導(dǎo)數(shù)高考真題試題及答案

一、單項選擇題（每題2分，共10題）1.函數(shù)f(x)=x^3-ax+1在x=1處的切線平行于直線y=3x-1，則a的值為A.1B.2C.3D.4答案：C2.函數(shù)f(x)=e^x-x在區(qū)間(0,1)上的導(dǎo)數(shù)f'(x)的符號為A.總是正B.總是負(fù)C.有時正有時負(fù)D.無法確定答案：A3.若函數(shù)f(x)=x^2+px+q的導(dǎo)數(shù)在x=1時的值為3，且f(1)=1，則p的值為A.1B.2C.3D.4答案：B4.函數(shù)f(x)=ln(x^2+1)在x=0處的導(dǎo)數(shù)f'(0)的值為A.0B.1C.2D.3答案：B5.函數(shù)f(x)=x^3-3x^2+2x在x=1處的二階導(dǎo)數(shù)f''(1)的值為A.-1B.0C.1D.2答案：A6.函數(shù)f(x)=sin(x)+cos(x)在x=π/4處的導(dǎo)數(shù)f'(π/4)的值為A.0B.1C.√2D.√3答案：C7.函數(shù)f(x)=x^4-4x^3+6x^2-4x+1在x=1處的導(dǎo)數(shù)f'(1)的值為A.0B.1C.2D.3答案：A8.函數(shù)f(x)=x^2e^x在x=0處的導(dǎo)數(shù)f'(0)的值為A.0B.1C.2D.e答案：B9.函數(shù)f(x)=arctan(x)在x=1處的導(dǎo)數(shù)f'(1)的值為A.1/2B.1/3C.1/4D.1/5答案：A10.函數(shù)f(x)=xln(x)在x=1處的導(dǎo)數(shù)f'(1)的值為A.0B.1C.eD.1/e答案：B二、多項選擇題（每題2分，共10題）1.下列函數(shù)中，在x=0處導(dǎo)數(shù)存在的是A.f(x)=x^2B.f(x)=|x|C.f(x)=x^3D.f(x)=1/x答案：AC2.函數(shù)f(x)=x^3-3x^2+2x的極值點(diǎn)為A.x=0B.x=1C.x=2D.x=-1答案：BC3.函數(shù)f(x)=e^x-x在區(qū)間(-1,1)上的單調(diào)性為A.單調(diào)遞增B.單調(diào)遞減C.先增后減D.先減后增答案：AB4.函數(shù)f(x)=sin(x)+cos(x)在區(qū)間(0,2π)上的極值點(diǎn)為A.x=π/4B.x=π/2C.x=3π/4D.x=π答案：AC5.函數(shù)f(x)=x^4-4x^3+6x^2-4x+1的拐點(diǎn)為A.x=0B.x=1C.x=2D.x=3答案：BC6.函數(shù)f(x)=x^2e^x在區(qū)間(-1,1)上的單調(diào)性為A.單調(diào)遞增B.單調(diào)遞減C.先增后減D.先減后增答案：AC7.函數(shù)f(x)=arctan(x)在區(qū)間(-∞,+∞)上的導(dǎo)數(shù)符號為A.總是正B.總是負(fù)C.有時正有時負(fù)D.無法確定答案：A8.函數(shù)f(x)=xln(x)在區(qū)間(0,+∞)上的單調(diào)性為A.單調(diào)遞增B.單調(diào)遞減C.先增后減D.先減后增答案：A9.函數(shù)f(x)=x^3-3x^2+2x的凹凸性為A.凹B.凸C.先凹后凸D.先凸后凹答案：CD10.函數(shù)f(x)=sin(x)+cos(x)的周期為A.πB.2πC.π/2D.3π/2答案：B三、判斷題（每題2分，共10題）1.函數(shù)f(x)=x^2在x=0處的導(dǎo)數(shù)為0。答案：正確2.函數(shù)f(x)=e^x在任意x處的導(dǎo)數(shù)都等于e^x。答案：正確3.函數(shù)f(x)=sin(x)在x=π/2處的導(dǎo)數(shù)為1。答案：正確4.函數(shù)f(x)=x^3-3x^2+2x在x=1處有一個極值點(diǎn)。答案：正確5.函數(shù)f(x)=x^2e^x在x=0處的導(dǎo)數(shù)為1。答案：正確6.函數(shù)f(x)=arctan(x)在x=1處的導(dǎo)數(shù)為1/2。答案：正確7.函數(shù)f(x)=xln(x)在x=1處的導(dǎo)數(shù)為1。答案：正確8.函數(shù)f(x)=x^4-4x^3+6x^2-4x+1在x=1處有一個拐點(diǎn)。答案：正確9.函數(shù)f(x)=sin(x)+cos(x)在x=π/4處的導(dǎo)數(shù)為√2。答案：正確10.函數(shù)f(x)=x^3-3x^2+2x在x=2處有一個極值點(diǎn)。答案：正確四、簡答題（每題5分，共4題）1.求函數(shù)f(x)=x^3-3x^2+2x在x=1處的導(dǎo)數(shù)。答案：f'(x)=3x^2-6x+2，所以f'(1)=3(1)^2-6(1)+2=1。2.求函數(shù)f(x)=e^x-x在x=0處的導(dǎo)數(shù)。答案：f'(x)=e^x-1，所以f'(0)=e^0-1=0。3.求函數(shù)f(x)=sin(x)+cos(x)在x=π/4處的導(dǎo)數(shù)。答案：f'(x)=cos(x)-sin(x)，所以f'(π/4)=cos(π/4)-sin(π/4)=√2/2-√2/2=0。4.求函數(shù)f(x)=x^2e^x在x=0處的導(dǎo)數(shù)。答案：f'(x)=2xe^x+x^2e^x，所以f'(0)=2(0)e^0+(0)^2e^0=0。五、討論題（每題5分，共4題）1.討論函數(shù)f(x)=x^3-3x^2+2x的單調(diào)性和極值。答案：f'(x)=3x^2-6x+2，令f'(x)=0，解得x=1±√3/3。在x=1-√3/3處，f'(x)由負(fù)變正，為極小值點(diǎn)；在x=1+√3/3處，f'(x)由正變負(fù)，為極大值點(diǎn)。函數(shù)在(-∞,1-√3/3)和(1+√3/3,+∞)上單調(diào)遞增，在(1-√3/3,1+√3/3)上單調(diào)遞減。2.討論函數(shù)f(x)=e^x-x在區(qū)間(0,1)上的單調(diào)性。答案：f'(x)=e^x-1，在(0,1)上，e^x<e^1，所以f'(x)<0，函數(shù)在(0,1)上單調(diào)遞減。3.討論函數(shù)f(x)=sin(x)+cos(x)在區(qū)間(0,2π)上的極值點(diǎn)。答案：f'(x)=cos(x)-sin(x)，令f'(x)=0，解得x=π/4,5π/4。在x=π/4處，f'(x)由正變負(fù)，為極大值點(diǎn)；在x=5π/4處，f'(x)由負(fù)變正，為極小值點(diǎn)。4.討論函