AP微积分2010真题

AP® Calculus BC 2010 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the College Board is composed of more than 5,700 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,800 colleges through major programs and services in college readiness, college admission, guidance, assessment, financial aid and enrollment. Among its widely recognized programs are the SAT®, the PSAT/NMSQT®, the Advanced Placement Program® (AP®), SpringBoard® and ACCUPLACER®. The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities and concerns. © 2010 The College Board. College Board, ACCUPLACER, Advanced Placement Program, AP, AP Central, SAT, SpringBoard and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service is a trademark owned by the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Permission to use copyrighted College Board materials may be requested online at: www.collegeboard.com/inquiry/cbpermit.html. Visit the College Board on the Web: www.collegeboard.com. AP Central is the official online home for the AP Program: apcentral.collegeboard.com. anny 文本框 满分网——AP真题 http://www.manfen.net 2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS © 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -2- CALCULUS BC SECTION II, Part A Time—45 minutes Number of problems—3 A graphing calculator is required for some problems or parts of problems. 1. There is no snow on Janet’s driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow accumulates on the driveway at a rate modeled by ( ) cos7 tf t te= cubic feet per hour, where t is measured in hours since midnight. Janet starts removing snow at 6 A.M. ( )6 .t = The rate ( ),g t in cubic feet per hour, at which Janet removes snow from the driveway at time t hours after midnight is modeled by ( ) 0 for 0 6 125 for 6 7 108 for 7 9 . t g t t t £ <ÏÔ= £ <ÌÔ £ £Ó (a) How many cubic feet of snow have accumulated on the driveway by 6 A.M.? (b) Find the rate of change of the volume of snow on the driveway at 8 A.M. (c) Let ( )h t represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time t hours after midnight. Express h as a piecewise-defined function with domain 0 9.t£ £ (d) How many cubic feet of snow are on the driveway at 9 A.M.? WRITE ALL WORK IN THE PINK EXAM BOOKLET. 2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS © 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -3- t (hours) 0 2 5 7 8 ( )E t (hundreds of entries) 0 4 13 21 23 2. A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon ( )0t = and 8 P.M. ( )8 .t = The number of entries in the box t hours after noon is modeled by a differentiable function E for 0 8.t£ £ Values of ( ),E t in hundreds of entries, at various times t are shown in the table above. (a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time 6.t = Show the computations that lead to your answer. (b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of ( )8 0 1 .8 E t dtÚ Using correct units, explain the meaning of ( )8 0 1 8 E t dtÚ in terms of the number of entries. (c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function P, where ( ) 3 230 298 976P t t t t= - + - hundreds of entries per hour for 8 12.t£ £ According to the model, how many entries had not yet been processed by midnight ( )12 ?t = (d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer. WRITE ALL WORK IN THE PINK EXAM BOOKLET. 2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS © 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. -4- 3. A particle is moving along a curve so that its position at time t is ( ) ( )( ), ,x t y t where ( ) 2 4 8x t t t= - + and ( )y t is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known that 3 1.tdy tedt -= - (a) Find the speed of the particle at time 3t = seconds. (b) Find the total distance traveled by the particle for 0 4t£ £ seconds. (c) Find the time t, 0 4,t£ £ when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time? Give a reason for your answer. (d) There is a point with x-coordinate 5 through which the particle passes twice. Find each of the following. (i) The two values of t when that occurs (ii) The slopes of the lines tangent to the particle’s path at that point (iii) The y-coordinate of that point, given ( ) 12 3y e = + WRITE ALL WORK IN THE PINK EXAM BOOKLET. END OF PART A OF SECTION II 2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS © 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -5- CALCULUS BC SECTION II, Part B Time—45 minutes Number of problems—3 No calculator is allowed for these problems. 4. Let R be the region in the first quadrant bounded by the graph of 2 ,y x= the horizontal line 6,y = and the y-axis, as shown in the figure above. (a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line 7.y = (c) Region R is the base of a solid. For each y, where 0 6,y£ £ the cross section of the solid taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid. WRITE ALL WORK IN THE PINK EXAM BOOKLET. 2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS © 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. -6- 5. Consider the differential equation 1 .dy ydx = - Let ( )y f x= be the particular solution to this differential equation with the initial condition ( )1 0.f = For this particular solution, ( ) 1f x < for all values of x. (a) Use Euler’s method, starting at 1x = with two steps of equal size, to approximate ( )0 .f Show the work that leads to your answer. (b) Find ( )31lim .1x f x xÆ - Show the work that leads to your answer. (c) Find the particular solution ( )y f x= to the differential equation 1dy ydx = - with the initial condition ( )1 0.f = ( ) 2 cos 1 for 0 1 for 02 x x xf x x -Ï πÔ= ÌÔ- =Ó 6. The function f, defined above, has derivatives of all orders. Let g be the function defined by ( ) ( ) 0 1 . x g x f t dt= + Ú (a) Write the first three nonzero terms and the general term of the Taylor series for cos x about 0.x = Use this series to write the first three nonzero terms and the general term of the Taylor series for f about 0.x = (b) Use the Taylor series for f about 0x = found in part (a) to determine whether f has a relative maximum, relative minimum, or neither at 0.x = Give a reason for your answer. (c) Write the fifth-degree Taylor polynomial for g about 0.x = (d) The Taylor series for g about 0,x = evaluated at 1,x = is an alternating series with individual terms that decrease in absolute value to 0. Use the third-degree Taylor polynomial for g about 0x = to estimate the value of ( )1 .g Explain why this estimate differs from the actual value of ( )1g by less than 1 .6! WRITE ALL WORK IN THE PINK EXAM BOOKLET. END OF EXAM anny 文本框 更多AP资料下载：http://www.manfen.net/thread-9638-1-1.html << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Gray Gamma 2.2) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJDFFile false /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends false /DetectCurves 0.0000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize false /OPM 1 /ParseDSCComments false /ParseDSCCommentsForDocInfo true /PreserveCopyPage false /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo true /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts false /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile (Color Management Off) /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /Description << /CHS /CHT /DAN /DEU /ESP /FRA /ITA /JPN /KOR /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. 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