26 28 Find an equation of the tangent line to the curve at the given point Discuss the curve under the guidelines of Section

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1 SECTION 5. THE NATURAL LOGARITHMIC FUNCTION 5. THE NATURAL LOGARITHMIC FUNCTION A Click here for answers. S Click here for solutions. 4 Use the Laws of Logarithms to epand the quantit.. ln ab. ln c. ln s 4. ln z 5 6 Epress the quantit as a single logarithm. 5. ln 0 ln 9 6. ln 4 ln 7 Differentiate the function. 7. f ln 8. f cos ln 6 8 Find an equation of the tangent line to the curve at the given point. 6. ln ln, e, 0 7. ln, 8. sin ln, 9 Discuss the curve under the guidelines of Section ln cos 0... ln tan ln ln s, ln, 0 9. f ln 0. F ln s. G s ln. h ln sin. ln 4. ln 5. ln ln 6. t t sin ln t 7. G sln 8. k r r sin r ln r 9. ln u t u ln u 0. ln sin. ln +. lns sin 45 Evaluate the integral.. 9 d d d d d e e d d 5 4 d d d ln 4 Differentiate f and find the domain of f.. f ln 4. f ln 5. If f ln, find f. sec tan d 45. ln 4 d sin cos d Copright 0, Cengage Learning. All rights reserved.

2 SECTION 5. THE NATURAL LOGARITHMIC FUNCTION 5. ANSWERS E Click here for eercises. S Click here for solutions.. ln a +lnb ln c. ln +ln +. ln + ln +ln 4. ln + ln ln z 5. ln 0 6. ln f / + 8. f sin ln / 9. f 0. F. G ln /. h +cot. + ln ln 6. g t coslnt /t 7. G ln 8. k r sin r ln r + r cos r ln r +sinr 9. g u u + ln u 0. lnsin cot. + ln +. +cot. f +,, 6. e 7. +ln 8. { 9. A. n <<n +, n an integer} B. -int. n n an integer, -int. 0 C. About the -ais. f has period and in parts D G we consider onl <<. D. VA ± E. Inc. on, 0, dec. on 0, F. Loc. ma. f 0 0 G. CD on, 0. A. { n/} Note: f is periodic with period,soin B-G we consider onl << B. -int. 4, 4 C. About the -ais, period D. VA 0, ± E. Inc. on 0,,dec.on, 0 F. None G. CD on, 0, 0, 4 4 ;CUon, 4,, 4.IP ±, 0 4. A. 0, B. -int 0.65 C. None D. VA 0 E. Inc. on 0, F. None G. CU on,,cdon 0,,IP, ln Copright 0, Cengage Learning. All rights reserved. 4. f ln,, 5.

3 SECTION 5. THE NATURAL LOGARITHMIC FUNCTION. A. R B. -int. 0, -int. 0 C. About the origin D. None E. Inc. on R F. None G. CU on, 0, CD on 0,,IP0, 0 0. ln ln 6. ln 5 + C ln C 4 ln C 9. ln ln + C ln + + C 4. ln ln + C 4. ln tan + C 44. ln + cos +C ln 5 + C 5 Copright 0, Cengage Learning. All rights reserved.

4 4 SECTION 5. THE NATURAL LOGARITHMIC FUNCTION 5. SOLUTIONS E Click here for eercises. Copright 0, Cengage Learning. All rights reserved.. ln ab c lnab ln c lna +lnb ln c lna +lnb ln c. ln + ln+ln + ln+ln +. ln ln / ln ln + ln +ln / 4. ln ln +ln z z ln +ln z ln + ln ln + ln ln z z 5. ln 0 + ln 9 ln 0 + ln 9/ ln0+lnln0 6. ln 4ln +ln / ln + 4 ln f ln + f / + 8. f cos ln f sin ln / 9. f ln f d d 0. F ln ln / ln F. G ln ln / G ln / ln /. h ln sin ln + ln sin h + sin cos +cot. ln + + / ln + + ln + + ln + 4. ln 5. ln +ln + + +ln +ln 6. g t sin ln t g t coslnt /t 7. G ln G ln 8. kr r sin r ln r k r sin r ln r + r cos r ln r +sinr 9. g u ln u +lnu g + ln u /u ln u/u u + ln u u + ln u 0. ln sin lnsin cos sin lnsin cot ln. ln + + / ln + + ln +. ln sin ln + ln +lnsin + + cos sin +cot. f ln + f + +. Dom f { +> 0},. 4. f ln f ln + ln Dom f { > 0 } { < },. 5. f ln f ln + ln + f ln + 6. f lnln f ln f e, so an equation of the tangent line at e, 0 is e 0 e e,or, or e e. e 7. f ln + f + f, so + an equation of the tangent line at, ln is ln,or +ln.

5 SECTION 5. THE NATURAL LOGARITHMIC FUNCTION 5 8. f sin ln f cosln/ f cos 0, so the equation of the tangent at, 0 is f lncos A. D { cos >0},, 5 { n <<n +, n an integer} B. -intercepts occur when ln cos 0 cos n n an integer, -intercept f 0 0. C. f f, so the curve is smmetric about the -ais. f + f, sof has period and in parts D G we consider onl <<. D. lim ln cos and / lim ln cos,so / + and are VA. No HA. E. f / cos sin tan >0 <<0, sof is increasing on, 0 and decreasing on 0,. F. f 0 0 is a local maimum. G. f sec <0 f is CD on,. No IP. 0. ln tan A. D { n/} B. -intercepts n +,no-intercept. 4 C. f f, so the curve is smmetric about the -ais. Also f + f, sof is periodic with period, and we consider D-G onl for <<. D. lim ln tan and lim ln tan, 0 / lim ln tan, so 0, ± are VA. / + E. f tansec tan sec tan > 0 tan >0 0 <<,sof is increasing on 0, and decreasing on, 0. F. No maimum or minimum G. f sin cos 4 sin f 8cos sin < 0 cos >0 4 << 4,sof is CD on 4, 0 and 0, 4 and CU on, 4 and 4,.IPare ± 4, 0.. f +ln A. D 0, B. -intercept 0.65, no-intercept C. No smmetr D. lim +ln, no HA. lim +ln, so 0is a VA. 0 + E. f +/ > 0, sof is increasing on 0,. F. No maimum or minimum G. f > 0 > >,sof is CU on, and CD on 0,. IP,. ln. f ln + + A. + + > 0 for all since + > + >, sod R. B. -intercept occurs where intercept f 0 0. C. ln + + ln + + since + + +, so the curve is smmetric about the origin. D. lim ln + +, lim ln + + lim ln +, no HA. No VA. E. f + + > so f is increasing on R. F. No maimum or minimum G. f f > 0 + / <0,sof is CU on, 0 and CD on 0,,andthere is an IP at 0, 0. Copright 0, Cengage Learning. All rights reserved. 0

6 6 SECTION 5. THE NATURAL LOGARITHMIC FUNCTION d 9 d [ln ]9 ln 9 ln ln 9 0ln9/ ln e d [ln ] e e e lne ln e 6 5. Let u +.Thendu d,so 0 d + 9 du u ln u] 9 ln 9 ln ln ln ln ln 44. Let u +cos. Thendu sin d,so sin du +cos d ln u + C u ln + cos +C 45. Let u +ln. Then du d,so + ln 4 d u 4 du u5 + ln 5 +C +C 5 5 ln or ln 6. Let u 5. Thendu d,so d 5 u du ln u + C ln 5 + C 7. Let u + +. Thendu + d,so d du u ln u + C ln C 8. Let u + 4.Thendu 4 d,so + d 4 4 u du ln u + C ln C 4 ln C since + 4 > 0 d ln 8 4 ln8 ln 4 ln 8 4 ln 40. Let u. Thendu d,so d du u ln u + C ln + C 4. Let u +. Thendu +d,so + + d du u ln u + C ln + + C Copright 0, Cengage Learning. All rights reserved. 4. Let u ln. Then du d, so d du ln ln u + C ln ln + C u 4. Let u tan. Thendu sec d,so sec du tan d ln u + C u ln tan + C