高三數(shù)學(xué)函數(shù)與導(dǎo)數(shù)試卷及答案

一、單項(xiàng)選擇題（每題2分，共20分）1.函數(shù)\(y=\log_{2}(x+1)\)的定義域是（）A.\((-1,+\infty)\)B.\([-1,+\infty)\)C.\((0,+\infty)\)D.\((1,+\infty)\)2.已知函數(shù)\(f(x)=x^{3}+ax^{2}+bx+c\)，若\(f(0)=0\)，則\(c\)的值為（）A.0B.1C.-1D.23.函數(shù)\(y=2^{x}\)與\(y=(\frac{1}{2})^{x}\)的圖象關(guān)于（）A.\(x\)軸對(duì)稱B.\(y\)軸對(duì)稱C.原點(diǎn)對(duì)稱D.直線\(y=x\)對(duì)稱4.函數(shù)\(f(x)=\lnx-x\)的單調(diào)遞增區(qū)間是（）A.\((0,1)\)B.\((1,+\infty)\)C.\((-\infty,1)\)D.\((-\infty,0)\)5.已知\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\gt0\)時(shí)，\(f(x)=x^{2}-2x\)，則\(f(-1)\)的值為（）A.-1B.1C.3D.-36.函數(shù)\(y=\frac{1}{x-1}\)在區(qū)間\([2,3]\)上的最大值為（）A.1B.\(\frac{1}{2}\)C.\(\frac{1}{3}\)D.\(\frac{1}{4}\)7.若函數(shù)\(f(x)=ax^{3}+3x^{2}+x+b\)（\(a\neq0\)）有極值點(diǎn)，則\(a\)的取值范圍是（）A.\((0,3)\)B.\((-\infty,0)\cup(0,3)\)C.\((3,+\infty)\)D.\((-\infty,0)\cup(3,+\infty)\)8.已知函數(shù)\(f(x)\)的導(dǎo)函數(shù)為\(f^\prime(x)\)，且滿足\(f(x)=2xf^\prime(1)+\lnx\)，則\(f^\prime(1)\)的值為（）A.-1B.1C.-eD.e9.函數(shù)\(y=\cosx+x\)在\([0,\frac{\pi}{2}]\)上的最大值是（）A.\(\frac{\pi}{2}\)B.1C.\(\frac{\pi}{2}+1\)D.010.若函數(shù)\(f(x)\)在區(qū)間\((a,b)\)內(nèi)有\(zhòng)(f^\prime(x)\gt0\)，且\(f(a)\geq0\)，則在\((a,b)\)內(nèi)有（）A.\(f(x)\gt0\)B.\(f(x)\lt0\)C.\(f(x)=0\)D.無法確定答案：1.A2.A3.B4.A5.A6.A7.B8.A9.C10.A二、多項(xiàng)選擇題（每題2分，共20分）1.下列函數(shù)中，是偶函數(shù)的有（）A.\(y=x^{2}\)B.\(y=\cosx\)C.\(y=\ln|x|\)D.\(y=e^{x}+e^{-x}\)2.函數(shù)\(y=x^{3}-3x\)的極值點(diǎn)有（）A.\(x=-1\)B.\(x=0\)C.\(x=1\)D.\(x=2\)3.下列關(guān)于函數(shù)\(y=\log_{a}x\)（\(a\gt0\)且\(a\neq1\)）的說法正確的是（）A.當(dāng)\(a\gt1\)時(shí)，函數(shù)在\((0,+\infty)\)上單調(diào)遞增B.當(dāng)\(0\lta\lt1\)時(shí)，函數(shù)在\((0,+\infty)\)上單調(diào)遞減C.函數(shù)的圖象過定點(diǎn)\((1,0)\)D.函數(shù)的值域是\(R\)4.已知函數(shù)\(f(x)\)在\(x=x_{0}\)處可導(dǎo)，且\(f^\prime(x_{0})=0\)，則下列說法正確的是（）A.\(x=x_{0}\)一定是函數(shù)\(f(x)\)的極值點(diǎn)B.若在\(x_{0}\)兩側(cè)\(f^\prime(x)\)的符號(hào)相反，則\(x=x_{0}\)是函數(shù)\(f(x)\)的極值點(diǎn)C.若在\(x_{0}\)兩側(cè)\(f^\prime(x)\)同號(hào)，則\(x=x_{0}\)不是函數(shù)\(f(x)\)的極值點(diǎn)D.\(f(x_{0})\)一定是函數(shù)\(f(x)\)的最值5.函數(shù)\(y=\sinx+\cosx\)可化簡(jiǎn)為（）A.\(y=\sqrt{2}\sin(x+\frac{\pi}{4})\)B.\(y=\sqrt{2}\cos(x-\frac{\pi}{4})\)C.\(y=\sqrt{2}\sin(x-\frac{\pi}{4})\)D.\(y=\sqrt{2}\cos(x+\frac{\pi}{4})\)6.若函數(shù)\(f(x)\)滿足\(f(x+2)=f(x)\)，則\(f(x)\)是周期函數(shù)，其周期可能為（）A.2B.4C.6D.87.下列函數(shù)中，在區(qū)間\((0,+\infty)\)上單調(diào)遞減的有（）A.\(y=\frac{1}{x}\)B.\(y=-x^{2}\)C.\(y=-x+1\)D.\(y=\log_{\frac{1}{2}}x\)8.已知函數(shù)\(f(x)=x^{2}-2ax+1\)，則（）A.函數(shù)圖象的對(duì)稱軸為\(x=a\)B.當(dāng)\(a\gt0\)時(shí)，函數(shù)在\((-\infty,a)\)上單調(diào)遞減C.當(dāng)\(a\lt0\)時(shí)，函數(shù)在\((a,+\infty)\)上單調(diào)遞增D.函數(shù)的最小值為\(1-a^{2}\)9.函數(shù)\(y=\frac{1}{x^{2}+1}\)的值域是（）A.\((0,1]\)B.\((0,+\infty)\)C.\([0,1]\)D.\((-\infty,1]\)10.若函數(shù)\(f(x)\)的導(dǎo)函數(shù)\(f^\prime(x)\)的圖象如圖所示，則下列說法正確的是（）A.函數(shù)\(f(x)\)在\((-\infty,-2)\)上單調(diào)遞增B.函數(shù)\(f(x)\)在\((-2,1)\)上單調(diào)遞減C.函數(shù)\(f(x)\)在\((1,+\infty)\)上單調(diào)遞增D.函數(shù)\(f(x)\)在\(x=-2\)處取得極大值答案：1.ABCD2.AC3.ABCD4.BC5.AB6.ABCD7.ABCD8.ABCD9.A10.ABCD三、判斷題（每題2分，共20分）1.函數(shù)\(y=x^{\frac{1}{2}}\)的定義域是\(R\)。（）2.若函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上的圖象是連續(xù)不斷的，且\(f(a)f(b)\lt0\)，則函數(shù)\(f(x)\)在\((a,b)\)內(nèi)至少有一個(gè)零點(diǎn)。（）3.函數(shù)\(y=\tanx\)在其定義域內(nèi)是單調(diào)遞增函數(shù)。（）4.函數(shù)\(y=e^{x}\)的導(dǎo)數(shù)是\(y^\prime=e^{x}\)。（）5.若函數(shù)\(f(x)\)為奇函數(shù)，則\(f(0)=0\)。（）6.函數(shù)\(y=\log_{a}x\)（\(a\gt1\)）的增長(zhǎng)速度比\(y=x\)快。（）7.函數(shù)\(y=\sin^{2}x\)的最小正周期是\(\pi\)。（）8.函數(shù)\(f(x)\)在某點(diǎn)處的切線斜率等于該點(diǎn)處的導(dǎo)數(shù)值。（）9.若函數(shù)\(f(x)\)在區(qū)間\((a,b)\)上單調(diào)遞增，則\(f^\prime(x)\gt0\)在\((a,b)\)上恒成立。（）10.函數(shù)\(y=\frac{1}{x}\)的圖象關(guān)于原點(diǎn)對(duì)稱。（）答案：1.×2.√3.×4.√5.×6.×7.√8.√9.×10.√四、簡(jiǎn)答題（每題5分，共20分）1.求函數(shù)\(y=x^{3}-3x^{2}+2\)的單調(diào)區(qū)間。答案：對(duì)\(y=x^{3}-3x^{2}+2\)求導(dǎo)得\(y^\prime=3x^{2}-6x=3x(x-2)\)。令\(y^\prime\gt0\)，得\(x\lt0\)或\(x\gt2\)，單調(diào)遞增區(qū)間為\((-\infty,0)\)和\((2,+\infty)\)；令\(y^\prime\lt0\)，得\(0\ltx\lt2\)，單調(diào)遞減區(qū)間為\((0,2)\)。2.已知函數(shù)\(f(x)=ax^{2}+bx+c\)，且\(f(1)=0\)，\(f(2)=3\)，\(f(-3)=28\)，求\(a\)，\(b\)，\(c\)的值。答案：由\(f(1)=0\)得\(a+b+c=0\)；由\(f(2)=3\)得\(4a+2b+c=3\)；由\(f(-3)=28\)得\(9a-3b+c=28\)。聯(lián)立求解得\(a=2\)，\(b=-3\)，\(c=1\)。3.求函數(shù)\(y=\lnx+x\)在點(diǎn)\((1,1)\)處的切線方程。答案：對(duì)\(y=\lnx+x\)求導(dǎo)得\(y^\prime=\frac{1}{x}+1\)，當(dāng)\(x=1\)時(shí)，\(y^\prime=2\)，切線斜率為2。由點(diǎn)斜式得切線方程為\(y-1=2(x-1)\)，即\(2x-y-1=0\)。4.已知函數(shù)\(f(x)=e^{x}-x\)，求其最小值。答案：對(duì)\(f(x)=e^{x}-x\)求導(dǎo)得\(f^\prime(x)=e^{x}-1\)。令\(f^\prime(x)=0\)，得\(x=0\)。當(dāng)\(x\lt0\)時(shí)，\(f^\prime(x)\lt0\)；當(dāng)\(x\gt0\)時(shí)，\(f^\prime(x)\gt0\)，所以\(x=0\)時(shí)\(f(x)\)取最小值\(f(0)=1\)。五、討論題（每題5分，共20分）1.討論函數(shù)\(y=\frac{1}{x-a}\)（\(a\)為常數(shù)）的圖象與性質(zhì)。答案：定義域?yàn)閈(x\neqa\)。當(dāng)\(x\gta\)時(shí)，函數(shù)單調(diào)遞減；當(dāng)\(x\lta\)時(shí)，函數(shù)也單調(diào)遞減。圖象是雙曲線，關(guān)于點(diǎn)\((a,0)\)對(duì)稱，漸近線為\(x=a\)和\(y=0\)。2.結(jié)合函數(shù)的單調(diào)性與導(dǎo)數(shù)關(guān)系，談?wù)勅绾卫脤?dǎo)數(shù)判斷函數(shù)的單調(diào)性。答案：若函數(shù)\(f(x)\)在某區(qū)間內(nèi)可導(dǎo)，當(dāng)\(f^\prime(x)\gt0\)時(shí)，\(f(x)\)在該區(qū)間單調(diào)遞增；當(dāng)\(f^\prime(x)\lt0\)時(shí)，\(f(x)\)在該區(qū)間單調(diào)遞減。通過求導(dǎo)，分析導(dǎo)數(shù)正負(fù)來確定函數(shù)單調(diào)性。3.已知函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上有最大值和最小值，討論如何求函數(shù)在該區(qū)間上的最值。答案：先求\(f(x)\)在\((a,b)\)內(nèi)的極值點(diǎn)，即令\(f^\prime(x)=0\)求出\(x\)值，再計(jì)算\(f(x)\)在這些極值點(diǎn)以及端點(diǎn)\(a\)、\(b\)處的函數(shù)值