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2 2 4 4 5 6 6 8 8 9 11221.(2024·山東青島·三模)定義[x[表示不超過x的最大整數(shù).例如:[1.2[=1,[-1,2[=-2,則()A.[x[+[y[=[x+y[B.?n∈Z,[x+n[=[x[+nC.f(x(=x-[x[是偶函數(shù)D.f(x(=x-[x[是增函數(shù)2.(2024·河南新鄉(xiāng)·二模)函數(shù)f(x(=[x[被稱為取整函數(shù),也稱高斯函數(shù),其中[x[表示不大于實(shí)數(shù)x的最大整數(shù).若?m∈(0,+∞(,滿足[x]2+[x[≤,則x的取值范圍是()A.[-1,2[B.(-1,2(C.[-2,2(D.(-2,2[(1)設(shè)f(x)=[x[+x+|-[2x[,x∈R,求證:是f(x(的一個(gè)周期,且f(x(=0恒成立;).A.函數(shù)y=在定義域上是奇函數(shù)B.函數(shù)y=的零點(diǎn)有無數(shù)個(gè)C.函數(shù)y=在定義域上的值域是(-1,1(D.不等式y(tǒng)=≤0解集是(-∞,0[5.(2024·河南開封·二模)(多選)高斯是德國(guó)著名整函數(shù)為f(x(=[x[,[x[表示不超過x的最大整數(shù),例如[-3.5[=-4,[2.1[=2.下列命題中正確的有 ()A.?x∈R,f(x(=x-1B.?x∈R,n∈Z,f(x+n(=f(x(+n33C.?x,y>0,f(lgx(+f(lgy(=f(lg(xy((*,f(lg1(+f(lg2(+f(lg3(+???+f(lgn(=92大整數(shù),例如:[3.9[=3,[-2.1[=-3.若在函數(shù)f(x(的定義域內(nèi),均滿足在區(qū)間[an,an+1(上,bn=[f(x([是一個(gè)常數(shù),則稱{bn{為f(x(的取整數(shù)列,稱{an{為f(x(的區(qū)間數(shù)列.下列說法正確的是 ()A.f(x(=log2x(x≥1(的區(qū)間數(shù)列的通項(xiàng)an=2nB.f(x(=log2x(x≥1(的取整數(shù)列的通項(xiàng)bn=n-1C.f(x(=log2(33x((x≥1(的取整數(shù)列的通項(xiàng)bn≥n+5D.若f(x(=log2x(1≤x<2n(,則數(shù)列{bn(an+1-an({的前n項(xiàng)和Sn=(n-2(2n+2xx-aA.=ad-bc.已知函數(shù)f(x(的定A.f(1(=1B.f(x(是偶函數(shù)C.f(x(是周期函數(shù)D.f(x(沒有極值點(diǎn)=ad-bc.已知函數(shù)f(θ(=.函數(shù)g(x(=范圍.44數(shù)D(x)={,,的結(jié)論正確的是()A.D(D(x))有零點(diǎn)B.D(x)是單調(diào)函數(shù)C.D(x)是奇函數(shù)D.D(x)是周期函數(shù)雷函數(shù)說法正確的是()A.D(D(e((=1B.它是偶函數(shù)他提出了著名的狄利克雷函數(shù):D(x(={,,,以下對(duì)D(x(的說法正確的是()A.D(D(x((=1B.D(x(的值域?yàn)閧0,1{C.存在x是無理數(shù),使得D(x+1(=D(x(+1D.?x∈R,總有D(x+1(=D(-x-1(13.(2024·重慶·一模)(多選)德國(guó)著名數(shù)學(xué)家狄利克雷在數(shù)學(xué)領(lǐng)域成就顯著,以其命名的函數(shù)f(x(=A.函數(shù)f(x(為偶函數(shù)C.對(duì)于任意的x∈R,都有f(f(x((=1D.在f(x(圖象上不存在不同的三個(gè)點(diǎn)A,B,C,使得△ABC為等邊三角形E.在f(x(圖象存在不同的三個(gè)點(diǎn)A,B,C,使得△ABC為等邊三角形()A.充分不必要條件B.必要不充分條件C.充要條件D.既不充分也不必要條件15.(2024·北京·模擬預(yù)測(cè))數(shù)學(xué)上的符號(hào)函數(shù)可以返回一個(gè)整型變量,用來指出參數(shù)的正負(fù)號(hào),一般用55④在[-2π,2π[上函數(shù)g(x(=xf(x(-1的零點(diǎn)個(gè)數(shù)為4.f(x(=max{-x2+4,-x+2,x+3{,則f(x(的最小值為()A.2.5B.3C.4D.5則M的最小值為()A.1B.2C.2D.419.(2024·湖北·一模)記max{f(x({,min{f(x({min{(max{A.3B.4C.5D.6示小于或等于n且與n互質(zhì)的正整數(shù)的數(shù)目.換句話說,φ(n(是所有不超過n且與n互素的數(shù)的總數(shù).66φ(n(=n0有無數(shù)解D.φ(n(≤n-1,當(dāng)且僅當(dāng)n是素?cái)?shù)時(shí)等號(hào)成立n=M(xe,n),x2和n求出x的值.7725.(23-24高三上·河南·階段練習(xí))(多選)黎曼函數(shù)(Riemannfunction)是一個(gè)特殊的函數(shù),由德國(guó)數(shù)學(xué)A.R(B.黎曼函數(shù)的定義域?yàn)閇0,1[C.黎曼函數(shù)的最大值為D.若f(x(是奇函數(shù),且f(1-x(=f(x(,當(dāng)x∈[0,1[時(shí),f(x(=R(x(,則f((+f(+6(=26.(2024·北京石景山·一模)黎曼函數(shù)在高等數(shù)學(xué)中有著廣泛應(yīng)用,其一種定義為:x∈[0,1[時(shí),R(x(=f(x0))處的曲率,其中f/是f的導(dǎo)函數(shù),f//(x(是f/(x)的導(dǎo)函數(shù).則拋物線x2=2py(p>0)上的各點(diǎn)處的曲率最大值為()A.2pB.pC.率表示曲線的彎曲程度.設(shè)函數(shù)y=f(x)的導(dǎo)函數(shù)為f/(x),f/(x)的導(dǎo)函數(shù)記為fⅡ(x),則函數(shù)y=f(x)的圖象在(x0,f(x0((的曲率證明:函數(shù)g=tanx,x圖象的曲率K(x)的極大值點(diǎn)位于區(qū)間.29.(22-23高三上·山東·階段練習(xí))(多選)曲線的曲率就是針對(duì)曲線上某個(gè)點(diǎn)的切線方向角對(duì)弧長(zhǎng)的轉(zhuǎn)動(dòng)88率,表明曲線偏離直線的程度,曲率越大,表示曲x(是f/(x(的導(dǎo)函數(shù).下面說法正確的是()A.若函數(shù)f(x)=x3,則曲線y=f(x)在點(diǎn)(-a,-a3)與點(diǎn)(a,a3)處的彎曲程度相同B.若f(x)是二次函數(shù),則曲線y=f(x)的曲率在頂點(diǎn)處取得最小值C.若函數(shù)f(x)=sinx,則函數(shù)K(x)的值域?yàn)閇0,1]D.若函數(shù)則曲線y=f(x)上任意一點(diǎn)的曲率的最大值為=f(x(在點(diǎn)x0處左可導(dǎo).當(dāng)函數(shù)y=f(x(在點(diǎn)x0處既右可導(dǎo)也左可導(dǎo)且導(dǎo)數(shù)值相等,則稱函數(shù)y=f(x(在點(diǎn)x0處可導(dǎo).(2)已知函數(shù)f(x(=x2eax+1-x3sinx-ex2.(ⅰ)求函數(shù)g(x(=eax+1-xsinx-e在x=0處的切線方程;9931.(2024·貴州·模擬預(yù)測(cè))定義:設(shè)f/(x)是f(x)的導(dǎo)函數(shù),fⅡ(x(是函數(shù)f/(x)的導(dǎo)數(shù),若方程fⅡ(x)=0有實(shí)數(shù)解x0,則稱點(diǎn)(x0,f(x0((為函數(shù)y=f(x)的“拐點(diǎn)”.經(jīng)過探究發(fā)現(xiàn):任何一個(gè)三次函數(shù)都有“拐點(diǎn)”且“拐點(diǎn)”就是三次函數(shù)圖象的對(duì)稱中心.已知函數(shù)f(x)=x3+bx2-x+a圖象的對(duì)稱中心為(0,1),則下列說法中正確的有()A.a=1,b=0B.函數(shù)f(x)C.函數(shù)f(x)有三個(gè)零點(diǎn)D.y=f(x)在區(qū)間(0,1)上單調(diào)遞減32.(2024·河南·三模)設(shè)函數(shù)f(x(的導(dǎo)函數(shù)為f/(x(,f/(x(的導(dǎo)函數(shù)為fⅡ(x(,fⅡ(x(的導(dǎo)函數(shù)為f川(x(.若f(x0(=0,f(x0((為曲線y=f(x(的拐點(diǎn).(2)已知函數(shù)f(x(=ax5-5x3,若(為曲線y=f(x(的一個(gè)拐點(diǎn),求f(x(的單調(diào)區(qū)間與極值.x-1的極限即為01696年提出洛必達(dá)法則:在一定條件下通過對(duì)分子、分母分別求導(dǎo)再求極限A.0B.C.1D.2f(x(,g(x(的導(dǎo)函數(shù)分別為f/(x(,g/(x(,且g(x)=0,則l=0,則稱函數(shù)f(x(為區(qū)間[0,a[上的k階無窮遞降函數(shù).(1)試判斷f(x(=x3-3x是否為區(qū)間[0,3[上的2階無窮遞降函數(shù);3<cosx要方法,其含義為:若函數(shù)f(x(和g(函數(shù)f(x(=1+x+判斷并說明f(x(的零點(diǎn)個(gè)數(shù);-36.(2024·黑龍江齊齊哈爾·三模)在數(shù)學(xué)中,布勞威爾不動(dòng)點(diǎn)定理是拓?fù)鋵W(xué)里的一個(gè)非常重要的不動(dòng)點(diǎn)定理,簡(jiǎn)單的講就是對(duì)于滿足一定條件的連續(xù)函數(shù)f(x),存在一個(gè)點(diǎn)x0,使得f(x0(=x0,那么我們稱該函數(shù)為“不動(dòng)點(diǎn)”函數(shù).函數(shù)f(x)=2x-sinx+cosx有個(gè)不動(dòng)點(diǎn).37.(2024·廣東廣州·二模)若x0是方程f(g(x((=g(f(x((的實(shí)數(shù)解,則稱x0是函數(shù)y=f(x(與y=g(x(的“復(fù)合穩(wěn)定點(diǎn)”.若函數(shù)f(x(=ax(a>0且a≠1)與g(x(=2x-2有且僅有兩個(gè)不同的“復(fù)合穩(wěn)定點(diǎn)”,則a的取值范圍為()A.B.C.(1,、2(D.(、2,+∞(空間并構(gòu)成了一般不動(dòng)點(diǎn)定理的基石,得名于荷蘭數(shù)學(xué)家魯伊茲·布勞威爾(L.E.J.Brouwer).簡(jiǎn)單地講0為該函數(shù)的不動(dòng)點(diǎn).(1)求函數(shù)f(x)=2x+x-3的不動(dòng)點(diǎn);(2)若函數(shù)g(x)=lnx-b有兩個(gè)不動(dòng)點(diǎn)x1,x2,且x1<x2,若x2-x1≤2,求實(shí)數(shù)b的取值范圍.39.(2024·內(nèi)蒙古呼和浩特·二模)對(duì)于函數(shù)f(x(,若實(shí)數(shù)x0滿足f(x0(=x0,則x0稱為f(x(的不動(dòng)點(diǎn).已知函數(shù)f(x(=ex-2x+ae-x(x≥0(.(1)當(dāng)a=-1時(shí),求證f(x(≥0;設(shè)n∈N*,證明>ln(n+1(.40.(2024·河北滄州·一模)對(duì)于函數(shù)y=f(x),x∈I,若存在∈I,使得f(f(x0((=x0,則稱x0為函數(shù)f(x)的二階不動(dòng)點(diǎn);依此類推,可以定義函xf(x)=x(L{,B={xf(f(x))=x(L{.若f(x(=(a+1(x-討論集合B的子集的個(gè)數(shù).41.(2024·江蘇蘇州·三模)對(duì)于函數(shù)f(x),若存在實(shí)數(shù)x0,使f(x0(f(x0+λ(=1,其中λ≠0,則稱f(x)為“可f(x)的可移λ倒數(shù)點(diǎn)”.設(shè)f若函數(shù)f(x)恰有3個(gè)“可移1倒數(shù)為“f(x)的可移λ倒數(shù)點(diǎn)”.已知g(x)=ex,h(x)=x+a(a>0).43.(2024·貴州貴陽·一模)英國(guó)數(shù)學(xué)家泰勒發(fā)現(xiàn)了如下公式:ex=1+x++?其中n!=1×2×3×4×?×n,e為自然對(duì)數(shù)的底數(shù),e=2.71828??.以上公式稱為泰勒公式.設(shè)f(x(=x≥1+x;<g(x(;設(shè)F(x(=g(x(-a(1+(,若x=0是F(x(的極小值點(diǎn),求實(shí)數(shù)a的取值范圍.可導(dǎo),則有如下公式xn=ffⅡf川x3+?+f(x(的四階導(dǎo)數(shù)??,一般地,函數(shù)f(x(的n-1階導(dǎo)數(shù)的導(dǎo)數(shù)叫做函數(shù)f(x(的n階導(dǎo)數(shù),記作f(n((x(=[fn-1(x([/,n≥4;f(x0(+(x-x0(+(x-x0(2+?+(x-x0(n+?,我們將g(x(稱為函數(shù)f(x(在點(diǎn)x=x0處的泰勒展開式.例如f1(x)=ex在點(diǎn)x=0處的泰勒展開式為=1+x+x2+?+xn+?(1)求出f(x)=cosx在點(diǎn)x=0處的泰勒展開式g(x(;?(1-((1+試求的值.展開式為:f(x(=f(0(+f/(0(x+x2+?+xn+?=xn,其中f(n((0(表示f(x(的n階導(dǎo)數(shù)在0處的取值,我們稱xn為f(x(麥克勞林展開式的第n+1項(xiàng).例如:ex=1+(2)數(shù)學(xué)競(jìng)賽小組發(fā)現(xiàn)ln(1+x(的麥克勞林展開式為ln(1+x(=x-這意味著:當(dāng)x>0時(shí),ln(1+x(>x-你能幫助數(shù)學(xué)競(jìng)賽小組完成對(duì)此不等式的證明嗎?x+lnx+>+mx,求整數(shù)m的最大值.47.(2024·河南周口·模擬預(yù)測(cè))已知函數(shù)f(x(=(x-1(ln(1-x(-x-cosx.(1)求函數(shù)f(x(在區(qū)間(0,1(上的極值點(diǎn)的個(gè)數(shù).(2”是一個(gè)求和符號(hào),例如i=1+2+?+n,=2x+2x2+?+2xn,等等.英國(guó)數(shù)學(xué)家布典應(yīng)用.48.(2024·全國(guó)·模擬預(yù)測(cè))已知函數(shù)f(x(=(x-a(e-x+x2-2x,g(x(=xe-x-ex-1-x3+ax2-f(x(,且f(x(在x=0處取得極大值.(1)求a的值與f(x(的單調(diào)區(qū)間.49.(2024·山西·三模)微分中值定理是微積分學(xué)中的重要定理,它是研究區(qū)間上函數(shù)值變化規(guī)律的有效工如果函數(shù)f(x)在閉區(qū)間[a,b[上連續(xù),在開區(qū)間(a,b)可導(dǎo),導(dǎo)數(shù)為f/(x),那么在開區(qū)間(a,b)內(nèi)至少存在一點(diǎn)c,使得f/,其中c叫做f在[a,b[上的“拉格朗日中值點(diǎn)”.已知函數(shù)f(x)=(2)若a=-1,b=1,求證:函數(shù)f(x)在區(qū)間(0,+∞)圖象上任意兩點(diǎn)A,B連線的斜率不大于18-e-6;50.(23-24高二下·江西九江·階段練習(xí))已知函數(shù)f(x(=x2-3x+alnx,a∈R.(1)當(dāng)a=1時(shí),求函數(shù)f(x(的在點(diǎn)(1,f(1((處的切線;(2)若函數(shù)f(x(在區(qū)間[1,2[上單調(diào)遞減,求a的取值(3)若函數(shù)g(x(的圖象上存在兩點(diǎn)A(x1,y1(,B(x2,y2(,且x1<x2,使得g/(則明理由.f/(c((b-a(成立.設(shè)f(x(=ex+x-4,其中e為自然對(duì)數(shù)的底數(shù),e≈2.71828.易知,f(x(在實(shí)數(shù)集Rf(x(<1;(2)從圖形上看,函數(shù)f(x(=ex+x-4的零點(diǎn)就是函數(shù)f(x(的圖象與x軸 中選定一個(gè)x0作為r的初始近似值,使得0<f(x0(<然后在點(diǎn)(x0,f(x0((處作曲線y=f(x(1是r的一次近似值;在點(diǎn)(x1,f(x1((處作曲線y=f(x(的,?,xn,?.①當(dāng)xn>r時(shí),證明:xn>xn+1>r;證明:0<xn-r<52.(22-23高二下·山東濟(jì)南·期中)帕德近似是法國(guó)數(shù)學(xué)家亨利帕德發(fā)明的用有理數(shù)多項(xiàng)式近似特定函數(shù)且滿足:f(0(=R(0(,f/(0(=R/(0(,fⅡ(0(=RⅡ(0(..f(m+n((0(=R(m+n((0(.已知f(x(=ln(x+1(在x=0處的[1,1[階帕德近似為R(x(=.注:fⅡ(x(=[f/(x([/,f川(x(=[fⅡ(x([/,f(4(x=[f川(x([/,f(5(x=[f(4(x[/...求證:(x+b(f((>1;算機(jī)數(shù)學(xué)中有著廣泛的應(yīng)用.已知函數(shù)f(x)在x=0處的[m,n[階帕德近似定義為:R(x)=中f(2)(x)=[f/(x)[/,f(3)(x)=[f(2)(x)[/,?,f(m+n)(x)=[f(m+n-1)(x)[/.已知f(x)=ln(x+1)在x=0處的(2)設(shè)h(x(=f(x(-R(x(,證明:xh(x)≥0;函數(shù)的方法.給定兩個(gè)正整數(shù)m,n,函數(shù)f(x(在x=0處的[m,n[階帕德近似定義為:R(x(=x(=[f/(x([/,f川(x(=[fⅡ(x([/,f(4((x(=[f川(x([/,f(5((x(=[f(4((x([/,??已知函數(shù)f(x(=ln(x+1(.②若f(x(-m(+1(R(x(≤1-cosx恒成立,求實(shí)數(shù)m的取值范圍.55.(23-24高二下·湖北·期中)帕fx(=[f/(x([/,f川(x(=[fⅡ(x([/,f(4((x(=[fⅡ(x([/,f(5((x已知f(x(=ln(x+1(在x=0處的[1,1[階帕德近似為g(x(=≥g(x(;似的我們可以定義雙曲正弦函數(shù)sh(x(=它們與正、余(1)求sh(x(與ch(x(的導(dǎo)數(shù);(2)證明:sh(x(≥x在x∈[0,+∞(上恒成立;求f=sh(x(-sinx-的零點(diǎn).n∈N+可表示為二進(jìn)制表達(dá)式0=1+???+ak-1+ak58.(22-23高一上·江蘇南通·期末)對(duì)于任意兩個(gè)正數(shù)u,v(u<v),記曲線與直線x=u,x=v,x軸圍成的曲邊梯形的面積為L(zhǎng)(u,v(,并約定L(u,u(=0和L(u,v(=-L(v,u(,德國(guó)數(shù)學(xué)家萊布尼茨(Leibniz)最早發(fā)現(xiàn)L(1,x(=lnx.關(guān)于L(u,v(,下列說法正確的是()A.L((=L(4,8(B.L(2100,3100(=100L(2,3(C.L(uu,vu(>v-uD.2L(u,v(<60.(2024·安徽·模擬預(yù)測(cè))給出定義:若函數(shù)f(x(在D上可導(dǎo),即f/(x(存在,且導(dǎo)函數(shù)f/(x(在D上也可導(dǎo),則稱f(x(在D上存在二階導(dǎo)數(shù),記fⅡ(x(=(f/(x((/.若fⅡ(x(<0在D上恒成立,則稱f(x(在D上為凸A.f(x(=sinx+cosxB.f(x(=lnx-2xC.f(x(=-x3+2x-1D.f(x(=-xe-x61.(23-24高三下·陜西安康·階段練習(xí))記函數(shù)f(x(的導(dǎo)函數(shù)為f/(x(,f/(x(的導(dǎo)函數(shù)為fⅡ(x(,設(shè)D是f(x(的定義域的子集,若在區(qū)間D上fⅡ(x(≤0,則稱f(x(在D上是“凸函數(shù)”.已知函數(shù)f(x(=asinx-x2.(2)若a=2,判斷g(x(=f(x(+1在區(qū)間(0,π(上的零點(diǎn)個(gè)數(shù).知x1,x2,?,xn>0,n≥2,且x1+x2+A.B.C.D.①設(shè)f/(x(為函數(shù)f(x(的導(dǎo)函數(shù).若f/(x(在區(qū)間D單調(diào)遞增;則稱f(x(為區(qū)D上的凹函數(shù);若f/(x(在區(qū)間D上單調(diào)遞減,則稱f(x(為區(qū)間D上的凸函數(shù).(1)已知函數(shù)f(x(=ax4+x3-3(2a+1(x2-x+3.64.(23-24高二下·上海閔行·期末)若函數(shù)y=f(x(的圖像上有兩個(gè)不同點(diǎn)P,Q處的切線重合,則稱該切線PQ為函數(shù)y=f(x(的圖像的“自公切線”.若g(x(=求函數(shù)y=g(x(的圖像的“自公切線”方程;(3)設(shè)f(x(=ax3+bx2+cx+d(a≠0(,求證:函數(shù)y=f(x(的圖像不存在“自公切線”(1)對(duì)于函數(shù)y=f(x(,分別在點(diǎn)(k,f(k(((k∈N,k≥1(處作函數(shù)y=f(x(的切線,記切線與x軸的交點(diǎn)分別為(xk,0((k∈N,k≥1(,記xk為數(shù)列{xn{的第k項(xiàng),則稱數(shù)列{xn{為函數(shù)y=f(x(的“切線-函數(shù)y=f(x(的“切線-y軸數(shù)列”.則Sn=a11+a22+?+an?bn,求{Sn{的通項(xiàng)公式.線y=f(x(在點(diǎn)(x1,f(x1((處的切線為l2,設(shè)l2與x軸交點(diǎn)的橫坐標(biāo)為x2,稱x2為r的2次近似值.一般地,曲線y=f(x(在點(diǎn)(xn,f(xn(((n∈N+(處的切線為ln+1,記ln+1與x軸交點(diǎn)的橫坐標(biāo)為xn+1,并稱xn+1為r的n+1次近似值.已知二次函數(shù)f(x(有兩個(gè)不相等的實(shí)根b,c,其中c>b.對(duì)函數(shù)y=f(x(持續(xù)實(shí)66.(2024·上海黃浦·二模)若函數(shù)y=f(x)的圖象上的兩個(gè)不同點(diǎn)處的切線互相重合,則稱該切線為函數(shù)y{xn}中的項(xiàng)”.67.(2024高三·全國(guó)·專題練習(xí))已知a為實(shí)數(shù),函數(shù)f(x)=alnx+x2-4x.(2)定義:若函數(shù)m(x)的圖象上存在兩點(diǎn)A,B,設(shè)線段AB的中點(diǎn)為P(x0,y0),若m(x)在點(diǎn)Q(x0,m(x0)68.(2024·浙江·三模)在平面直角坐標(biāo)系中,如果將函數(shù)y=f(x)的圖象繞坐標(biāo)原點(diǎn)逆時(shí)針旋轉(zhuǎn)α(0<α≤所得曲線仍然是某個(gè)函數(shù)的圖象,則稱f(x(為“α旋轉(zhuǎn)函數(shù)”.(2)已知函數(shù)f(x(=ln(2x+1((x>0(是“α旋轉(zhuǎn)函數(shù)”,求tanα的最大值;(3)若函數(shù)g(x(=m(x-1(ex-xlnx-是“旋轉(zhuǎn)函數(shù)”,求m的取值范圍.69.(2024·黑龍江·三模)若函數(shù)y=f(x(滿足:對(duì)任意的實(shí)數(shù)s,t∈(0,+∞(,有f(s+t(>f(s(+f(t(恒成立,則稱函數(shù)y=增函數(shù)”.(3)設(shè)g(x(=ex-ln(x+1(-1,若曲線y=g(x(在x=x0處的切線方程為y=0,求x0的值,并證明函數(shù)70.(2024·貴州六盤水·三模)若函數(shù)f(x(在[a,b[上有定義,且對(duì)于任意不同的x1,x2∈[a,b[,都有|f(x1(-f(x2(|<k|x1-x2|,則稱f(x(為[a,b[上的“k類函數(shù)”(1)若f(x(=x2,判斷f(x(是否為[1,2[上的“4類函數(shù)”;若f(x(=lnx+(a+1(x+為[1,e[上的“2類函數(shù)”,求實(shí)數(shù)a的取值范圍;(3)若f(x(為[1,2[上的“2類函數(shù)”且f(1(=f(2(,證明:?x1,x2∈[1,2[,|f(x1(-f(x2(|<1.71.(2024·新疆喀什·三模)已知定義域?yàn)镽的函數(shù)f(x(滿足:對(duì)于任意的x∈R,都有f(x+2π(=f(x(+f(2π(,則稱函數(shù)f(x(具有性質(zhì)P.(1)判斷函數(shù)g(x(=x,h(x(=sinx是否具有性質(zhì)P;(直接寫出結(jié)論)(2)已知函數(shù)f(x(=sin(ωx+φ(<φ<,|φ|<,判斷是否存在ω,φ,使函數(shù)f(x(具有性質(zhì)P?(3)設(shè)函數(shù)f(x(具有性質(zhì)P,且在區(qū)間[0,2π[上的值域?yàn)閇f(0(,f(2π([.函數(shù)g(x(=sin(f(x((,滿足已知函數(shù)f(x(=3ax+a,,且f(x(滿足:對(duì)?x1,x2∈R有(x1-x2((f(x1(-f(x2((≤0,則aB.0C.[-3.5[=-4,[2.1[=2.下列結(jié)論正確的有()A.函數(shù)f(x(=[x[與函數(shù)h(x(=x-1無公共點(diǎn)f(-x(+=-(f(x(+D.所有滿足f(m(=f(n((m,n(的點(diǎn)(m,n(組成區(qū)域的面積為義在R上的函數(shù)D(x(=后來數(shù)學(xué)家研究發(fā)現(xiàn)該函數(shù)在其定義域上處處不連續(xù)、處處不A.D(x+y(≤D(x(+D(y(B.D(x(的圖象關(guān)于OyO軸對(duì)稱C.D2(x(=D(D(x((的圖象關(guān)于OyO軸對(duì)稱D.存,?,xn的最小數(shù)為min{x1,x2,?,xn{,若f(x(=min{x+1,x2-2x+1,-x+8{,則函數(shù)f(x(的最大值為.f(x(=min{x2-3x+3,3-|x-3|{,則()A.f(x(有且僅有一個(gè)極小值點(diǎn)為B.f(x(有且僅有一個(gè)極大值點(diǎn)為3f(x(≤k恒成立f(x(=min{2x-a,-x+6-a{,x∈R.若函數(shù)y=f(x(+ax有兩個(gè)零點(diǎn),則實(shí)數(shù)a的取值范圍是()1-1-,?,pr是n的所有不重復(fù)的質(zhì)因數(shù)(質(zhì)因正整數(shù)k使得xk=x0,且當(dāng)0<j<k時(shí),xj≠x0,則稱x0是f(x)的一個(gè)周期為k的周期點(diǎn).若f(x(=如果函數(shù)y=f(x)滿足如下條件:函數(shù)y=在區(qū)間上的拉格朗日中值m為; .①y=x2-1;②y=x3;③y=ex-1;④y=sinx;⑤y=ln(x+1),其中dx=F如果平面圖形由兩條曲線圍成(1)如圖,連續(xù)函數(shù)y=f(x(在區(qū)間[-3,-2[與[2,3[的圖形分別為直徑為1的上、下半圓周,在區(qū)間[-2,0[與[0,2[的圖形分別為直徑為2的下、上半圓周,設(shè)F(x(=dt.求F(2(-F(3(的值;(2)在曲線f(x(=x2(x≥0)上某一個(gè)點(diǎn)處作切線,便之與曲線和x軸所圍成的面積為求切線方程;(x(為函數(shù)f(x(的n階導(dǎo)數(shù),f(n((x(=[f(n-1((x([/(n≥2,n∈N*(,若f(n((x(存在,則稱f(x(n階可導(dǎo).英國(guó)數(shù)學(xué)家泰勒發(fā)現(xiàn):若f(x(在x0附近n+1階可導(dǎo),則可構(gòu)造Tn(x(=f(x0(+(x-x0(+(x-x0(2+???+(x-x0(n(稱其為f(x(在x0處的n次泰勒多項(xiàng)式)來逼近f(x(在x0附近的函數(shù)值.下列說法正確的是()A.若f(x(=sinx,則f(n((x(=sin(x+B.若f(x(=,則f(n((x(=(-1(n(n!(x-(n+1(C.f(x(=ex在x0=0處的3次泰勒多項(xiàng)式為(x(=1+x+(1)求函數(shù)f(x,y)=x2y2+2xy+xy2關(guān)于變量y的導(dǎo)數(shù)并求當(dāng)x=1處的導(dǎo)數(shù)值.(2)利用拉格朗日乘數(shù)法求:設(shè)實(shí)數(shù)x,y滿足g(x,值.②設(shè)a>b>c>0,求2a2++-10ac+25c2的最小值.A,均有|f(x(-l(x(|≥|f(x(-l0(x(|,則稱l0(x(為函數(shù)f(x(在x∈[a,b[上的最佳逼近直線.(2)求函數(shù)g(x(=x2+2x-3在x∈[-1,1[上的最佳逼近直線.15.(2024·上海·模擬預(yù)測(cè))設(shè)定義域?yàn)镽的函數(shù)y=f(x(在R上可導(dǎo),導(dǎo)函數(shù)為y=f/(x(.若區(qū)間I及實(shí)數(shù)t滿足:f(x+t(≥t?f/(x(對(duì)任意x∈I成立,則稱函數(shù)y=f(x(為I上的“M(t(函數(shù)”.數(shù)n,y=f(x(都是R上的M(n(函數(shù),問:P是否為Q的充分條件?P是否為Q的必要條件?證明你的結(jié)論.有函數(shù)y=f(x(圖像上的點(diǎn)列M1,M2,?,Mn,?,使得對(duì)任意正整數(shù)n,點(diǎn)Mn都是點(diǎn)Mn+1的一個(gè)“上數(shù)列dT、dT+1、?(x(為函數(shù)f(x(的n階導(dǎo)數(shù),f(n((x(=[f(n-1((x([/(n≥2,n∈N*(,若f(n((x(存在,則稱f(x(n階可導(dǎo).英國(guó)數(shù)學(xué)家泰勒發(fā)現(xiàn):若f(x(在x0附近n+1階可導(dǎo),則可構(gòu)造(x(=f(x0(+(x-x0(+(x-x0(2+???+(x-x0(n(稱其為f(x(在x0處的n次泰勒多項(xiàng)式)來逼近f(x(在x0附近的函數(shù)值.下列說法正確的是()A.若f(x(=sin
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