高考真題講解導(dǎo)數(shù)合集及答案

一、單項選擇題（每題2分，共10題）1.函數(shù)f(x)=x^3-3x^2+2在區(qū)間[-1,3]上的最大值是A.2B.3C.4D.5答案：C2.函數(shù)f(x)=e^x-x在x=0處的導(dǎo)數(shù)是A.-1B.0C.1D.e答案：C3.函數(shù)f(x)=x^2-4x+3的導(dǎo)數(shù)f'(x)等于A.2x-4B.2x+4C.-2x+4D.-2x-4答案：A4.函數(shù)f(x)=sin(x)+cos(x)的導(dǎo)數(shù)f'(x)是A.cos(x)-sin(x)B.sin(x)+cos(x)C.-sin(x)-cos(x)D.-cos(x)+sin(x)答案：A5.函數(shù)f(x)=log(x)在x=1處的導(dǎo)數(shù)是A.1B.0C.log(1)D.不存在答案：B6.函數(shù)f(x)=x^2e^x的導(dǎo)數(shù)f'(x)是A.2xe^xB.x^2e^xC.(x^2+2x)e^xD.(x^2-2x)e^x答案：C7.函數(shù)f(x)=1/x在x=2處的導(dǎo)數(shù)是A.1/2B.-1/4C.-1/2D.1/4答案：D8.函數(shù)f(x)=tan(x)的導(dǎo)數(shù)f'(x)是A.sec^2(x)B.-sec^2(x)C.cot(x)D.-cot(x)答案：A9.函數(shù)f(x)=arctan(x)的導(dǎo)數(shù)f'(x)是A.1/(1+x^2)B.-1/(1+x^2)C.x/(1+x^2)D.-x/(1+x^2)答案：A10.函數(shù)f(x)=csc(x)的導(dǎo)數(shù)f'(x)是A.-cot(x)csc(x)B.cot(x)csc(x)C.-csc(x)cot(x)D.csc(x)cot(x)答案：A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+2的導(dǎo)數(shù)f'(x)等于A.3x^2-6xB.3x^2-3xC.3x^2+6xD.3x^2+3x答案：A3.函數(shù)f(x)=sin(x)+cos(x)的導(dǎo)數(shù)f'(x)是A.cos(x)-sin(x)B.sin(x)+cos(x)C.-sin(x)-cos(x)D.-cos(x)+sin(x)答案：AD4.函數(shù)f(x)=log(x)在x=1處的導(dǎo)數(shù)是A.1B.0C.log(1)D.不存在答案：B5.函數(shù)f(x)=x^2e^x的導(dǎo)數(shù)f'(x)是A.2xe^xB.x^2e^xC.(x^2+2x)e^xD.(x^2-2x)e^x答案：CD6.函數(shù)f(x)=1/x在x=2處的導(dǎo)數(shù)是A.1/2B.-1/4C.-1/2D.1/4答案：AD7.函數(shù)f(x)=tan(x)的導(dǎo)數(shù)f'(x)是A.sec^2(x)B.-sec^2(x)C.cot(x)D.-cot(x)答案：A8.函數(shù)f(x)=arctan(x)的導(dǎo)數(shù)f'(x)是A.1/(1+x^2)B.-1/(1+x^2)C.x/(1+x^2)D.-x/(1+x^2)答案：A9.函數(shù)f(x)=csc(x)的導(dǎo)數(shù)f'(x)是A.-cot(x)csc(x)B.cot(x)csc(x)C.-csc(x)cot(x)D.csc(x)cot(x)答案：AC10.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：AC三、判斷題（每題2分，共10題）1.函數(shù)f(x)=x^2在x=1處的導(dǎo)數(shù)是2。答案：正確2.函數(shù)f(x)=sin(x)的導(dǎo)數(shù)是cos(x)。答案：正確3.函數(shù)f(x)=e^x的導(dǎo)數(shù)是e^x。答案：正確4.函數(shù)f(x)=log(x)的導(dǎo)數(shù)是1/x。答案：正確5.函數(shù)f(x)=x^3的導(dǎo)數(shù)是3x^2。答案：正確6.函數(shù)f(x)=1/x的導(dǎo)數(shù)是-1/x^2。答案：正確7.函數(shù)f(x)=tan(x)的導(dǎo)數(shù)是sec^2(x)。答案：正確8.函數(shù)f(x)=arctan(x)的導(dǎo)數(shù)是1/(1+x^2)。答案：正確9.函數(shù)f(x)=csc(x)的導(dǎo)數(shù)是-cot(x)csc(x)。答案：正確10.函數(shù)f(x)=|x|在x=0處導(dǎo)數(shù)不存在。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+2的導(dǎo)數(shù)，并找出其在區(qū)間[-1,3]上的最大值和最小值。答案：f'(x)=3x^2-6x。令f'(x)=0，得x=0或x=2。計算f(-1)=5，f(0)=2，f(2)=-2，f(3)=2。最大值為5，最小值為-2。2.求函數(shù)f(x)=sin(x)+cos(x)的導(dǎo)數(shù)，并找出其在區(qū)間[0,π/2]上的最大值和最小值。答案：f'(x)=cos(x)-sin(x)。令f'(x)=0，得x=π/4。計算f(0)=1，f(π/4)=√2/2，f(π/2)=0。最大值為√2/2，最小值為0。3.求函數(shù)f(x)=x^2e^x的導(dǎo)數(shù)，并找出其在區(qū)間[-1,1]上的最大值和最小值。答案：f'(x)=(x^2+2x)e^x。令f'(x)=0，得x=0或x=-2。由于x=-2不在區(qū)間[-1,1]內(nèi)，計算f(-1)=e^-1，f(0)=0，f(1)=e。最大值為e，最小值為0。4.求函數(shù)f(x)=1/x在區(qū)間[1,2]上的最大值和最小值。答案：f'(x)=-1/x^2。在區(qū)間[1,2]上，f'(x)始終小于0，因此函數(shù)單調(diào)遞減。最大值為f(1)=1，最小值為f(2)=1/2。五、討論題（每題5分，共4題）1.討論函數(shù)f(x)=x^3-3x^2+2的單調(diào)性和極值。答案：f'(x)=3x^2-6x。令f'(x)=0，得x=0或x=2。在區(qū)間(-∞,0)上，f'(x)>0，函數(shù)單調(diào)遞增；在區(qū)間(0,2)上，f'(x)<0，函數(shù)單調(diào)遞減；在區(qū)間(2,+∞)上，f'(x)>0，函數(shù)單調(diào)遞增。極小值為f(2)=-2，極大值為f(0)=2。2.討論函數(shù)f(x)=sin(x)+cos(x)的單調(diào)性和極值。答案：f'(x)=cos(x)-sin(x)。令f'(x)=0，得x=π/4+kπ，k為整數(shù)。在區(qū)間[0,π/2]上，f'(x)在x=π/4處由正變負(fù)，因此x=π/4為極大值點(diǎn)，極大值為√2/2。在區(qū)間(π/4,π/2)上，f'(x)<0，函數(shù)單調(diào)遞減。最小值為f(π/2)=0。3.討論函數(shù)f(x)=x^2e^x的單調(diào)性和極值。答案：f'(x)=(x^2+2x)e^x。令f'(x)=0，得x=0或x=-2。在區(qū)間(-∞,-2)上，f'(x)>0，函數(shù)單調(diào)遞增；在區(qū)間(-2,0)上，f'(x)<0，函數(shù)單調(diào)遞減；在區(qū)間(0,+∞)上