2025年函數(shù)以及圖像題目及答案一、單項選擇題1.函數(shù)f(x)=x^2-4x+3的圖像是開口向上的拋物線，其頂點坐標是（A）A.(2,-1)B.(2,1)C.(-2,-1)D.(-2,1)2.函數(shù)f(x)=|x-1|在x=1處的導數(shù)是（C）A.1B.-1C.不存在D.03.函數(shù)f(x)=1/x在x=1處的導數(shù)是（A）A.-1B.1C.0D.不存在4.函數(shù)f(x)=e^x的圖像關于（B）對稱A.x軸B.y軸C.原點D.直線y=x5.函數(shù)f(x)=sin(x)在x=π/2處的導數(shù)是（A）A.1B.-1C.0D.π6.函數(shù)f(x)=cos(x)在x=π處的導數(shù)是（B）A.1B.-1C.0D.π7.函數(shù)f(x)=log(x)在x=1處的導數(shù)是（A）A.1B.-1C.0D.不存在8.函數(shù)f(x)=x^3-3x在x=0處的導數(shù)是（C）A.0B.1C.-3D.39.函數(shù)f(x)=tan(x)在x=π/4處的導數(shù)是（A）A.1B.-1C.0D.π10.函數(shù)f(x)=arcsin(x)在x=0處的導數(shù)是（B）A.1B.1/√2C.0D.-1/√2二、多項選擇題1.函數(shù)f(x)=x^2-4x+3的圖像與x軸的交點是（A,B）A.(1,0)B.(3,0)C.(2,0)D.(-1,0)2.函數(shù)f(x)=|x-1|在x=1處的導數(shù)是（C）A.1B.-1C.不存在D.03.函數(shù)f(x)=1/x在x=1處的導數(shù)是（A）A.-1B.1C.0D.不存在4.函數(shù)f(x)=e^x的圖像關于（B）對稱A.x軸B.y軸C.原點D.直線y=x5.函數(shù)f(x)=sin(x)在x=π/2處的導數(shù)是（A）A.1B.-1C.0D.π6.函數(shù)f(x)=cos(x)在x=π處的導數(shù)是（B）A.1B.-1C.0D.π7.函數(shù)f(x)=log(x)在x=1處的導數(shù)是（A）A.1B.-1C.0D.不存在8.函數(shù)f(x)=x^3-3x在x=0處的導數(shù)是（C）A.0B.1C.-3D.39.函數(shù)f(x)=tan(x)在x=π/4處的導數(shù)是（A）A.1B.-1C.0D.π10.函數(shù)f(x)=arcsin(x)在x=0處的導數(shù)是（B）A.1B.1/√2C.0D.-1/√2三、判斷題1.函數(shù)f(x)=x^2-4x+3的圖像是開口向上的拋物線，其頂點坐標是（2,-1）（正確）2.函數(shù)f(x)=|x-1|在x=1處的導數(shù)是0（錯誤）3.函數(shù)f(x)=1/x在x=1處的導數(shù)是-1（正確）4.函數(shù)f(x)=e^x的圖像關于y軸對稱（正確）5.函數(shù)f(x)=sin(x)在x=π/2處的導數(shù)是1（正確）6.函數(shù)f(x)=cos(x)在x=π處的導數(shù)是-1（正確）7.函數(shù)f(x)=log(x)在x=1處的導數(shù)是1（正確）8.函數(shù)f(x)=x^3-3x在x=0處的導數(shù)是0（錯誤）9.函數(shù)f(x)=tan(x)在x=π/4處的導數(shù)是1（正確）10.函數(shù)f(x)=arcsin(x)在x=0處的導數(shù)是1/√2（正確）四、簡答題1.簡述函數(shù)f(x)=x^2-4x+3的圖像特點。答：函數(shù)f(x)=x^2-4x+3的圖像是一條開口向上的拋物線，其頂點坐標為(2,-1)，與x軸的交點為(1,0)和(3,0)。2.簡述函數(shù)f(x)=|x-1|的圖像特點。答：函數(shù)f(x)=|x-1|的圖像是一個V字形，頂點在(1,0)，在x=1處導數(shù)不存在。3.簡述函數(shù)f(x)=1/x的圖像特點。答：函數(shù)f(x)=1/x的圖像是一條雙曲線，在x=1處的導數(shù)為-1，圖像關于原點對稱。4.簡述函數(shù)f(x)=e^x的圖像特點。答：函數(shù)f(x)=e^x的圖像是一條指數(shù)增長曲線，關于y軸對稱，在x=0處的導數(shù)為1。五、討論題1.討論函數(shù)f(x)=sin(x)和f(x)=cos(x)的圖像特點及導數(shù)關系。答：函數(shù)f(x)=sin(x)和f(x)=cos(x)的圖像都是周期為2π的波形曲線。f(x)=sin(x)在x=π/2處的導數(shù)為1，f(x)=cos(x)在x=π處的導數(shù)為-1。兩者導數(shù)之間的關系是f'(x)=cos(x)-sin(x)。2.討論函數(shù)f(x)=log(x)和f(x)=e^x的圖像特點及導數(shù)關系。答：函數(shù)f(x)=log(x)的圖像是對數(shù)曲線，在x=1處的導數(shù)為1。函數(shù)f(x)=e^x的圖像是指數(shù)增長曲線，關于y軸對稱，在x=0處的導數(shù)為1。兩者導數(shù)之間的關系是f'(x)=1/(xln(e))=1/x。3.討論函數(shù)f(x)=tan(x)和f(x)=cot(x)的圖像特點及導數(shù)關系。答：函數(shù)f(x)=tan(x)的圖像是正切曲線，在x=π/4處的導數(shù)為1。函數(shù)f(x)=cot(x)的圖像是余切曲線，在x=π/4處的導數(shù)為-1。兩者導數(shù)之間的關系是f'(x)=sec^2(x)-csc^2(x)。4.討論函數(shù)f(x)=arcsin(x)和f(x)=arccos(x)的圖像特點及導數(shù)關系。答：函數(shù)f(x)=arcsin(x)的圖像是反正