定积分与微积分基本定理-专题

定积分与微积分基本定理/r/n-专题/r/n【/r/n学习目标/r/n】/r/n1./r/n通过实例（如求曲边梯形的面积、变力做功等），从问题情境中了解定积分的实际背景；借助几何直观体会定积分的基本思想，初步了解定积分的概念/r/n并会解一些简单的积分问题/r/n。/r/n2./r/n了解微积分基本定理的含义/r/n与应用/r/n。/r/n【/r/n课堂讲解/r/n】/r/n1．（2021·江西·贵溪市实验中学高二阶段练习（理））给出以下命题：（1）/r/nx/r/n2/r/ne/r/nx/r/n′/r/n=2x/r/ne/r/nx/r/n；（2）/r/n0/r/n2π/r/ncos/r/nx/r/ndx=4/r/n；（3）/r/nf/r/nx/r/n的原函数为/r/nA．1/r/n /r/nB．2/r/n /r/nC．3/r/n /r/nD．4/r/n【分析】/r/n对于（1）：运用乘法的求导法则可判断；/r/n对于（2）：将原式变形为/r/n0/r/n2π/r/n对于（3）：根据积分的定义和周期函数的应用可得/r/n0/r/na/r/nf/r/nx/r/n对于（4）：利用在某一点的导函数的定义可判断./r/n【解】/r/n对于（1）：/r/nx/r/n2/r/n对于（2）：/r/n0/r/n=/r/n=/r/nsin/r/n对于（3）：因为/r/nf/r/nx/r/n的原函数为/r/nF/r/nx/r/n，且/r/n所以/r/n0/r/na/r/nf/r/nx/r/n所以/r/n0/r/na/r/n对于（4）：设函数/r/nf/r/nx/r/n可导，令/r/nt=/r/n1/r/nΔx/r/n所以.其中正确命题的个数为3，/r/n故选：C./r/n【/r/n考点分析/r/n】/r/n本题考查求导函数求积分的定义和运算法则./r/n2/r/n．（2020·全国·高三专题练习（理））二项式/r/nmx−1/r/n3/r/nm>0/r/n展开式的第二项的系数为-3，则/r/n−2/r/nm/r/nA．3/r/n /r/nB．/r/n7/r/n3/r/n /r/nC．/r/n8/r/n3/r/n【分析】/r/n二项式/r/nmx−1/r/n3/r/nm>0/r/n的展开式的通项公式得/r/nT/r/n2/r/n=/r/n∁/r/n3/r/n1/r/n(mx)/r/n2/r/n解：二项式/r/nmx−1/r/n3/r/nm>0/r/n的展开式的通项公式得/r/n∵/r/n第二项的系数为/r/n−3/r/n，/r/n∴/r/n/r/n−3/r/nm/r/n∴/r/nm/r/n2/r/n=1/r/n，/r/nm>0/r/n，解得/r/n当/r/nm=1/r/n时，则/r/n−2/r/nm/r/n故选：/r/nA/r/n．/r/n【/r/n考点分析/r/n】/r/n本题考查了二项式定理与微积分基本定理的应用，考查了推理能力与计算能力．/r/n3/r/n计算/r/n1/r/ne/r/n1/r/nx/r/nA．0/r/n /r/nB．/r/n1/r/n /r/nC．2/r/n /r/nD．/r/n−/r/n1/r/n【分析】/r/n找到/r/ny=/r/n1/r/nx/r/n的原函数/r/n由题意，/r/n1/r/n故选：B/r/n4./r/n计算:/r/n(1)/r/n1/r/n2/r/n(2)/r/n0/r/n2/r/n分析：（1）将/r/ny=/r/n3−2x/r/n,x∈/r/n1,2/r/n试题解析：(1)/r/n1/r/n2/r/n(2)∵(/r/n-/r/ncos/r/nx/r/n)/r/n'/r/n=sin/r/nx/r/n,∴/r/n0/r/n2/r/nπ/r/n【/r/n自主思考/r/n】/r/n山东省荣成市第六中学阶段练习/r/n在/r/n3/r/nx/r/n−2/r/n3/r/nx/r/n11/r/n答案：/r/n6/r/n【/r/n同步练习/r/n】/r/n一、选择题/r/n1．（2021·安徽省宣城市）/r/n1/r/n2/r/n2−x+/r/nx/r/nA．/r/n2/r/nln/r/n2+/r/n1/r/n2/r/n /r/nB．/r/n2/r/nln/r/n2−/r/n2．（2021·山西阳泉（理））若/r/na>0,/r/n /r/nb>0/r/n，二项式/r/n(ax+b)/r/n6/r/n的展开式中/r/nx/r/nA．0/r/n /r/nB．1/r/n /r/nC．2/r/n /r/nD．3/r/n3．曲线/r/ny=/r/nsin/r/nx/r/n，/r/nx∈[0,2π]/r/n与/r/nx/r/n轴所围成的面积是（/r/nA．0/r/n /r/nB．2/r/n /r/nC．4/r/n /r/nD．/r/nπ/r/n4．/r/n0/r/n1/r/n1−/r/nx/r/nA．/r/nπ+1/r/n4/r/n /r/nB．/r/nπ+1/r/n2/r/n /r/nC．/r/nπ/r/n2/r/n5．（2020·安徽·高三阶段练习（理））定积分/r/n−1/r/n1/r/n3/r/nx/r/nA．/r/n1+/r/nπ/r/n2/r/n /r/nB．/r/n2+/r/nπ/r/n2/r/n /r/nC．/r/n6．（2021·江西·（理））给出以下命题：（1）/r/nx/r/n2/r/ne/r/nx/r/n′/r/n=2x/r/ne/r/nx/r/n；（2）/r/n0/r/n2π/r/ncos/r/nx/r/ndx=4/r/n；（3）/r/nf/r/nx/r/n的原函数为/r/nA．1/r/n /r/nB．2/r/n /r/nC．3/r/n /r/nD．4/r/n7．若/r/nS/r/n1/r/n=/r/n0/r/n1/r/nx/r/nA．/r/nS/r/n1/r/n</r/nS/r/nC．/r/nS/r/n2/r/n</r/nS/r/n8．下列各式错误的是（/r/n

/r/n）/r/nA．/r/n0/r/nπ/r/n2/r/nsin/r/nφdφ/r/n＝1/r/n /r/nB．/r/n0/r/nπ/r/nC．/r/n1/r/ne/r/ne/r/nx/r/ndx/r/n＝－1/r/n9．（2021·江西赣州·高三期中（理））/r/n0/r/nπ/r/n2/r/nx+/r/nA．/r/n1−/r/nπ/r/n2/r/n8/r/n /r/nB．/r/nπ/r/n2/r/n8/r/n−1/r/n10．/r/n0/r/n1/r/n1−/r/nx/r/nA．/r/nπ+1/r/n4/r/n /r/nB．/r/nπ+1/r/n2/r/n /r/nC．/r/nπ/r/n2/r/n二、填空题/r/n11/r/n．/r/n1/r/n2/r/n12/r/n．/r/n−4/r/n4/r/n13/r/n．（2020·海南华侨中学高三阶段练习）设函数/r/nf/r/nx/r/n=a/r/nx/r/n2/r/n+b/r/n14/r/n．（2021·安徽·安庆市白泽湖中学高二期中（理））/r/n1/r/n2/r/n三、解答题/r/n1/r/n5/r/n．(1)已知/r/n，求f(a)的最大值./r/n(2)已知f(x)=ax/r/n2/r/n+bx+c(a≠0)，且/r/n=2，f′(0)=0，/r/n，求a，b，c的值./r/n16/r/n．（20/r/n22/r/n·北京朝阳·高三期中（文））已知函数/r/nf(x)=x−/r/nsin/r/n（I）求证：当/r/nx∈[0,/r/nπ/r/n2/r/n]/r/n（II）设/r/ng(x)=/r/nx/r/ntan/r/nx/r/n（i）试判断函数/r/ng(x)/r/n的单调性并证明；/r/n（ii）若/r/ng(x)<a/r/n恒成立，求实数/r/na/r/n的最小值./r/n17/r/n（全国·高三专题练习）已知/r/nf(x)=/r/n3/r/n（Ⅰ）写出/r/nf(x)/r/n的最小正周期/r/nT/r/n；/r/n（Ⅱ）求由/r/ny=f(x)(0≤x≤/r/n5π/r/n6/r/n),y=0(0≤x≤/r/n18/r/n．（河北廊坊·高三阶段练习（文））已知函数/r/nf(x)=/r/nx/r/n3/r/n−(a+2)/r/n（1）曲线/r/ny=f(x)/r/n在点/r/n(1,f(1))/r/n处的切线斜率是否为定值？/r/n（2）若/r/nf(x)>0/r/n，证明：/r/nln/r/n(a+3)</r/n19/r/n．已知函数/r/nf/r/nx/r/n=/r/nln/r/n（1）若函数/r/nf/r/nx/r/n的图象与直线/r/nx+2y−4=0/r/n相切，求/r/nm/r/n（2）求/r/nf/r/nx/r/n在区间/r/n1,2/r/n（3）若函数/r/nf/r/nx/r/n有两个不同的零点/r/nx/r/n1/r/n，/r/n/r/nx/r/n2/r/n20/r/n．已知/r/n−1/r/n1/r/n(/r/nx/r/n求a，b．/r/n答案/r/n1/r/n2/r/n3/r/n4/r/n5/r/n6/r/n7/r/n8/r/n9/r/n10/r/nA/r/nC/r/nC/r/nA/r/nB/r/nC/r/nB/r/nC/r/nD/r/nA/r/n11/r/n．/r/ne/r/n12/r/n．/r/n8π+ln2−/r/n13/r/n．/r/n±/r/n14/r/n．/r/nπ−2/r/n15/r/n．(1)/r/n(2)a=6，b=0，/r/n16/r/n．（2）（i）/r/ng(x)/r/n在/r/n(0,/r/nπ/r/n2/r/n)/r/n17/r/n．(1)/r/nπ/r/n

/r/n(2)/r/n2−/r/n3/r/n18/r/n（1）∵/r/nf'(x)=3/r/nx/r/n∴/r/nf'(1)=3−(a+2)+a+3=4/r/n，/r/n故曲线/r/ny=f(x)/r/n在点/r/n(1,f(1))/r/n处的切线斜率/r/nk=4/r/n为定值./r/n（2）证明：∵/r/nf(x)>0/r/n，/r/nx∈(0,+∞)/r/n，∴/r/nx−(a+2)/r/nln/r/n设/r/nℎ(x)=x−(a+2)/r/nln/r/nx+/r/n当/r/n0<x<a+3/r/n时，/r/nℎ'(x)<0/r/n；当/r/nx>a+3/r/n时，/r/nℎ'(x)>0/r/n从而/r/nℎ/r/n(x)/r/n即/r/nln/r/n(a+3)</r/n19/r/n．(1)/r/nm=/r/n3/r/n2/r/n（2）/r/nf/r/n（1）设切点/r/nP/r/nx/r/n0/r/n,/r/n所以/r/nk=−/r/n1/r/n2/r/n/r/n又/r/nln/r/nx/r/n由①得/r/nm/r/nx/r/n0/r/n=1+/r/n所以/r/nx/r/n0/r/n=1/r/n，因为/r/ng/r/nx/r/n0/r/n=/r/n所以切点/r/nP/r/n1,m/r/n，代入切线方程得/r/nm=/r/n（2）因为/r/nf/r/nx/r/n所以/r/nf'/r/nx/r/n=/r/n1/r/nx/r/n−/r/n当/r/nm≤0/r/n时，/r/n/r/nf'/r/nx/r/n>0/r/n，则/r/nf/r/nx/r/n所以/r/nf/r/nx/r/n在/r/n1,2/r/n递增，则/r/nf/r/n当/r/nm>0/r/n时，/r/n/r/nx∈/r/n0,m/r/n有/r/nf'/r/nx/r/n<0/r/n，/r/n/r/nx∈/r/n所以/r/nf/r/nx/r/n在/r/n0,m/r/n上单调递减，在/r/nm,+∞/r/n则当/r/nm≥2/r/n时，/r/n/r/nf/r/nx/r/n在/r/n1,2/r/n递减，则/r/nf/r/n当/r/n0<m≤1/r/n时，/r/n/r/nf/r/nx/r/n在/r/n1,2/r/n递增，则/r/nf/r/n当/r/n1<m<2/r/n时，/r/n/r/nf/r/nx/r/n在/r/n1,m/