LARSON CALCULUS Ch03 Applications of the Derivative

1. Use ithe igraph iof i ito iidentify iat iwhich iof ithe iindicated ipoints ithe iderivative i ichanges ifrom ipositive ito inegative. A) (5,6) B) (-1,2), i(5,6) C) (2,4) D) (2,4), i(5,6) E) (-1,2) Ans: A 2. Use ithe igraph iof i ito iidentify iat iwhich iof ithe iindicated ipoints ithe iderivative i ichanges ifrom inegative ito ipositive. A) (2,4) B) (-1,2) C) (-1,2), i(5,6) D) (5,6) E) (2,4), i(5,6) Ans: B 3. Identify ithe iopen iintervals iwhere ithe ifunction i iis iincreasing ior idecreasing. A) decreasing: i i; iincreasing: i B) increasing: i i; idecreasing: i C) increasing ion i D) decreasing ion i E) none iof ithe iabove Ans: A 4. Both ia ifunction iand iits iderivative iare igiven. iUse ithem ito ifind iall icritical inumbers. A) B) C) D) E) Ans: D 5. Identify ithe iopen iintervals iwhere ithe ifunction i iis iincreasing ior idecreasing. A) increasing: i i; idecreasing: i B) decreasing: i i; iincreasing: i C) increasing ion i D) decreasing ion i E) none iof ithe iabove Ans: A 6. For ithe igiven ifunction, ifind iall icritical inumbers. A) B) iand i C) iand i D) iand i E) iand i Ans: D 7. Find iany icritical inumbers iof ithe ifunction i i, it i< i5. A) 0 B) C) D) both iA iand iB E) both iA iand iC Ans: C 8. Identify ithe iopen iintervals iwhere ithe ifunction i iis iincreasing ior idecreasing. A) decreasing: i i; iincreasing: i B) increasing: i i; idecreasing: i C) increasing: i i; idecreasing: i D) increasing: i ; idecreasing: i E) decreasing ifor iall ix Ans: B 9. For ithe igiven ifunction, ifind ithe icritical inumbers. A) B) C) D) E) Ans: A 10. Find ithe iopen iintervals ion iwhich ithe ifunction i iis iincreasing ior idecreasing. A) The ifunction iis iincreasing ion ithe iinterval i , iand idecreasing ion ithe iintervals i iand i . B) The ifunction iis iincreasing ion ithe iinterval i , iand idecreasing ion ithe iintervals i iand i . C) The ifunction iis iincreasing ion ithe iinterval i , iand idecreasing ion ithe iintervals i iand i . D) The ifunction iis idecreasing ion ithe iinterval i , iand iincreasing ion ithe iintervals i iand i . E) The ifunction iis idecreasing ion ithe iinterval i , iand iincreasing ion ithe iintervals i iand i . Ans: A 11. Find ithe iopen iintervals ion iwhich ithe ifunction i iis iincreasing ior idecreasing. A) The ifunction iis iincreasing ion ithe iinterval i iand idecreasing ion ithe iinterval i . B) The ifunction iis iincreasing ion ithe iinterval i iand idecreasing ion ithe iinterval i . C) The ifunction iis iincreasing ion ithe iinterval i iand idecreasing ion ithe iinterval i . D) The ifunction iis iincreasing ion ithe iinterval i iand idecreasing ion ithe iinterval i . E) The ifunction iis iincreasing ion ithe iinterval i iand idecreasing ion ithe iinterval i . Ans: E 12. Suppose ithe inumber iy iof imedical idegrees iconferred iin ithe iUnited iStates ican ibe imodeled iby i ifor i , iwhere it iis ithe itime iin iyears, iwith i icorresponding ito i1975. iUse ithe itest ifor iincreasing iand idecreasing ifunctions ito iestimate ithe iyears iduring iwhich ithe inumber iof imedical idegrees iis iincreasing iand ithe iyears iduring iwhich iit iis idecreasing. A) The inumber iof imedical idegrees iis iincreasing ifrom i1975 ito i1992 iand i2000 ito i2005, iand idecreasing iduring i1992 ito i2000. B) The inumber iof imedical idegrees iis iincreasing ifrom i1975 ito i1991 iand i1999 ito i2005, iand idecreasing iduring i1991 ito i1999. C) The inumber iof imedical idegrees iis iincreasing ifrom i1975 ito i1992 iand i1999 ito i2005, iand idecreasing iduring i1992 ito i1999. D) The inumber iof imedical idegrees iis iincreasing ifrom i1975 ito i1993 iand i1999 ito i2005, iand idecreasing iduring i1993 ito i1999. E) The inumber iof imedical idegrees iis iincreasing ifrom i1975 ito i1992 iand i1998 ito i2005, iand idecreasing iduring i1992 ito i1998. Ans: C 13. A ifast-food irestaurant idetermines ithe icost imodel, i iand irevenue imodel, i ifor i iwhere ix iis ithe inumber iof ihamburgers isold. iDetermine ithe iintervals ion iwhich ithe iprofit ifunction iis iincreasing iand ion iwhich iit iis idecreasing. A) The iprofit ifunction iis iincreasing ion ithe iinterval i iand idecreasing ion ithe iinterval i . B) The iprofit ifunction iis iincreasing ion ithe iinterval i iand idecreasing ion ithe iinterval i . C) The iprofit ifunction iis iincreasing ion ithe iinterval i iand idecreasing ion ithe iinterval i . D) The iprofit ifunction iis iincreasing ion ithe iinterval i iand idecreasing ion ithe iinterval i . E) The iprofit ifunction iis iincreasing ion ithe iinterval i iand idecreasing ion ithe iinterval i . Ans: C 14. For ithe igiven ifunction, ifind ithe irelative iminima. A) B) C) D) E) no irelative iminima Ans: B 15. Find ithe ix-values iof iall irelative imaxima iof ithe igiven ifunction. A) B) C) D) E) no irelative imaxima Ans: D 16. For ithe ifunction i : (a) i iFind ithe icritical inumbers iof if i(if iany); (b) i iFind ithe iopen iintervals iwhere ithe ifunction iis iincreasing ior idecreasing; iand (c) i iApply ithe iFirst iDerivative iTest ito iidentify iall irelative iextrema. Then iuse ia igraphing iutility ito iconfirm iyour iresults. A) (a) i ix i= i0 i, i6 (b) i iincreasing: i i; idecreasing: i (c) i irelative imax: i i; irelative imin: i B) (a) i ix i= i0 i, i6 (b) i idecreasing: i i; iincreasing: i (c) i irelative imin: i i; irelative imax: i C) (a) i ix i= i0 i, i2 (b) i iincreasing: i i; idecreasing: i (c) i irelative imax: i i; irelative imin: i D) (a) i ix i= i0 i, i2 (b) i idecreasing: i i; iincreasing: i (c) i irelative imin: i i; irelative imax: i E) (a) i ix i= i0 i, i2 (b) i iincreasing: i i; idecreasing: i (c) i irelative imax: i i; ino irelative imin. Ans: A 17. Find iall irelative imaxima iof ithe igiven ifunction. A) B) C) D) , E) no irelative imaxima Ans: B 18. Find iall irelative iminima iof ithe igiven ifunction. A) B) C) D) , E) no irelative iminima Ans: D 19. Locate ithe iabsolute iextrema iof ithe ifunction i ion ithe iclosed iinterval i . A) no iabsolute imax; iabsolute imin: if(–1) i= i5 B) absolute imax: if(2) i= i–22 i; iabsolute imin: if(–1) i= i5 C) absolute imax: if(–1) i= i5 i; ino iabsolute imin D) absolute imax: if(–1) i= i5 i; iabsolute imin: if(2) i= i–22 E) no iabsolute imax ior imin Ans: D 20. Locate ithe iabsolute iextrema iof ithe ifunction i ion ithe iclosed iinterval i[0,5]. A) absolute imax: if(5) i= i65 i; iabsolute imin: if(2) i= i–16 B) absolute imax: if(2) i= i–16 i; iabsolute imin: if(5) i= i65 C) absolute imax: if(5) i= i65 i; ino iabsolute imin D) no iabsolute imax; iabsolute imin: if(5) i= i65 E) no iabsolute imax ior imin Ans: A 21. Find ithe ix-value iat iwhich ithe iabsolute iminimum iof if i(x) ioccurs ion ithe iinterval i[a, ib]. A) B) C) D) E) Ans: A 22. Locate ithe iabsolute iextrema iof ithe igiven ifunction ion ithe iclosed iinterval i[–36,36]. A) absolute imax: if(6) i= i3 B) absolute imin: if(-6) i= i–3 C) no iabsolute imax D) no iabsolute imin E) both iA iand iD F) both iA iand iB Ans: F 23. Find ithe iabsolute iextrema iof ithe ifunction i ion ithe iclosed iinterval i . iRound iyour ianswer ito itwo idecimal iplaces. A) The imaximum iof ithe ifunction iis i1 iand ithe iminimum iof ithe ifunction iis i0. B) The imaximum iof ithe ifunction iis i2.92 iand ithe iminimum iof ithe ifunction iis i1. C) The imaximum iof ithe ifunction iis i2.92 iand ithe iminimum iof ithe ifunction iis i0. D) The imaximum iof ithe ifunction iis1 iand ithe iminimum iof ithe ifunction iis i2.08. E) The imaximum iof ithe ifunction iis i0 iand ithe iminimum iof ithe ifunction iis i2.08. Ans: C 24. Approximate ithe icritical inumbers iof ithe ifunction ishown iin ithe igraph iand idetermine iwhether ithe ifunction ihas ia irelative imaximum, ia irelative iminimum, ian iabsolute imaximum, ian iabsolute iminimum, ior inone iof ithese iat ieach icritical inumber ion ithe iinterval ishown. A) The icritical inumber i iyields ian iabsolute imaximum iand ithe icritical inumber i iyields ian iabsolute iminimum.. B) Both ithe icritical inumbers i i& i iyield ian iabsolute imaximum. C) The icritical inumber i iyields ian iabsolute iminimum iand ithe icritical inumber i iyields ian iabsolute imaximum. D) Both ithe icritical inumbers i iand i iyield ian iabsolute iminimum. E) The icritical inumber i iyields ia irelative iminimum iand ithe icritical inumber i iyields ia irelative imaximum. Ans: C 25. Find ithe iabsolute iextrema iof ithe ifunction i ion ithe iinterval i . A) The imaximum iof ithe ifunction iis i1 iand ithe iminimum iof ithe ifunction iis i0. B) The imaximum iof ithe ifunction iis i0 iand ithe iminimum iof ithe ifunction iis i–10. C) The imaximum iof ithe ifunction iis i–10 iand ithe iminimum iof ithe ifunction iis i0. D) The imaximum iof ithe ifunction iis i10 iand ithe iminimum iof ithe ifunction iis i0. E) The imaximum iof ithe ifunction iis i0 iand ithe iminimum iof ithe ifunction iis i10. Ans: D 26. Graph ia ifunction ion ithe iinterval i ihaving ithe ifollowing icharacteristics. Absolute imaximum iat i Absolute iminimum iat i Relative imaximum iat i Relative iminimum iat i A) B) C) D) E) Ans: A 27. Medication. iThe inumber iof imilligrams ix iof ia imedication iin ithe ibloodstream it ihours iafter ia idose iis itaken ican ibe imodeled iby i i i i . iFind ithe it-value iat iwhich ix iis imaximum. iRound iyour ianswer ito itwo idecimal iplaces. A) 0 ihours B) 2.24 ihours C) 894.43 ihours D) 4.24 ihours E) 5.46 ihours Ans: B 28. Medication. iThe inumber iof imilligrams ix iof ia imedication iin ithe ibloodstream it ihours iafter ia idose iis itaken ican ibe imodeled iby i i i i . iFind ithe imaximum ivalue iof ix. iRound iyour ianswer ito itwo idecimal iplaces. A) 2.65 img B) 755.93 img C) 1663.04 img D) 8.20 img E) 1500.40 img Ans: B 29. Suppose ithe iresident ipopulation iP(in imillions) iof ithe iUnited iStates ican ibe imodeled iby i , iwhere i icorresponds ito i1800. iAnalytically ifind ithe iminimum iand imaximum ipopulations iin ithe iU.S. ifor i . A) The ipopulation iis iminimum iat i iand imaximum iat i . B) The ipopulation iis iminimum iat i iand imaximum iat i . C) The ipopulation iis iminimum iat i iand imaximum iat i . D) The ipopulation iis iminimum iat i iand imaximum iat i . E) The ipopulation iis iminimum iat i iand imaximum iat i . Ans: D 30. Determine ithe iopen iintervals ion iwhich ithe igraph iof i iis iconcave idownward ior iconcave iupward. A) concave iupward ion i ; iconcave idownward ion i B) concave idownward ion i C) concave iupward ion i D) concave idownward ion i ; iconcave iupward ion i E) concave iupward ion i ; iconcave idownward ion i Ans: C 31. Determine ithe iopen iintervals ion iwhich ithe igraph iof i iis iconcave idownward ior iconcave iupward. A) concave idownward ion i B) concave idownward ion i i; iconcave iupward ion i C) concave iupward ion i i; iconcave idownward ion i D) concave idownward ion i i; iconcave iupward ion i E) concave iupward ion i i; iconcave idownward ion i Ans: E 32. Find iall irelative iextrema iof ithe ifunction i . iUse ithe iSecond iDerivative iTest iwhere iapplicable. A) relative imin: i B) relative imax: i C) no irelative imax D) no irelative imin E) both iA iand iC F) both iB iand iD Ans: E 33. Find iall irelative iextrema iof ithe ifunction i iUse ithe iSecond iDerivative iTest iwhere iapplicable. A) relative imax: i ; ino irelative imin B) relative imax: i ; ino irelative imin C) no irelative imax ior imin D) relative imin: i ; ino irelative imax E) relative imin: i ; ino irelative imax Ans: E 34. Find iall irelative iextrema iof ithe ifunction i . iUse ithe iSecond iDerivative iTest iwhere iapplicable. A) relative imax: if(1) i= i–6 B) relative imin: if(0) i= i–7 C) no irelative imax ior imin D) both iA iand iB E) none iof ithe iabove Ans: B 35. Find iall irelative iextrema iof ithe ifunction i . iUse ithe iSecond-Derivative iTest iwhen iapplicable. A) The irelative iminimum iis i iand ithe irelative imaximum iis i . B) The irelative imaximum iis i . C) The irelative iminimum iis i . D) The irelative imaximum iis i iand ithe irelative iminima iare i iand i . E) The irelative iminimum iis i iand ithe irelative imaximum iis i . Ans: B 36. Find iall irelative iextrema iof ithe ifunction i . iUse ithe iSecond-Derivative iTest iwhen iapplicable. A) The irelative imaximum iis i . B) The irelative iminimum iis i . C) The irelative imaximum iis i . D) The irelative iminimum iis i . E) The irelative imaximum iis i . Ans: C 37. State ithe isigns iof i iand i ion ithe iinterval i(0, i2). A) = i0 > i0 B) < i0 < i0 C) > i0 > i0 D) < i0 > i0 E) > i0 < i0 Ans: B 38. Find ithe ix-value iat iwhich ithe igiven ifunction ihas ia ipoint iof iinflection. A) B) C) D) E) no ipoint iof iinflection Ans: C 39. Find ithe ipoints iof iinflection iand idiscuss ithe iconcavity iof ithe ifunction. A) inflection ipoint iat i ; iconcave iupward ion i i; iconcave idownward ion i B) inflection ipoint iat i ; iconcave idownward ion i i; iconcave iupward ion i C) inflection ipoint iat i ; iconcave iupward ion i i; iconcave idownward ion i D) inflection ipoint iat i ; iconcave idownward ion i i; iconcave iupward ion i E) none iof ithe iabove Ans: A 40. A ifunction iand iits igraph iare igiven. iUse ithe isecond iderivative ito ilocate iall ix-values iof ipoints iof iinflection ion ithe igraph iof i . iCheck ithese iresults iagainst ithe igraph ishown. A) B) C) D) , i E) , i , i Ans: D 41. Sketch ia igraph iof ia ifunction if ihaving ithe ifollowing icharacteristics. A) B) C) D) E) Ans: C 42. The igraph iof if iis ishown iin ithe ifigure. iSketch ia igraph iof ithe iderivative iof if. A) B) C) D) E) Ans: E 43. The igraph iof if iis ishown iin ithe ifigure. iSketch ia igraph iof ithe iderivative iof if. A) B) C) D) E) Ans: C 44. The igraph iof if iis ishown iin ithe ifigure. iSketch ia igraph iof ithe iderivative iof if. A) B) The iderivative iof if idoes inot iexist. C) D) E) Ans: D 45. The igraph iof if iis ishown. iGraph if, if' iand if'' ion ithe isame iset iof icoordinate iaxes.