-to-one, we have ea e b ln eab ln b eab. e

27 Section 5.5 Bases Other than e and Applications 9 Section 5.5 Bases Other than e and Applications. log 8 log. log 7 5. (a) 8 7. (a) log. log.5 8 log 8 log h (a) log log. 7. (a) log log (a) log log 6 log log 6 6 OR. 75. ln ln 75 ln ln z 65 z ln ln 65 z ln 65 ln z ln 65 ln t 7. t ln.9 ln t ln ln.9.5 log 5 5

28 5 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 9. log.5. g 6 5. hs log s 5.5 Zero:.59 Zero: s. ±.5 ±.85 (.59, ) (., ) 8 5. g log 6 6 g g ln gt t t gt t ln t t t t t t ln t t t ln. h cos h sin ln cos ln cos sin. log log log ln ln ln 5. log 5 log 5 7. d d ln 5 ln 5 gt log t t ln ln t t gt tt ln t ln t ln t t ln 5 ln t t ln 9.,, 5. ln At,, ln. Tangent line: ln ln ln log, 7, At ln 7,, 7 ln. Tangent line: 7 7 ln 7 ln ln

29 Section 5.5 Bases Other than e and Applications ln ln d d ln ln d d ln ln ln ln d d ln d d ln ln 57. ln sin ln At sin,, sin, : cos ln Tangent line: 59. ln cos lnln ln cos, e, cos sin lnln ln At e,, cose. e Tangent line: cose e e cose cose e 6. d C 6. ln 5 d 5 d 5 ln 5 C ln 5 5 C 65. u, du ln d d, 67. ln d ln ln ln C d ln ln 7 ln 7 ln d 5 ln 5 ln 5 ln 5 ln ln 5 ln 7. Area d ln ln 5 ln 6 7 ln ln

30 5 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions d 7. d., (a) (, (,. d. d ln.. C ln.5. C 6 6 ln.5 C C ln.5 ln.5. ln.5. ln (, ) (, ) (8, ) 6 8 (a) is an eponential unction o : False is a logarithmic unction o : True; log (c) is an eponential unction o : True, (d) is a linear unction o : False log ln g g ln Note: Let g. Then: ln ln ln ln ln ln g h h k k ln From greatest to smallest rate o growth: g, k, h, 79. Ct P.5 t (a) C.95.5 $.6 dc dt When When Pln.5.5t (c) t, t 8, dc dt dc dt.5p..7p. dc dt ln.5p.5t ln.5ct The constant o proportionalit is ln P $, r %.5, t n 65 Continuous A.5 n n A e A P $, r 5%.5, t A.5 n n A e n 65 Continuous A

31 Section 5.5 Bases Other than e and Applications 5 85., Pe.5t P,e.5t t 5 P 95,.9 6,65.7 6,787.9,., , P.5 t P,.5 t t 5 P 95,.8 6,76. 6,86.5,8.66, (a) A, $,. A $, (c) (d) A , $,98.9 5,.8 $8,5.57 A $,985. Take option (c). 9. (a) lim t 6.7e8.t 6.7e 6.7 million t V.7 t e 8.t V.7 million t r V6. million t r 9. 7e.65 (d) (a) 7e.65 I ( egg masses), %. (c) I 66.67%, then 8.8 or 8,8 egg masses. 8.75e.65 7e e.65 7e.65 7e.65 7e or 7,8 egg masses. 95. (a) (c) B d.759e.9d 97. (a) t dt 5.67 Bd 9.95e.9d B.8. tonsinch B tonsinch gt dt 5.67 ht dt 5.67 (c) The unctions appear to be equal: Analticall, t 8 t 8 t 9 t gt ht e.65886t e t.5 t gt 9 t.5 t t gt ht. No. The deinite integrals over a given interval ma be equal when the unctions are not equal. 6 5

32 5 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 99. t False. e is an irrational number. Ck t When t, C. k t ,.6,.6, Let k.6..6 t. True g e ln g ln e ln e 5. True d d e e and e e when. e e d d e e 7. ln ln 5 5 t C d dt 8 5 5, d dt 5 5 d 8 5 dt ln 5 5 t C 5 e5tc C e 5t C e 5t 5 e 5t 5 5e5t e 5t e 5t 5e5t e 5t 5 e.5.t.5e.t 9. (a) ln (i) At ln ln ln ln ln ln ln ln ln (c) (ii) At (iii) At At c, c:, :, : is undeined or c c ln c ln ln ln e. e, e, is undeined., c, e c c ln c 6 8 ln 8 ln ln.77 ln 8 ln ln ln ln

33 Section 5.6 Inverse Trigonometric Functions: Dierentiation 55. Let ln ln For n 8, e < n < n, 8.88 and so letting n, n, we have n n > n n. Note: ln, >. ln or > e is decreasing or e. Hence, or e < : < > > ln > ln ln > ln > and Note: This same argument shows e > e. Section 5.6 Inverse Trigonometric Functions: Dierentiation. arcsin (a) (c) (d) Smmetric about origin: π π arcsin arcsin Intercept:,. arccos,,, because 6 because cos cos because cos 6 5. arcsin 6 7. arccos 9. arctan 6. arccsc. arccos arcsec.69 arccos.69.66

34 56 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 7. (a) sin arctan 5 9. (a) cot arcsin cot 6 5 θ θ sec arcsin 5 5 csc arctan θ θ 5. cosarcsin. arcsin cos θ sinarcsec arcsec, sin, The absolute value bars on are necessar because o the restriction and sin or this domain must alwas be nonnegative.,, θ 5. tan arcsec 7. 9 csc arctan + arcsec θ arctan θ tan 9 csc 9. (a) Let = g arctan (c) Asmptotes: ± tan sin sinarctan. +. arcsin. arcsin arccos sin sinarccos sin.7, θ

35 Section 5.6 Inverse Trigonometric Functions: Dierentiation (a) arccsc arcsin, arctan arctan, > Let arccsc. Then or Let arctan arctan. Then, and < <, tan tanarctan tanarctan tanarctan tanarctan csc sin. Thus, arcsin. Thereore, arccsc arcsin. (which is undeined). Thus,. Thereore, arctan arctan. 7. arcsin 9. sin Domain: Range: sin,, is the graph o arcsin shited unit to the right. π π (, π ) π (, π ) π arcsec Domain: Range: sec sec,,,,,, (, π ( π (, (. arcsin. g arccos 5. arctan a g a a a a 7. g arcsin 9. ht sinarccos t t g 9 arcsin 9 arcsin 9 ht t t t t 5. arccos 5. arccos arccos ln arctan ln ln arctan d d 55. arcsin d d arcsin arcsin

36 58 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions arcsin arctan arcsin,, 6. At,,. Tangent line: arctan,, At,,. Tangent line: 65. arccos,, At,,. arccos Tangent line: 67. arctan, a,.5 P = P ,..5 P P 69. arcsin, a P P 6 6 P P

37 Section 5.6 Inverse Trigonometric Functions: Dierentiation arcsec 7. when 5 when or 5 ± ±.7 Relative maimum:.7,.66 Relative minimum:.7,.77 arctan arctan 8 6 B the First Derivative Test,,. is a relative maimum. 75. arctan At,, Tangent line: arctan, arctan 8 arctan , 77. At arcsin arcsin,,,, Tangent line: 79. The trigonometric unctions are not one-to-one on,, so their domains must be restricted to intervals on which the are one-to-one. 8. arccot, < < cot 8. False arccos 85. True d arctan d > or all. tan since the range is,. So, graph the unction arctan or > and arctan or <. 87. True d arctantan sec d tan sec sec

38 6 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 89. (a) cot 5 9. (a) ht 6t 56 d dt I arccot 5 d 5 dt 5 d 5 dt d dt 5 and, d 6 radhr. dt d I and, d 58.8 radhr. dt dt 6t 56 when t sec tan h 5 6t 56 5 arctan 6 d dt 5 t 6 8t5 5t 6 t 5,65 66 t h 5 θ When t, ddt.5 radsec. When t, ddt.6 radsec. 9. (a) tanarctan arctan Thereore, tanarctan tanarctan tanarctan tanarctan, arctan arctan arctan,. Let and. arctan arctan arctan 56 arctan 6 arctan 56 arctan k sin k cos or k k cos or k Thereore, k sin is strictl monotonic and has an inverse or k or k. 97. (a) arccos arcsin (c) Let u arccos and v arcsin cos u and sin v. The graph o is the constant unction. u v sinu v sin u cos v sin v cos u Hence, u v. Thus, arccos arcsin.

39 Section 5.7 Inverse Trigonometric Functions: Integration sec, <, < (, ) (, ) θ θ θ (c, ) (, ) π π tan c, tan c, < c < To maimize, we minimize c c arctan c arctan c. (a) arcsec, or < or < c c c c c c c 8c 6 π π 8c c B the First Derivative Test, c is a minimum. Hence, c, c, is a relative maimum or the angle. Checking the endpoints: c : tan.7 arcsec sec sec tan sec tan c : tan.7 c :.578 tan sec tan ±sec Thus,, is the absolute maimum. On < and <, tan. Section 5.7 Inverse Trigonometric Functions: Integration. 5 9 d 5 arcsin C. 7 6 d 7 arctan C 5. d d arcsec C 7. d d d d ln C (Use long division.) 9. d arcsin C. Let u t, du t dt. t t dt t t dt arcsint C

40 6 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions. Let u e, du e d e d e e d e arctan e C d d d ln arctan C d, u, u, d u du u u u du du arcsin u C u arcsin C d 9 d 8 9 d 9 8 arcsin C 6 8 arcsin C. Let u, du d. 6 9 d 5. Let u arcsin, du 6 d arcsin 6 d. arcsin d arcsin 8.8. Let u, du d. d 7. Let d arctan. 6 u, du d. d d 9. Let u cos, du sin d.. sin cos d sin cos d. arctancos ln 6 arctan C d d arctan 6 d 6 6 d 6 6 d 6 6 d 6 d 5. d d arcsin C 7. Let u, du d. d d C

41 Section 5.7 Inverse Trigonometric Functions: Integration 6 9. d d d d d arcsin Let u, du d. d d arctan C. Let u e t. Then u e t, u du e t dt, and et dt u u du du 6 u du u du dt. u u u arctan C et arctan e t C 5. Let u, u, u du d, u. u du u u u du 9. (a) d arctanu d C, u 6 7. (a) d arcsin C, u d C, u (c) d cannot be evaluated using the basic integration rules. Let u. Then u and d u du. d u uu du u u du u 5 5 u u 5 C 5 5 C 5 C (c) Let u. Then u and d u du. d u u du u u du u u C uu C C Note: In and (c), substitution was necessar beore the basic integration rules could be used. 5 u C 5. Area 5. Matches (c),, ) C d arcsin C arcsin

42 6 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 55. (a) 57. (a) d d,, d arctan C, : arctan C C arctan (, ),, d arcsec C arcsec C C arcsec, d d, 6. d d 6, A arctan arctan arctan 8 5 d d 65. Area d arcsin arcsin arcsin Area cos sin d cos d sin arctansin arctan arctan

43 Section 5.7 Inverse Trigonometric Functions: Integration (a) d d ln ln arctan C Thus, A arctan d ln ln arctan ln arctan ln ln arctan ln ln 9.85 arctan arctan arctan arctan d ln ln arctan C. 7. (a) π Shaded area is given b arcsin d. arcsin d.578 (c) Divide the rectangle into two regions. Area rectangle baseheight Area rectangle arcsin d sin d arcsin d cos Hence, arcsin d arcsin d, F t dt (a) F represents the average value o over the interval,. Maimum at, since the graph is greatest on,. F arctan t F arctan arctan when. d 75. False,9 6 arcsec C 77. True d d arccos C 79. d d arcsin u a C u u du Thus, a a u arcsin u C. a a u a u

44 66 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 8. Assume u >. d d a arcsec u a C a ua uaua a The case u < is handled in a similar manner. Thus, du uu a u uu a d a arcsec u C. a u uu a a u uu a. 8. (a) vt t 5 st vt dt t 5 dt (c) (e) 55 kv dv dt k arctan k v t C arctan k v kt C k v tanc k t v k tanc k t When t, v 5, C arctan5k, and we have vt k tan arctan 5 k k t. h 6.86 Simpson s Rule:, tanarctan5.5. t dt n ; h 88 eet () Air resistance lowers the maimum height. s 6 5 C C st 6t 5t When the object reaches its maimum height, vt. s (d) When k.: v(t, tanarctan5.5. t 5 6t 5t C vt t 5 t 5 t t Maimum height vt when t 6.86 sec. 7 Section 5.8 Hperbolic Functions. (a) sinh e e.8 tanh sinh cosh e e e.96 e. (a) cschln e ln e ln cothln 5 coshln 5 5 sinhln 5 eln e ln 5 e ln 5 e ln

45 Section 5.8 Hperbolic Functions (a) cosh ln.7 sech ln e e tanh sech e e e e e e e e e e e e 9. sinh cosh cosh sinh e e e e e e e e e e e e e e e e e e e e sinh. sinh sinh sinh sinh e e e e e e e e e e e e e e e e e e e e sinh. sinh cosh cosh tanh csch sech coth cosh 5. sech 7. lnsinh 9. ln tanh sech tanh cosh coth sinh tanh sech sinh cosh csch sinh. h sinh. t arctansinh t h cosh cosh sinh t cosh t cosh t sech t sinh t cosh t

46 68 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 5. sinh,, 7. cosh Tangent line: cosh sinh,, cosh sinh sinh cosh At,,. Tangent line: 9. sin sinh cos cosh, ( π, cosh π) ( π, cosh π) sin cosh cos sinh cos sinh sin cosh sin cosh when, ±. Relative maima: ±, cosh Relative minimum:, (, ). g sech. a sinh g sech sech tanh sech tanh tanh a cosh a sinh a cosh Using a graphing utilit, ±.997. Thereore,. B the First Derivative Test,.997,.667 is a relative maimum and.997,.667 is a relative minimum. (.,.66) (.,.66) 5. tanh, 7. (a) sech, sech tanh, P P P 5 cosh, At ±5, 5 cosh.6. P At, 5 cosh 5. sinh (c) At 5, sinh Let u, du d. sinh d sinh d. Let u cosh, du sinh d. cosh sinh d cosh C cosh C

47 Section 5.8 Hperbolic Functions 69. Let u sinh, du cosh d. cosh sinh d ln sinh C 5. Let u, du d. csch d csch d coth C 7. Let u, du d. csch coth d csch coth d 9. Let u, du d. d d arctan C csch C 5. ln sinh tanh d ln d, u cosh 5. cosh lncosh ln lncoshln lncosh ln 5 ln 5 Note: coshln eln e ln 5 5 d 5 ln 5 5 d 5 d ln 9 5 ln 55. Let u, du d. 57. d d arcsin cosh sinh tan 6. tan sec sec tanh sin cos sec sin 6. sinh sinh sinh 65. Answers will var. 67. lim sinh 69. lim sech sinh e 7. lim lim e 7. e d e e e d csch e C ln e e C

48 7 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 75. Let u, du d. d d sinh C ln C 77. d d ln ln C 79. d d d 6 ln ln C C 8. Let u, du d d 8 d arcsin 9 C d d 6 ln C ln 5 C 5 d 5 d ln 5 C 85. A sech d 87. e e e d e d 8 arctane 8 arctane d 5 d A 5 ln 5 ln (a) ln.7 d sinh sinh.7 d ln

49 Section 5.8 Hperbolic Functions 7 9. k 6 dt d kt 6 d 6 ln 6 When : t C ln When : k 5 ln C ln 8 t k 6 ln 7 ln ln 7 6 C When t : 6 5 ln ln 8 ln 7 6 ln kg 9. a sech a a, a > d d 95. Let a a tanh u. sinh u cosh u eu e u e u eu e u e u e u e u e u e u e u e u u ln u ln, < < a a a u tanh, < < 97. a a a a b e t dt et b b b eb eb eb e b sinhb 99. sech tanh sech. sech sech tanh sech sech sinh sinh cosh cosh sinh

50 7 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions. c cosh c Let P, be a point on the catenar. P(, ) sinh c The slope at P is sinh c. The equation o line L is (, c) L c. sinh c When c The length o L is sinh c c sinh, c. c sinh c c c cosh c, the ordinate o the point P. Review Eercises or Chapter 5. ln Vertical shit units upward Vertical asmptote: 5 = 5. ln 5 ln 5 5 ln ln ln 5. ln ln ln ln ln ln 7. ln ln e e e g ln ln. g ln ln ln ln ln ln ln. ba b a lna b 5. d d bb ab a b a b ln,, Tangent line:

51 7. u 7, du 7 d d 7 7 d 7 ln 7 C. d d ln ln Review Eercises or Chapter 5 7 sin cos d sin cos d. ln cos C sec d ln sec tan ln 5. (a) 7 7. (a) , (c) 6 (c) 6 6 or. 9. (a) (c). 5 or tan 5. (a) ln 6 sec 6 6 (c) ln e e e e ln e ln e ln e ln e ln e

52 7 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 7. e 9. 6 gt t e t gt t e t te t te t t e e. e e e e e e e e lne,, 7. Tangent line: e d 6 e 6 d 5. 6 e 6 e 6 e e d e d 55. e C g e g e e e e ln ln d d d d ln d d d d ln e e d e e e e d e e e C e e e Let u e, du e d. e e d e e d ln e lne lne ln e e lne e C 57. e a cos b sin e a sin b cos e a cos b sin e a b sin a b cos e a b cos a b sin e a b sin a b cos e 6a 8b sin 8a 6b cos e 6a 8b a b b sin 8a 6b a b a cos

53 Review Eercises or Chapter Area e d e e log 65. ln ln ln 69. ln ln ln g log log g ln ln 7. 5 d ln 5 5 C 7. t 5 log 8, 8, h (a) Domain: t 8 h < 8, (c) t 5 log 8, 8, h t5 8, 8, h 6 h,, Vertical asmptote: h 8, 8, h 8, t5 h 8, t5 dh dt 6 ln t5 is greatest when t. 75. arctan 6

54 76 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 77. (a) Let arcsin Let arcsin sin sin arcsin sin. θ sin cos arcsin cos. 79. tanarcsin 8. arcsec arcsec 8. arcsin arcsin arcsin arcsin arcsin arcsin 85. Let u e, du e d. e e d e e d e e d arctane C 87. Let u, du d. d d arcsin C u arctan, du d. 89. Let arctan d arctan arctan C d 9. A d 9. d d arcsin arcsin.86 d A k m dt arcsin A k m t C Since when t, ou have C. Thus, sin k m t A A sin k m t. 95. tanh tanh tanh 97. Let u, du d. sech d sech d tanh C

55 Problem Solving or Chapter 5 77 Problem Solving or Chapter 5. tan tan 6 Minimize a a.5 Endpoints: : a : a : or Maimum is.76 at a arctan arctan ± 8 ± 8 θ θ θ a 6. (a) ln, (c) Let g ln, g, and g. From the deinition o derivative g g ln g lim lim. Thus, lim. lim 5..5 and. intersect. does not intersect. Suppose is tangent to a at,. a a. a ln a ln ln e, a e e For < a e e.5, the curve a intersects. =.5 6 = = =. 6 6

56 78 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions 7. (a) arcsin sin Area A 6 Area B sin d cos 6.68 arcsin d AreaC 8 6 A B (c) (d) Area A ln e d e ln Area B ln d ln A ln ln tan Area A tan d ln cos ln ln ln ln π A Area C arctan d ln.688 ln π B = ln ln A e = B = arctan C 9. e e b e a a e a ae a b I : e a ae a b b ab b a c a Tangent line b e a Thus, a c a a.. Let u tan, du sec d. Area sin cos d du u sec tan d arctan u arctan. (a) u , t v 985.9, t u v 5 The larger part goes or interest. The curves intersect when t 7.7 ears. (c) The slopes are negatives o each other. Analticall, (d) u v du dt dv dt u5 v5.6. t.7 ears Again, the larger part goes or interest.