高考导数培优试题及答案

高考导数培优试题及答案

一、单项选择题(每题2分,共10题)1.函数\(f(x)=x^2\)的导数是(）A.\(x\)B.\(2x\)C.\(x^2\)D.\(2\)2.曲线\(y=\sinx\)在点\((0,0)\)处的切线斜率为()A.\(-1\)B.\(0\)C.\(1\)D.\(2\)3.若\(f(x)=e^x\),则\(f^\prime(1)\)等于()A.\(e\)B.\(1\)C.\(0\)D.\(e^2\)4.函数\(f(x)=x^3-3x\)的单调递增区间是()A.\((-\infty,-1)\)B.\((-1,1)\)C.\((1,+\infty)\)D.\((-\infty,-1)\)和\((1,+\infty)\)5.函数\(f(x)\)在\(x=x_0\)处可导,且\(f^\prime(x_0)=2\),则\(\lim\limits_{\Deltax\to0}\frac{f(x_0+\Deltax)-f(x_0)}{2\Deltax}\)等于()A.\(1\)B.\(2\)C.\(4\)D.\(8\)6.已知函数\(f(x)\)的导函数\(f^\prime(x)\)的图象如图所示,则\(f(x)\)()A.有极小值点\(m\),极大值点\(n\)B.有极小值点\(n\),极大值点\(m\)C.有极小值点\(a\),极大值点\(b\)D.有极小值点\(b\),极大值点\(a\)7.函数\(y=x+\frac{1}{x}(x\gt0)\)的最小值是(）A.\(2\)B.\(-2\)C.\(1\)D.\(0\)8.若函数\(f(x)=ax^3+3x^2+x+b(a\neq0)\)在\(R\)上有极值点,则\(a\)的取值范围是()A.\((0,3)\)B.\((-\infty,0)\cup(0,3)\)C.\((-\infty,0)\cup(3,+\infty)\)D.\((3,+\infty)\)9.已知函数\(f(x)\)满足\(f(x)=xf^\prime(1)+2\lnx\),则\(f^\prime(1)\)等于()A.\(-1\)B.\(1\)C.\(-2\)D.\(2\)10.曲线\(y=\frac{1}{3}x^3-2x+3\)在点\((1,\frac{4}{3})\)处的切线方程为()A.\(x+y-\frac{7}{3}=0\)B.\(x-y-\frac{7}{3}=0\)C.\(x+y+\frac{7}{3}=0\)D.\(x-y+\frac{7}{3}=0\)二、多项选择题(每题2分,共10题)1.下列求导正确的是(）A.\((x^3)^\prime=3x^2\)B.\((\sinx)^\prime=\cosx\)C.\((e^x)^\prime=e^x\)D.\((\lnx)^\prime=\frac{1}{x}\)2.函数\(f(x)=x^3-3x^2+2\)的极值点有()A.\(x=0\)B.\(x=1\)C.\(x=2\)D.\(x=3\)3.若函数\(f(x)\)在区间\((a,b)\)内可导,且\(f^\prime(x)\gt0\),则()A.\(f(x)\)在\((a,b)\)内单调递增B.\(f(x)\)在\((a,b)\)内单调递减C.\(f(x)\)在\((a,b)\)内有极大值D.\(f(x)\)在\((a,b)\)内无极值4.函数\(y=\frac{1}{x}\)的导数的说法正确的是()A.\(y^\prime=-\frac{1}{x^2}\)B.函数在\((0,+\infty)\)上导数小于\(0\)C.函数在\((-\infty,0)\)上导数小于\(0\)D.函数在定义域内导数恒小于\(0\)5.已知函数\(f(x)=ax^2+bx+c(a\neq0)\),其导函数\(f^\prime(x)\),则()A.当\(a\gt0\)时,\(f(x)\)开口向上B.\(f^\prime(x)=2ax+b\)C.若\(f^\prime(x)=0\)的根为\(x_0\),则\(f(x)\)在\(x=x_0\)处取得极值D.当\(a\lt0\)时,\(f(x)\)在对称轴左侧单调递增6.对于函数\(f(x)=x^3-3x\),以下说法正确的是（）A.\(f(x)\)是奇函数B.\(f(x)\)的极大值为\(2\)C.\(f(x)\)的极小值为\(-2\)D.\(f(x)\)的单调递减区间是\((-1,1)\)7.函数\(f(x)\)的导函数\(f^\prime(x)\)的图象可能是()A.一条直线B.一条抛物线C.一个反比例函数图象D.一个正弦函数图象8.若函数\(f(x)\)在\(x=1\)处的导数\(f^\prime(1)=3\),则()A.\(\lim\limits_{\Deltax\to0}\frac{f(1+\Deltax)-f(1)}{\Deltax}=3\)B.曲线\(y=f(x)\)在点\((1,f(1))\)处的切线斜率为\(3\)C.函数\(f(x)\)在\(x=1\)处的切线方程为\(y-f(1)=3(x-1)\)D.函数\(f(x)\)在\(x=1\)附近单调递增9.已知函数\(f(x)=e^x-x\),则()A.\(f^\prime(x)=e^x-1\)B.\(f(x)\)在\((-\infty,0)\)上单调递减C.\(f(x)\)在\((0,+\infty)\)上单调递增D.\(f(x)\)的最小值为\(1\)10.函数\(y=\lnx-x\)的性质正确的是()A.定义域为\((0,+\infty)\)B.\(y^\prime=\frac{1}{x}-1\)C.有极大值\(-1\)D.在\((1,+\infty)\)上单调递减三、判断题(每题2分,共10题)1.常数函数\(y=C\)(\(C\)为常数)的导数是\(0\)。()2.若\(f(x)\)在\(x=x_0\)处可导,则\(f(x)\)在\(x=x_0\)处一定连续。（)3.函数\(f(x)\)的导数\(f^\prime(x)\gt0\)的区间就是\(f(x)\)的单调递增区间。()4.函数\(y=x^2\)在\(R\)上有极值。()5.若\(f^\prime(x_0)=0\),则\(x=x_0\)一定是\(f(x)\)的极值点。(）6.函数\(y=\sinx\)的导数\(y^\prime=\cosx\),\(y^\prime\)的最大值为\(1\)。（)7.曲线\(y=f(x)\)在点\((x_0,f(x_0))\)处的切线与曲线\(y=f(x)\)只有一个公共点。()8.函数\(f(x)=x^3\)的导数\(f^\prime(x)=3x^2\),\(f^\prime(x)\)恒大于\(0\)。（)9.函数\(y=\frac{1}{x}\)在定义域内没有极值。()10.若函数\(f(x)\)的导函数\(f^\prime(x)\)是偶函数,则\(f(x)\)是奇函数。（）四、简答题(每题5分,共4题)1.求函数\(f(x)=x^3-4x^2+5x\)的导数。答案:根据求导公式\((x^n)^\prime=nx^{n-1}\),\(f^\prime(x)=(x^3)^\prime-(4x^2)^\prime+(5x)^\prime=3x^2-8x+5\)。2.求函数\(y=\ln(2x+1)\)的导数。答案:令\(u=2x+1\),\(y=\lnu\),先对\(y\)关于\(u\)求导得\(y^\prime_{u}=\frac{1}{u}\),再对\(u\)关于\(x\)求导得\(u^\prime_{x}=2\),根据复合函数求导法则\(y^\prime=y^\prime_{u}\cdotu^\prime_{x}=\frac{2}{2x+1}\)。3.求函数\(f(x)=x^3-3x\)的极值。答案:\(f^\prime(x)=3x^2-3\),令\(f^\prime(x)=0\),得\(x=\pm1\)。当\(x\lt-1\)时,\(f^\prime(x)\gt0\);当\(-1\ltx\lt1\)时,\(f^\prime(x)\lt0\);当\(x\gt1\)时,\(f^\prime(x)\gt0\)。所以极大值\(f(-1)=2\),极小值\(f(1)=-2\)。4.已知函数\(f(x)=ax^2+bx+c\)在\(x=1\)处取得极值\(2\),且\(f(0)=1\),求\(a\)、\(b\)、\(c\)的值。答案：\(f(0)=1\),则\(c=1\)。\(f^\prime(x)=2ax+b\),因为在\(x=1\)处取得极值\(2\),所以\(f(1)=a+b+c=2\),\(f^\prime(1)=2a+b=0\),将\(c=1\)代入\(a+b+c=2\)得\(a+b=1\),联立\(\begin{cases}2a+b=0\\a+b=1\end{cases}\),解得\(a=-1\),\(b=2\)。五、讨论题(每题5分,共4题)1.讨论函数\(f(x)=x^3-3ax^2+3a^2x-a^3\)的单调性。答案:\(f^\prime(x)=3x^2-6ax+3a^2=3(x-a)^2\)。当\(x\inR\)时,\(f^\prime(x)\geq0\)恒成立,仅当\(x=a\)时\(f^\prime(x)=0\)。所以\(f(x)\)在\((-\infty,+\infty)\)上单调递增。2.讨论函数\(y=\frac{1}{x^2+1}\)的最值情况。答案:\(y^\prime=\frac{-2x}{(x^2+1)^2}\)。令\(y^\prime=0\),得\(x=0\)。当\(x\lt0\)时,\(y^\prime\gt0\);当\(x\gt0\)时,\(y^\prime\lt0\)。所以\(x=0\)时\(y\)有极大值也是最大值\(y(0)=1\),无最小值。3.讨论函数\(f(x)=e^x-ax\)(\(a\inR\))的极值情况。答案：\(f^\prime(x)=e^x-a\)。当\(a\leq0\)时,\(f^\prime(x)\gt0\),\(f(x)\)无极值；当\(a\gt0\)时,令\(f^\prime(x)=0\),得\(x=\lna\)。\(x\lt\lna\)时,\(f^\prime(x)\lt0\)；\(x\gt\lna\)时,\(f^\prime(x)\gt0\),此时\(f(x)\)在\(x=\lna\)处有极小值\(a-a\lna\)。4.讨论函数\(f(x)=\lnx-\frac{1}{2}x^2\)的零点个数。答案:\(f^\prime(x)=\frac{1}{x}-x=\frac{1-x^2}{x}(x\gt0)\)。令\(f^\prime(x)=0\),得\(x=1\)。\(f(x)\)在\((0,1)\)递增,在\((1,+\infty)\)递减,\(f(x)\)极大值\(f(1)=-\frac{1}{2}\lt0\),且\(x\to0^+\)