定積分與微積分基本定理-專題

定積分與微積分基本定理/r/n-專題/r/n【/r/n學(xué)習(xí)目標(biāo)/r/n】/r/n1./r/n通過實例（如求曲邊梯形的面積、變力做功等），從問題情境中了解定積分的實際背景；借助幾何直觀體會定積分的基本思想，初步了解定積分的概念/r/n并會解一些簡單的積分問題/r/n。/r/n2./r/n了解微積分基本定理的含義/r/n與應(yīng)用/r/n。/r/n【/r/n課堂講解/r/n】/r/n1．（2021·江西·貴溪市實驗中學(xué)高二階段練習(xí)（理））給出以下命題：（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的原函數(shù)為/r/nA．1/r/n /r/nB．2/r/n /r/nC．3/r/n /r/nD．4/r/n【分析】/r/n對于（1）：運用乘法的求導(dǎo)法則可判斷；/r/n對于（2）：將原式變形為/r/n0/r/n2π/r/n對于（3）：根據(jù)積分的定義和周期函數(shù)的應(yīng)用可得/r/n0/r/na/r/nf/r/nx/r/n對于（4）：利用在某一點的導(dǎo)函數(shù)的定義可判斷./r/n【解】/r/n對于（1）：/r/nx/r/n2/r/n對于（2）：/r/n0/r/n=/r/n=/r/nsin/r/n對于（3）：因為/r/nf/r/nx/r/n的原函數(shù)為/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對于（4）：設(shè)函數(shù)/r/nf/r/nx/r/n可導(dǎo)，令/r/nt=/r/n1/r/nΔx/r/n所以.其中正確命題的個數(shù)為3，/r/n故選：C./r/n【/r/n考點分析/r/n】/r/n本題考查求導(dǎo)函數(shù)求積分的定義和運算法則./r/n2/r/n．（2020·全國·高三專題練習(xí)（理））二項式/r/nmx?1/r/n3/r/nm>0/r/n展開式的第二項的系數(shù)為-3，則/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二項式/r/nmx?1/r/n3/r/nm>0/r/n的展開式的通項公式得/r/nT/r/n2/r/n=/r/n?/r/n3/r/n1/r/n(mx)/r/n2/r/n解：二項式/r/nmx?1/r/n3/r/nm>0/r/n的展開式的通項公式得/r/n∵/r/n第二項的系數(shù)為/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當(dāng)/r/nm=1/r/n時，則/r/n?2/r/nm/r/n故選：/r/nA/r/n．/r/n【/r/n考點分析/r/n】/r/n本題考查了二項式定理與微積分基本定理的應(yīng)用，考查了推理能力與計算能力．/r/n3/r/n計算/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的原函數(shù)/r/n由題意，/r/n1/r/n故選：B/r/n4./r/n計算:/r/n(1)/r/n1/r/n2/r/n(2)/r/n0/r/n2/r/n分析：（1）將/r/ny=/r/n3?2x/r/n,x∈/r/n1,2/r/n試題解析：(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山東省榮成市第六中學(xué)階段練習(xí)/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同步練習(xí)/r/n】/r/n一、選擇題/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·山西陽泉（理））若/r/na>0,/r/n?/r/nb>0/r/n，二項式/r/n(ax+b)/r/n6/r/n的展開式中/r/nx/r/nA．0/r/n /r/nB．1/r/n /r/nC．2/r/n /r/nD．3/r/n3．曲線/r/ny=/r/nsin/r/nx/r/n，/r/nx∈[0,2π]/r/n與/r/nx/r/n軸所圍成的面積是（/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·高三階段練習(xí)（理））定積分/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））給出以下命題：（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的原函數(shù)為/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．下列各式錯誤的是（/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·江西贛州·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、填空題/r/n11/r/n．/r/n1/r/n2/r/n12/r/n．/r/n?4/r/n4/r/n13/r/n．（2020·海南華僑中學(xué)高三階段練習(xí)）設(shè)函數(shù)/r/nf/r/nx/r/n=a/r/nx/r/n2/r/n+b/r/n14/r/n．（2021·安徽·安慶市白澤湖中學(xué)高二期中（理））/r/n1/r/n2/r/n三、解答題/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·北京朝陽·高三期中（文））已知函數(shù)/r/nf(x)=x?/r/nsin/r/n（I）求證：當(dāng)/r/nx∈[0,/r/nπ/r/n2/r/n]/r/n（II）設(shè)/r/ng(x)=/r/nx/r/ntan/r/nx/r/n（i）試判斷函數(shù)/r/ng(x)/r/n的單調(diào)性并證明；/r/n（ii）若/r/ng(x)<a/r/n恒成立，求實數(shù)/r/na/r/n的最小值./r/n17/r/n（全國·高三專題練習(xí)）已知/r/nf(x)=/r/n3/r/n（Ⅰ）寫出/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·高三階段練習(xí)（文））已知函數(shù)/r/nf(x)=/r/nx/r/n3/r/n?(a+2)/r/n（1）曲線/r/ny=f(x)/r/n在點/r/n(1,f(1))/r/n處的切線斜率是否為定值？/r/n（2）若/r/nf(x)>0/r/n，證明：/r/nln/r/n(a+3)</r/n19/r/n．已知函數(shù)/r/nf/r/nx/r/n=/r/nln/r/n（1）若函數(shù)/r/nf/r/nx/r/n的圖象與直線/r/nx+2y?4=0/r/n相切，求/r/nm/r/n（2）求/r/nf/r/nx/r/n在區(qū)間/r/n1,2/r/n（3）若函數(shù)/r/nf/r/nx/r/n有兩個不同的零點/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故曲線/r/ny=f(x)/r/n在點/r/n(1,f(1))/r/n處的切線斜率/r/nk=4/r/n為定值./r/n（2）證明：∵/r/nf(x)>0/r/n，/r/nx∈(0,+∞)/r/n，∴/r/nx?(a+2)/r/nln/r/n設(shè)/r/n?(x)=x?(a+2)/r/nln/r/nx+/r/n當(dāng)/r/n0<x<a+3/r/n時，/r/n?'(x)<0/r/n；當(dāng)/r/nx>a+3/r/n時，/r/n?'(x)>0/r/n從而/r/n?/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）設(shè)切點/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，因為/r/ng/r/nx/r/n0/r/n=/r/n所以切點/r/nP/r/n1,m/r/n，代入切線方程得/r/nm=/r/n（2）因為/r/nf/r/nx/r/n所以/r/nf'/r/nx/r/n=/r/n1/r/nx/r/n?/r/n當(dāng)/r/nm≤0/r/n時，/r/n/r/nf'/r/nx/r/n>0/r/n，則/r/nf/r/nx/r/n所以/r/nf/r/nx/r/n在/r/n1,2/r/n遞增，則/r/nf/r/n當(dāng)/r/nm>0/r/n時，/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上單調(diào)遞減，在/r/nm,+∞/r/n則當(dāng)/r/nm≥2/r/n時，/r/n/r/nf/r/nx/r/n在/r/n1,2/r/n遞減，則/r/nf/r/n當(dāng)/r/n0<m≤1/r/n時，/r/n/r/nf/r/nx/r/n在/r/n1,2/r/n遞增，則/r/nf/r/n當(dāng)/r/n1<m<2/r/n時，/r/n/r/nf/r/nx/r/n在/r/n1,m/