60_00R.q //0 :5 PM Pge 58 58 CHAPTER Differentition In Eercises, fin the erivtive of the function b using the efinition of the erivtive.. f. f. f. f In Eercises 5 n 6, escribe the -vlues t which ifferentible. 7. Sketch the grph of () Is f continuous t? (b) Is f ifferentible t? Eplin. 8. Sketch the grph of f,, () Is f continuous t? (b) Is f ifferentible t? Eplin. In Eercises 9 n 0, fin the slope of the tngent line to the grph of the function t the given point. 9. 0. Review Eercises for Chpter 5. f 6. f g 6, h 8, f., 5 6, 5 In Eercises n, () fin n eqution of the tngent line to the grph of f t the given point, (b) use grphing utilit to grph the function n its tngent line t the point, n (c) use the erivtive feture of the grphing utilit to confirm our results.. f. f,, 0,, In Eercises n, use the lterntive form of the erivtive to fin the erivtive t c (if it eists).. g c. f, c, In Eercises 5 0, fin the erivtive of the function. 8 5. 5 6. 7. f 8 8. g 8 <. 8 f is See www.clccht.com for worke-out solutions to o-numbere eercises. 9. ht t 0. f t 8t 5. f. gs s 5s. h 6. f 5. gt 6. h t 7. f sin 8. g cos 6 9. 0. g 5 sin f cos sin Writing In Eercises n, the figure shows the grphs of function n its erivtive. Lbel the grphs s f or n write short prgrph stting the criteri use in mking the selection. To print n enlrge cop of the grph, go to the website www.mthgrphs.com.... Vibrting String When guitr string is plucke, it vibrtes with frequenc of F 00T, where F is mesure in vibrtions per secon n the tension T is mesure in pouns. Fin the rtes of chnge of F when () T n (b) T 9.. Verticl Motion A bll is roppe from height of 00 feet. One secon lter, nother bll is roppe from height of 75 feet. Which bll hits the groun first? 5. Verticl Motion To estimte the height of builing, weight is roppe from the top of the builing into pool t groun level. How high is the builing if the splsh is seen 9. secons fter the weight is roppe? 6. Verticl Motion A bomb is roppe from n irplne t n ltitue of,00 feet. How long will it tke for the bomb to rech the groun? (Becuse of the motion of the plne, the fll will not be verticl, but the time will be the sme s tht for verticl fll.) The plne is moving t 600 miles per hour. How fr will the bomb move horizontll fter it is relese from the plne? 7. Projectile Motion A bll thrown follows pth escribe b 0.0. () Sketch grph of the pth. (b) Fin the totl horizontl istnce the bll is thrown. (c) At wht -vlue oes the bll rech its mimum height? (Use the smmetr of the pth.) () Fin n eqution tht gives the instntneous rte of chnge of the height of the bll with respect to the horizontl chnge. Evlute the eqution t 0, 0, 5, 0, n 50. (e) Wht is the instntneous rte of chnge of the height when the bll reches its mimum height? π π f
60_00R.q //0 :5 PM Pge 59 REVIEW EXERCISES 59 8. Projectile Motion The pth of projectile thrown t n ngle of 5 with level groun is v 0 where the initil velocit is v 0 feet per secon. () Fin the -coorinte of the point where the projectile strikes the groun. Use the smmetr of the pth of the projectile to locte the -coorinte of the point where the projectile reches its mimum height. (b) Wht is the instntneous rte of chnge of the height when the projectile is t its mimum height? (c) Show tht oubling the initil velocit of the projectile multiplies both the mimum height n the rnge b fctor of. () Fin the mimum height n rnge of projectile thrown with n initil velocit of 70 feet per secon. Use grphing utilit to grph the pth of the projectile. 9. Horizontl Motion The position function of prticle moving long the -is is t t t for < t <. () Fin the velocit of the prticle. (b) Fin the open t-intervl(s) in which the prticle is moving to the left. (c) Fin the position of the prticle when the velocit is 0. () Fin the spee of the prticle when the position is 0. 0. Moeling Dt The spee of cr in miles per hour n the stopping istnce in feet re recore in the tble. Spee, 0 0 0 50 60 Stopping Distnce, 5 55 05 88 00 () Use the regression cpbilities of grphing utilit to fin qurtic moel for the t. (b) Use grphing utilit to plot the t n grph the moel. (c) Use grphing utilit to grph. () Use the moel to pproimte the stopping istnce t spee of 65 miles per hour. (e) Use the grphs in prts (b) n (c) to eplin the chnge in stopping istnce s the spee increses. In Eercises 5, fin the erivtive of the function.. f 7. g. h sin. f t t cos t 5. 6 5 f 6. f 7. 9 f 8. f 9. 50. cos sin 5. sec 5. tn 5. cos sin 5. g sin cos In Eercises 55 58, fin n eqution of the tngent line to the grph of f t the given point. 55. f,, 56. f,, sin 57. f tn, 0, 0 58. f, sin, 59. Accelertion The velocit of n object in meters per secon is vt 6 t, 0 t 6. Fin the velocit n ccelertion of the object when t. 60. Accelertion An utomobile s velocit strting from rest is where v is mesure in feet per secon. Fin the vehicle s velocit n ccelertion t ech of the following times. () secon (b) 5 secons (c) 0 secons In Eercises 6 6, fin the secon erivtive of the function. 6. gt t t 6. f 6. f tn 6. ht sin t 5 cos t In Eercises 65 n 66, show tht the function stisfies the eqution. 65. 66. In Eercises 67 78, fin the erivtive of the function. 67. h 68. f 5 69. f s s 5 s 5 70. h 7. cos 7. cos cos 7. 7. sec7 sec5 sin 7 5 75. 76. f sin 7 sin7 77. sin 78. In Eercises 79 8, fin the erivtive of the function t the given point. 79. vt Function sin cos 90t t 0 0 cos f,, 80. f,, Eqution 0 sin cos
60_00R.q //0 :5 PM Pge 60 60 CHAPTER Differentition 8. 8. csc,, csc cot, 6, In Eercises 0 06, use implicit ifferentition to fin /. 0. 0 0. 9 0 0. 6 0. 05. sin cos 06. cos In Eercises 8 86, use computer lgebr sstem to fin the erivtive of the function. Use the utilit to grph the function n its erivtive on the sme set of coorinte es. Describe the behvior of the function tht correspons to n zeros of the grph of the erivtive. 8. g 8. f 85. f t t t 86. In Eercises 87 90, () use computer lgebr sstem to fin the erivtive of the function t the given point, (b) fin n eqution of the tngent line to the grph of the function t the point, n (c) grph the function n its tngent line on the sme set of coorinte es. 87. 88. 89. 90. f t t t 5, g, tn, csc,,, 0, tn, csc In Eercises 9 9, fin the secon erivtive of the function. 9. 9. sin tn In Eercises 07 n 08, fin the equtions of the tngent line n the norml line to the grph of the eqution t the given point. Use grphing utilit to grph the eqution, the tngent line, n the norml line. 07. 0,, 08. 6, 09. A point moves long the curve in such w tht the -vlue is incresing t rte of units per secon. At wht rte is chnging for ech of the following vlues? () (b) (c) 0. Surfce Are The eges of cube re epning t rte of 5 centimeters per secon. How fst is the surfce re chnging when ech ege is.5 centimeters?. Depth The cross section of five-meter trough is n isosceles trpezoi with two-meter lower bse, three-meter upper bse, n n ltitue of meters. Wter is running into the trough t rte of cubic meter per minute. How fst is the wter level rising when the wter is meter eep?. Liner n Angulr Velocit A rotting becon is locte kilometer off stright shoreline (see figure). If the becon rottes t rte of revolutions per minute, how fst (in kilometers per hour) oes the bem of light pper to be moving to viewer who is kilometer own the shoreline? 5, 9. f cot 9. sin In Eercises 95 98, use computer lgebr sstem to fin the secon erivtive of the function. t 6 5 95. f t 96. g ( t 97. g tn sin 98. h 99. Refrigertion The temperture T of foo put in freezer is T 700 t t 0 where t is the time in hours. Fin the rte of chnge of T with respect to t t ech of the following times. () t (b) t (c) t 5 () t 0 00. Flui Flow The emergent velocit v of liqui flowing from hole in the bottom of tnk is given b v gh, where g is the ccelertion ue to grvit ( feet per secon per secon) n h is the epth of the liqui in the tnk. Fin the rte of chnge of v with respect to h when () h 9 n (b) h. (Note tht g feet per secon per secon. The sign of g epens on how problem is moele. In this cse, letting g be negtive woul prouce n imginr vlue for v. ) θ km. Moving Show A snbg is roppe from blloon t height of 60 meters when the ngle of elevtion to the sun is 0 (see figure). Fin the rte t which the show of the snbg is trveling long the groun when the snbg is t height of 5 meters. Hint: The position of the snbg is given b st 60.9t. Position: s(t) = 60.9t 0 km Show s pth 60 m rev min Not rwn to scle Rs
60_00R.q //0 :5 PM Pge 6 P.S. Problem Solving 6 P.S. Problem Solving See www.clccht.com for worke-out solutions to o-numbere eercises.. Consier the grph of the prbol. () Fin the rius r of the lrgest possible circle centere on the -is tht is tngent to the prbol t the origin, s shown in the figure. This circle is clle the circle of curvture (see Section.5). Fin the eqution of this circle. Use grphing utilit to grph the circle n prbol in the sme viewing winow to verif our nswer. (b) Fin the center 0, b of the circle of rius centere on the -is tht is tngent to the prbol t two points, s shown in the figure. Fin the eqution of this circle. Use grphing utilit to grph the circle n prbol in the sme viewing winow to verif our nswer. 5. Fin thir-egree polnomil p tht is tngent to the line t the point,, n tngent to the line 5 t the point,. 6. Fin function of the form f b cos c tht is tngent to the line t the point 0,, n tngent to the line t the point,. 7. The grph of the eight curve,, 0, (0, b) is shown below. r Figure for () Figure for (b). Grph the two prbols n 5 in the sme coorinte plne. Fin equtions of the two lines simultneousl tngent to both prbols.. () Fin the polnomil P 0 whose vlue n slope gree with the vlue n slope of f cos t the point 0. (b) Fin the polnomil P 0 whose vlue n first two erivtives gree with the vlue n first two erivtives of f cos t the point 0. This polnomil is clle the secon-egree Tlor polnomil of f cos t 0. (c) Complete the tble compring the vlues of f n P. Wht o ou observe? () Eplin how ou coul use grphing utilit to grph this (b) Use grphing utilit to grph the curve for vrious vlues of the constnt. Describe how ffects the shpe of the (c) Determine the points on the curve where the tngent line is horizontl. 8. The grph of the per-shpe qurtic, b,, b > 0, is shown below..0 0. 0.00 0 0.00 0..0 cos P () Fin the thir-egree Tlor polnomil of f sin t 0.. () Fin n eqution of the tngent line to the prbol t the point,. (b) Fin n eqution of the norml line to t the point,. (The norml line is perpeniculr to the tngent line.) Where oes this line intersect the prbol secon time? (c) Fin equtions of the tngent line n norml line to t the point 0, 0. () Prove tht for n point, b 0, 0 on the prbol, the norml line intersects the grph secon time. () Eplin how ou coul use grphing utilit to grph this (b) Use grphing utilit to grph the curve for vrious vlues of the constnts n b. Describe how n b ffect the shpe of the (c) Determine the points on the curve where the tngent line is horizontl.
60_00R.q //0 :5 PM Pge 6 6 CHAPTER Differentition 9. A mn 6 feet tll wlks t rte of 5 feet per secon towr streetlight tht is 0 feet high (see figure). The mn s -foot-tll chil follows t the sme spee, but 0 feet behin the mn. At times, the show behin the chil is cuse b the mn, n t other times, b the chil. () Suppose the mn is 90 feet from the streetlight. Show tht the mn s show etens beon the chil s show. (b) Suppose the mn is 60 feet from the streetlight. Show tht the chil s show etens beon the mn s show. (c) Determine the istnce from the mn to the streetlight t which the tips of the two shows re ectl the sme istnce from the streetlight. () Determine how fst the tip of the show is moving s function of, the istnce between the mn n the street light. Discuss the continuit of this show spee function. 0 ft 6 ft Not rwn to scle ft 0 ft Figure for 9 Figure for 0 0. A prticle is moving long the grph of (see figure). When 8, the -component of its position is incresing t the rte of centimeter per secon. () How fst is the -component chnging t this moment? (b) How fst is the istnce from the origin chnging t this moment? (c) How fst is the ngle of inclintion chnging t this moment?. Let L be ifferentible function for ll. Prove tht if L b L Lb for ll n b, then L L0 for ll. Wht oes the grph of L look like?. Let E be function stisfing E0 E0. Prove tht if E b EEb for ll n b, then E is ifferentible n E E for ll. Fin n emple of function stisfing E b EEb. sin. The funmentl limit lim ssumes tht is mesure 0 in rins. Wht hppens if ou ssume tht is mesure in egrees inste of rins? () Set our clcultor to egree moe n complete the tble. z (in egrees) 0. 0.0 0.000 sin z z θ (8, ) 6 8 0 (b) Use the tble to estimte sin z lim z 0 z for z in egrees. Wht is the ect vlue of this limit? (Hint: 80 rins) (c) Use the limit efinition of the erivtive to fin z sin z for z in egrees. () Define the new functions Sz sincz n Cz coscz, where c 80. Fin S90 n C80. Use the Chin Rule to clculte z Sz. (e) Eplin wh clculus is me esier b using rins inste of egrees.. An stronut stning on the moon throws rock into the ir. The height of the rock is s 7 0 t 7t 6 where s is mesure in feet n t is mesure in secons. () Fin epressions for the velocit n ccelertion of the rock. (b) Fin the time when the rock is t its highest point b fining the time when the velocit is zero. Wht is the height of the rock t this time? (c) How oes the ccelertion of the rock compre with the ccelertion ue to grvit on Erth? 5. If is the ccelertion of n object, the jerk j is efine b j t. () Use this efinition to give phsicl interprettion of j. (b) Fin j for the slowing vehicle in Eercise 7 in Section. n interpret the result. (c) The figure shows the grph of the position, velocit, ccelertion, n jerk functions of vehicle. Ientif ech grph n eplin our resoning. b c