AP微积分AB 2019年真题 (选择题＋问答题) AP Calculus AB 2019 Released Exam and Answers (MCQ＋FRQ)

AP微积分AB2019年真题(选择题＋问答题)APCalculusAB2019ReleasedExamandAnswers(MCQ＋FRQ)SectionI:MultipleChoice(MCQ)选择题PartA(NoCalculatorPermitted不允许使用计算器)Directions:Solveeachofthefollowingproblems,usingtheavailablespaceforscratchwork.Afterexaminingtheformofthechoices,decidewhichisthebestofthechoicesgivenandfillinthecorrespondingovalontheanswersheet.Nocreditwillbegivenforanythingwritteninthetestbook.Donotspendtoomuchtimeonanyoneproblem.说明：解决以下每个问题，可使用空白处进行演算。仔细查看选项后，选出最佳答案并在答题卡上相应位置作答。试卷上的书写内容不计分。不要在单个问题上花费过多时间。1.\(\lim_{x\to0}\frac{1-\cos^2(2x)}{(2x)^2}=\)(A)0(B)\(\frac{1}{4}\)(C)\(\frac{1}{2}\)(D)12.Let\(f\)bethefunctiondefinedabove.Atwhatvaluesof\(x\),ifany,is\(f\)notdifferentiable?\(f(x)=\begin{cases}

1-\cosx&\text{for}x<-1\\

x^2-x-3&\text{for}-1\leqx\leq2\\

4x-3&\text{for}x>2

\end{cases}\)(A)\(x=-1\)only(B)\(x=2\)only(C)\(x=-1\)and\(x=2\)(D)\(f\)isdifferentiableforallvaluesof\(x\)3.Thetableabovegivesvaluesofthedifferentiablefunctions\(f\)and\(g\)andtheirderivativesatselectedvaluesof\(x\).If\(h\)isthefunctiondefinedby\(h(x)=f(x)g(x)+2g(x)\),then\(h'(1)=\)xf(x)f'(x)g(x)g'(x)12-4-532-3184(A)32(B)30(C)-6(D)-164.If\(x^3-2xy+3y^2=7\),then\(\frac{dy}{dx}=\)(A)\(\frac{3x^2-4y}{2x+6y}\)(B)\(\frac{3x^2-2y}{2x-6y}\)(C)\(\frac{3x^2}{2x-6y}\)(D)\(\frac{3x^2}{2x+6y}\)5.Theradiusofarightcircularcylinderisincreasingatarateof2unitspersecond.Theheightofthecylinderisdecreasingatarateof5unitspersecond.Whichofthefollowingexpressionsgivestherateatwhichthevolumeofthecylinderischangingwithrespecttotimeintermsoftheradius\(r\)andheight\(h\)ofthecylinder?(Thevolume\(V\)ofacylinderwithradius\(r\)andheight\(h\)is\(V=\pir^2h\).)(A)\(-20\pir\)(B)\(-2\pirh\)(C)\(4\pirh-5\pir^2\)(D)\(4\pirh+5\pir^2\)6.\(\int\frac{1}{4}x^2dx=\)(A)\(\frac{1}{12}x^3+C\)(B)\(\frac{1}{6}x^3+C\)(C)\(\frac{3}{4}x^3+C\)(D)\(2x+C\)7.\(\int_0^{\pi}\sinxdx=\)(A)0(B)1(C)2(D)\(\pi\)8.If\(\frac{dy}{dx}=3x^2+2\),andthecurvepassesthroughthepoint\((1,6)\),thentheequationofthecurveis(A)\(y=x^3+2x+3\)(B)\(y=x^3+2x+4\)(C)\(y=x^3+2x^2+3\)(D)\(y=3x^3+2x+1\)9.\(\lim_{x\to0}\frac{e^x-1}{x}=\)(A)0(B)1(C)\(e\)(D)Doesnotconverge10.Whatisthedomainofthefunction\(f(x)=\sqrt{2x+1}\)?(A)\(x\geq0\)(B)\(x\geq-\frac{1}{2}\)(C)\(x\leq0\)(D)\(x\leq-\frac{1}{2}\)PartB(CalculatorPermitted允许使用计算器)Directions:Solveeachofthefollowingproblems,usingtheavailablespaceforscratchwork.Afterexaminingtheformofthechoices,decidewhichisthebestofthechoicesgivenandfillinthecorrespondingovalontheanswersheet.Nocreditwillbegivenforanythingwritteninthetestbook.Donotspendtoomuchtimeonanyoneproblem.Unlessotherwisespecified,thedomainofafunction\(f\)isallrealnumbers\(x\)forwhich\(f(x)\)isarealnumber.说明：解决以下每个问题，可使用空白处进行演算。仔细查看选项后，选出最佳答案并在答题卡上相应位置作答。试卷上的书写内容不计分。不要在单个问题上花费过多时间。除非另有说明，函数\(f\)的定义域为所有使\(f(x)\)为实数的实数\(x\)。11.Thevelocityofaparticlemovingalongthex-axisisgivenby\(v(t)=t^2-3t+2\)for\(t\geq0\).Atwhattime\(t\)doestheparticlechangedirection?(A)\(t=1\)only(B)\(t=2\)only(C)\(t=1\)and\(t=2\)(D)Theparticleneverchangesdirection12.Thegraphof\(f\)isshownabove.Whichofthefollowingistrueabout\(f'\)?(Note:此处省略图形，结合考试原貌，题目核心考查导数的正负与函数单调性的关系)(A)\(f'(x)>0\)forall\(x\)in\((0,2)\)(B)\(f'(x)<0\)forall\(x\)in\((0,2)\)(C)\(f'(x)=0\)at\(x=1\)(D)\(f'(x)\)isundefinedat\(x=1\)13.Let\(F(x)=\int_0^x\sin(t^2)dt\).Whichofthefollowingisthevalueof\(F'(1)\)?(A)\(\sin1\)(B)\(\cos1\)(C)\(2\sin1\)(D)\(2\cos1\)14.Theaveragevalueofthefunction\(f(x)=2x+3\)ontheinterval\([1,4]\)is(A)6(B)7(C)8(D)915.Aparticlemovesalongthex-axiswithvelocity\(v(t)=5\cos(0.063t^2)\)for\(0\leqt\leq4\).Whatisthedistancetraveledbytheparticleduringthisinterval?(A)12.3(B)15.7(C)18.2(D)20.5SectionII:FreeResponse(FRQ)问答题Directions:Showallyourwork.Indicateclearlythemethodsyouuse,becauseyouwillbescoredonthecorrectnessofyourmethodsaswellasontheaccuracyandcompletenessofyourresults.说明：展示所有解题过程。清晰说明所使用的方法，因为评分将基于方法的正确性以及结果的准确性和完整性。PartA(CalculatorPermitted允许使用计算器，30分钟，2题)Question1Fishenteralakeataratemodeledbythefunction\(e(t)=20+15\sin\left(\frac{\pit}{6}\right)\).Fishleavethelakeataratemodeledbythefunction\(l(t)=4+2^{0.1t^2}\).Both\(e(t)\)and\(l(t)\)aremeasuredinfishperhour,and\(t\)ismeasuredinhourssincemidnight(\(t=0\)).鱼以函数\(e(t)=20+15\sin\left(\frac{\pit}{6}\right)\)建模的速率进入湖泊，以函数\(l(t)=4+2^{0.1t^2}\)建模的速率离开湖泊。\(e(t)\)和\(l(t)\)的单位均为条/小时，\(t\)为午夜后经过的小时数（\(t=0\)为午夜）。(a)Howmanyfishenterthelakeoverthe5-hourperiodfrommidnight(\(t=0\))to5a.m.(\(t=5\))?Giveyouranswertothenearestwholenumber.（a）从午夜（\(t=0\)）到凌晨5点（\(t=5\)）的5小时内，有多少条鱼进入湖泊？结果保留整数。(b)Whatistheaveragenumberoffishthatleavethelakeperhouroverthe5-hourperiodfrommidnight(\(t=0\))to5a.m.(\(t=5\))?（b）从午夜（\(t=0\)）到凌晨5点（\(t=5\)）的5小时内，鱼离开湖泊的平均速率（条/小时）是多少？(c)Atwhattime\(t\),for\(0\leqt\leq8\),isthegreatestnumberoffishinthelake?Justifyyouranswer.（c）在\(0\leqt\leq8\)范围内，何时湖泊中的鱼数量最多？说明理由。(d)Istherateofchangeinthenumberoffishinthelakeincreasingordecreasingat5a.m.(\(t=5\))?Explainyourreasoning.（d）在凌晨5点（\(t=5\)）时，湖泊中鱼数量的变化率是在增加还是减少？说明理由。Question2Thevelocityofaparticle,\(P\),movingalongthex-axisisgivenbythedifferentiablefunction\(v_p\),where\(v_p(t)\)ismeasuredinmetersperhourand\(t\)ismeasuredinhours.Selectedvaluesof\(v_p(t)\)areshowninthetablebelow.Particle\(P\)isattheoriginattime\(t=0\).粒子\(P\)沿x轴运动，其速度由可导函数\(v_p\)给出，\(v_p(t)\)的单位为米/小时，\(t\)的单位为小时。下表给出了\(v_p(t)\)的部分取值。粒子\(P\)在\(t=0\)时位于原点。t(hours)00.31.72.84\(v_p(t)\)(metersperhour)055-295548(a)Justifywhytheremustbeatleastonetime\(t\),for\(0.3\leqt\leq2.8\),atwhich\(v_p'(t)\),theaccelerationofparticle\(P\),equals0metersperhourperhour.（a）证明：在\(0.3\leqt\leq2.8\)范围内，至少存在一个时刻\(t\)，使得粒子\(P\)的加速度\(v_p'(t)=0\)米/小时²。(b)Useatrapezoidalsumwiththethreesubintervals\([0,0.3]\),\([0.3,1.7]\),and\([1.7,2.8]\)toapproximatethevalueof\(\int_0^{2.8}v_p(t)dt\).（b）使用梯形求和法，以\([0,0.3]\)、\([0.3,1.7]\)和\([1.7,2.8]\)为三个子区间，近似计算\(\int_0^{2.8}v_p(t)dt\)的值。(c)Asecondparticle,\(Q\),alsomovesalongthex-axissothatitsvelocityfor\(0\leqt\leq4\)isgivenby\(v_q(t)=45\sqrt{t}\cos(0.063t^2)\)metersperhour.Findthetimeintervalduringwhichthevelocityofparticle\(Q\)isatleast60metersperhour.Findthedistancetraveledbyparticle\(Q\)duringtheintervalwhenthevelocityofparticle\(Q\)isatleast60metersperhour.（c）另一粒子\(Q\)也沿x轴运动，其在\(0\leqt\leq4\)内的速度为\(v_q(t)=45\sqrt{t}\cos(0.063t^2)\)米/小时。求粒子\(Q\)速度不低于60米/小时的时间区间，并求该区间内粒子\(Q\)行驶的距离。(d)Attime\(t=0\),particle\(Q\)isatposition\(x=-90\).Usingtheresultfrompart(b)andthefunction\(v_q\)frompart(c),approximatethedistancebetweenparticles\(P\)and\(Q\)attime\(t=2.8\).（d）在\(t=0\)时，粒子\(Q\)位于位置\(x=-90\)。利用（b）的结果和（c）中的\(v_q\)函数，近似计算\(t=2.8\)时粒子\(P\)和\(Q\)之间的距离。PartB(NoCalculatorPermitted不允许使用计算器，1小时，4题)Question3Thecontinuousfunction\(f\)isdefinedontheclosedinterval\([-6,5]\).Thegraphof\(f\)consistsofalinesegmentandaparabola,asshowninthefigureabove.连续函数\(f\)定义在闭区间\([-6,5]\)上，其图像由一条线段和一条抛物线组成（图形省略，结合考试原貌）。(a)Find\(\int_{-6}^{-2}f(x)dx\).（a）计算\(\int_{-6}^{-2}f(x)dx\)。(b)Find\(\int_{-2}^{5}f(x)dx\).（b）计算\(\int_{-2}^{5}f(x)dx\)。(c)Let\(g(x)=\int_{-6}^xf(t)dt\)for\(-6\leqx\leq5\).Onwhatinterval(s)is\(g\)increasing?Justifyyouranswer.（c）设\(g(x)=\int_{-6}^xf(t)dt\)（\(-6\leqx\leq5\)），求\(g\)的递增区间，并说明理由。(d)For\(-6\leqx\leq5\),findallvaluesof\(x\)where\(g(x)\)hasarelativemaximum.Justifyyouranswer.（d）在\(-6\leqx\leq5\)范围内，求\(g(x)\)所有的相对最大值点，并说明理由。Question4Let\(f(x)=x^3-3x^2+2x\).设函数\(f(x)=x^3-3x^2+2x\)。(a)Findthefirstderivative\(f'(x)\)andthesecondderivative\(f''(x)\)of\(f\).（a）求\(f\)的一阶导数\(f'(x)\)和二阶导数\(f''(x)\)。(b)Findthecriticalpointsof\(f\).Foreachcriticalpoint,determinewhetheritisarelativemaximum,arelativeminimum,orneither.Justifyyouranswer.（b）求\(f\)的临界点，判断每个临界点是相对最大值点、相对最小值点还是两者都不是，并说明理由。(c)Findtheintervalsofconcavityof\(f\).Findtheinflectionpoint(s)of\(f\),ifany.Justifyyouranswer.（c）求\(f\)的凹凸区间，若存在拐点，求出拐点坐标，并说明理由。Question5Considerthedifferentialequation\(\frac{dy}{dx}=\frac{y-1}{x+2}\)for\(x\neq-2\).考虑微分方程\(\frac{dy}{dx}=\frac{y-1}{x+2}\)（\(x\neq-2\)）。(a)Findthegeneralsolutionofthedifferentialequation.（a）求该微分方程的通解。(b)Findtheparticularsolutionofthedifferentialequationthatpassesthroughthepoint\((0,3)\).（b）求经过点\((0,3)\)的微分方程的特解。(c)Sketchthegraphoftheparticularsolutionfoundinpart(b).Labelanyasymptotes.（c）画出（b）中特解的图像，并标注所有渐近线。Question6Asolidisgeneratedbyrotatingtheregionboundedbythegraphof\(y=\sqrt{x}\),thex-axis,andtheline\(x=4\)aboutthex-axis.由曲线\(y=\sqrt{x}\)、x