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导数专题 导数的几何意义

时间:2022-12-06 21:55:53浏览次数:63  
标签:专题 数的几何 导数 ln dfrac sqrt right qquad left

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基础知识

函数\(y=f(x)\)在点\(x=x_0\)处的导数的几何意义是曲线\(y=f(x)\)在点\(P(x_0 ,f(x_0))\)处的切线的斜率,
即:曲线\(y=f(x)\)在点\(P(x_0 ,f(x_0))\)处的切线l的斜率\(k=f'(x_0)\),
切线\(l\)的方程为\(y-f(x_0)=f'(x_0)(x-x_0)\).
image.png
解释
"过点\(x=x_0\)”与"在点\(x=x_0\)处"的区别
曲线\(C:y=f(x)\)在点\(P(x_0 ,y_0)\)处的切线指的是\(P\)为切点的切线,如图一;
过点\(P(x_0 ,y_0)\)的切线是指切线过点\(P\),点\(P\)是否切点均可,切线可多条,如图二.
image.png \(\qquad \qquad\) image.png
 

基本方法

【题型1】在某点处的切线

【典题1】 (多选)若函数\(f(x)\)的图象上存在两个不同的点\(A\),\(B\),使得曲线\(y=f(x)\)在这两点处的切线重合,称函数\(f(x)\)具有\(T\)性质.下列函数中具有\(T\)性质的有(  )
 A.\(y=e^x-x\) \(\qquad \qquad \qquad \qquad\) B.\(y=x^4-x^2\) \(\qquad \qquad \qquad \qquad\) C.\(y=x^3\) \(\qquad \qquad \qquad \qquad\) D.\(y=x+\sin ⁡x\)
解析 由题意可得,性质T指函数\(f(x)\)图象上有两个不同点的切线是重合的,即两个不同点所对应的导数值相等,且该点处函数的切线方程也相同.
对于\(A\)选项,\(y=e^x-x\),则\(y'=e^x-1\),导函数为增函数,不存在不同的两个\(x\)使得导数值相等,故\(A\)不符合;
对于\(B\)选项,\(y'=4x^3-2x\),
设两切点分别为\((x_1,x_1^4-x_1^2 )\),\((x_2,x_2^4-x_2^2 )\),且\(4x_1^3-2x_1=4x_2^3-2x_2\),
取\(x_1=-\dfrac{\sqrt{2}}{2}\),\(x_2=\dfrac{\sqrt{2}}{2}\),
则\(y_1=-\dfrac{1}{4}=y_2\),两切点处的导数值为\(y'=0\),
两切点连线的直线斜率为\(k=\dfrac{y_2-y_1}{x_2-x_1}=0\),
所以两切点处的导数值等于两切点连线的斜率,
符合性质\(T\),所以\(B\)选项符合;
对于\(C\)选项,设两切点分别为\((x_1,x_1^3 )\)和\((x_2,x_2^3 )\),
则两切点处的导数值相等有:\(3x_1^2=3x_2^2\),解得:\(x_1=-x_2\),
令\(x_1=a\),则\(x_2=-a\),两切点处的导数\(y'=3a^2\),
两切点连线的斜率为 \(k=\dfrac{a^3-\left(-a^3\right)}{a-(-a)}=a^2\),
则\(3a^2=a^2\),得\(a=0\),两切点重合,不符合题意,所以\(C\)选项不符合;
对于\(D\)选项,\(y'=1+\cos ⁡x\),设两切点的横坐标分别为\(x_1\)和\(x_2\),
则\(1+\cos ⁡x_1=1+\cos ⁡x_2\),所以\(\cos ⁡x_1=\cos ⁡x_2\),
取\(x_1=\dfrac{\pi}{2}\),\(x_2=\dfrac{5\pi}{2}\),则\(y_1=\dfrac{\pi}{2}+1\),\(y_2=\dfrac{5\pi}{2}+1\),
两切点处的导数值为\(y'=1\),两切点连线的直线斜率为 \(k=\dfrac{y_2-y_1}{x_2-x_1}=1\),
所以两切点处的导数值等于两切点连线的斜率,符合性质\(T\),所以\(D\)选项符合.
故选:\(BD\).
 

【典题2】 已知函数 \(f(x)=\dfrac{1}{2} \sin \left(2 x+\dfrac{\pi}{3}\right)\)的图像在\((x_1,f(x_1 ))\)处的切线与在\((x_2,f(x_2 ))\)处的切线相互垂直,那么\(|x_1-x_2 |\)的最小值是(  )
 A. \(\dfrac{\pi}{4}\) \(\qquad \qquad \qquad \qquad\) B.\(\dfrac{\pi}{2}\)\(\qquad \qquad \qquad \qquad\) C.\(π\) \(\qquad \qquad \qquad \qquad\) D.\(2π\)
解析 由函数 \(f(x)=\dfrac{1}{2} \sin \left(2 x+\dfrac{\pi}{3}\right)\)得: \(f^{\prime}(x)=\cos \left(2 x+\dfrac{\pi}{3}\right)\),
显然若图象在\((x_1,f(x_1 ))\)处的切线与在\((x_2,f(x_2 ))\)处的切线相互垂直,
只需\(f' (x_1 ) f' (x_2 )=-1\),
不妨设\(2x_1+\dfrac{\pi}{3}=2nπ\),\(n∈Z\),\(2x_2+\dfrac{\pi}{3}=2mπ+π\),\(m∈Z\),
显然\(x_1-x_2=(n-m)π-\dfrac{\pi}{2}\),
显然当\(n=m\)时,\(|x_1-x_2 |\)的最小值是\(\dfrac{\pi}{2}\).
故选:\(B\).
 

【巩固练习】

1.若函数\(f(x)=2\ln x+4x^2+bx+5\)的图象上的任意一点的切线斜率都大于\(0\),则\(b\)的取值范围是(  )
 A.\((-∞,-8)\) \(\qquad \qquad\) B.\((-8,+∞)\) \(\qquad \qquad\) C.\((-∞,8)\) \(\qquad \qquad\) D.\((8,+∞)\)
 

2.(多选)若以曲线\(y=f(x)\)上任意一点\(M(x,y)\)为切点作切线\(l\),曲线上总存在异于点\(M\)的点\(N(x',y')\),使得以点\(N\)为切点的切线l'满足\(l∥l'\),则称曲线\(y=f(x)\)具有“可平行性”.下列曲线具有“可平行性”的是(  )
 A.\(y=x+\dfrac{1}{x}\) \(\qquad \qquad \qquad \qquad\) B.\(y=x^3-x\)
 C.\(y=\sin x\) \(\qquad \qquad \qquad \qquad\) D.\(y=(x-2)^2+\ln x\)
 

3.设对于曲线\(f(x)=-e^x-x\)上任一点处的切线\(l_1\),总存在曲线\(g(x)=kx+\cos ⁡x\)上一点处的切线\(l_2\),使得\(l_1⊥l_2\),则实数\(k\)的取值范围是\(\underline{\quad \quad}\) .
 

4.若函数\(f(x)=ax+\sin x\)的图象上存在互相垂直的切线,则实数\(a\)的值为\(\underline{\quad \quad}\).
 
 

参考答案

  1. 答案 \(B\)
    解析 根据题意,函数\(f(x)=2\ln x+4x^2+bx+5\),其定义域为\((0,+∞)\),
    其导数\(f'(x)=\dfrac{2}{x}+8x+b\),
    若函数\(f(x)\)的图象上的任意一点的切线斜率都大于\(0\),
    则有\(f'(x)=\dfrac{2}{x}+8x+b>0\)在\((0,+∞)\)上恒成立,
    变形可得\(b>-\left(\dfrac{2}{x}+8 x\right)\)在\((0,+∞)\)上恒成立,
    又由\(\dfrac{2}{x}+8 x \geq 2 \times \sqrt{\dfrac{2}{x} \times 8 x}=8\),
    当且仅当\(x=\dfrac{1}{2}\)时等号成立,即\(\dfrac{2}{x}+8x\)有最小值\(8\),
    若 \(b>-\left(\dfrac{2}{x}+8 x\right)\)在\((0,+∞)\)上恒成立,
    必有\(b>-8\),即\(b\)的取值范围为\((-8,+∞)\);
    故选\(B\).

  2. 答案 \(AC\)
    解析 设\(f'(x)\)的值域为\(Q\),
    由题意,曲线\(y=f(x)\)具有“可平行性”等价于对任意的\(a∈Q\),
    方程\(y'=a\)至少有两个根,
    对于\(A\),由\(y'=1-\dfrac{1}{x^2} =a\)(\(x≠0\)且\(a<1\)),得\(\dfrac{1}{x^2} =1-a\),
    此方程有两个不同的根,符合题意;
    对于\(B\),由\(y'=-1\)时,\(x\)的取值唯一,只有\(0\),不合题意;
    对于\(C\),由\(y'=\cos x\)和三角函数的周期性可知,\(\cos x=a(-1≤a≤1)\)的解有无穷多个,符合题意;
    对于\(D\),\(y'=2x-4+\dfrac{1}{x}\),
    令\(2x-4+\dfrac{1}{x}=a(a≥2\sqrt{2}-4)\),则有\(2x^2-(4+a)x+1=0\),
    当\(△=0\)时解唯一,不合题意.
    故选:\(AC\).

  3. 答案 \([0,1]\)
    解析 \(f(x)=-e^x-x\),其导数\(f' (x)=-e^x-1\),
    有\(f' (x)<-1\),则\(-f' (x)>1\),则\(\dfrac{1}{-f^{\prime}(x)} \in(0,1)\),
    由\(g(x)=kx+\cos ⁡x\),得\(g' (x)=k-\sin ⁡x\),
    \(\because -\sin ⁡x∈[-1,1]\),\(\therefore k-\sin ⁡x∈[-1+k,1+k]\),
    要使过曲线\(f(x)=-e^x-x\)上任一点处的切线\(l_1\),总存在曲线\(g(x)=kx+\cos ⁡x\)上一点处的切线\(l_2\),使得\(l_1⊥l_2\),
    则有\(\left\{\begin{array}{l} -1+k \leqslant 0 \\ 1+k \geqslant 1 \end{array}\right.\),解得\(0⩽k⩽1\).
    \(\therefore\)实数\(k\)的取值范围是\([0,1]\).

  4. 答案 \(0\)
    解析 \(\because f(x)=ax+\sin x\),\(\therefore f'(x)=a+\cos x\),
    假设函数\(f(x)=ax+\sin x\)的图象上存在互相垂直的切线,
    不妨设在\(x=m\)与\(x=n\)处的切线互相垂直
    则\((a+\cos m)(a+\cos n)=-1\)
    \(\therefore a^2+(\cos m+\cos n)a+(\cos m\cos n+1)=0 \quad (*)\)
    因为\(a\)的值必然存在,即方程\((*)\)必然有解,所以
    判别式\(△=(\cos m+\cos n)^2-4(\cos m\cos n+1)≥0\)
    所以\(\cos ^2⁡m+\cos ^2⁡n-2\cos m\cos n=(\cos m-\cos n)^2≥4\)
    解得\(\cos m-\cos n≥2\)或\(\cos m-\cos n≤-2\)
    由于\(|\cos x|≤1\),
    所以有\(\cos m=1\),\(\cos n=-1\)或\(\cos m=-1\),\(\cos n=1\),且\(△=0\)
    所以\((*)\)变为:\(a^2=0\)所以\(a=0\).
    故答案为:\(0\) .
     

【题型2】过某点处的切线

【典题1】 已知函数\(f(x)=2x^3-ax\),若\(a=1\)时,直线\(y=k(x-1)+1\)与曲线\(y=f(x)\)相切,则\(k\)的所有可能的取值为\(\underline{\quad \quad}\);若\(a∈R\)时,直线\(y=k(x-2)\)与曲线\(y=f(x)\)相切,且满足条件的\(k\)的值有且只有\(3\)个,则\(a\)的取值范围为\(\underline{\quad \quad}\).
解析 当\(a=1\)时,\(f(x)=2x^3-x\),求导得\(f' (x)=6x^2-1\),
设直线\(y=k_1 (x-1)+1\)与曲线\(y=f(x)\)相切的切点为\((x_0,2x_0^3-x_0 )\),
则\(k_1=f' (x_0 )=6x_0^2-1\),且\(2x_0^3-x_0=k_1 (x_0-1)+1\),
即\(2x_0^3-x_0=(6x_0^2-1)(x_0-1)+1\),
整理得\((x_0-1)^2 (2x_0+1)=0\),解得\(x_0=-\dfrac{1}{2}\)或\(x_0=1\),
则\(k_1=f' \left(-\dfrac{1}{2}\right)=\dfrac{1}{2}\)或\(k_1=f' (1)=5\),
所以\(k_1\)的所有可能的取值为\(\dfrac{1}{2}\),\(5\);
由\(f(x)=2x^3-ax\),求导得\(f' (x)=6x^2-a\),
设直线\(y=k(x-2)\)与曲线\(y=f(x)\)相切的切点为\((t,2t^3-at)\),
于是得\(k=f' (t)=6t^2-a\),且\(2t^3-at=k(t-2)\),则\(k=2t^3\),
显然函数\(y=2t^3\)在\(R\)上单调递增,
因直线\(y=k(x-2)\)与曲线\(y=f(x)\)相切的\(k\)的值有且只有\(3\)个,
则有直线\(y=k(x-2)\)与曲线\(y=f(x)\)相切的切点横坐标值有且只有\(3\)个,
即方程\(a=6t^2-2t^3\)有\(3\)个不等实根,
令\(g(t)=2t^3-6t^2+a\),求导得\(g' (t)=6t^2-12t=6t(t-2)\),
当\(t<0\)或\(t>2\)时,\(g' (t)>0\),当\(0<t<2\)时,\(g' (t)<0\),
即函数\(g(t)\)在\((-∞,0)\),\((2,+∞)\)上递增,在\((0,2)\)上递减,
当\(t=0\)时,\(g(t)\)取得极大值\(g(0)=a\),
当\(t=2\)时,\(g(t)\)取得极小值\(g(2)=a-8\),
方程\(a=6t^2-2t^3\)有\(3\)个不等实根,当且仅当函数\(g(t)\)有\(3\)个不同的零点,
因此 \(\left\{\begin{array}{l} a>0 \\ a-8<0 \end{array}\right.\),解得\(0<a<8\),
所以\(a\)的取值范围为\((0,8)\).
故答案为:\(\dfrac{1}{2}\),\(5\);\((0,8)\).
 

【典题2】 已知函数\(f(x)=x+\dfrac{1}{x}\),过点\(P(1,0)\)作函数\(y=f(x)\)图像的两条切线,切点分别为\(M\),\(N\).则下列说法正确的是(  )
 A.\(PM⊥PN\) \(\qquad \qquad \qquad \qquad\) B.直线MN的方程为\(2x-y+1=0\)
 C. \(|M N|=2 \sqrt{10}\) \(\qquad \qquad \qquad \qquad\) D.\(△PMN\)的面积为\(3\sqrt{2}\)
解析 因为\(f(1)=1+1=2\),所以\(P(1,0)\)没有在函数的图象上,
\(f^{\prime}(x)=1-\dfrac{1}{x^2}=\dfrac{x^2-1}{x^2}\),
设切点坐标为\((a,b)(a≠0)\),
当\(a=1\)时,\(f(1)=2\),\(x=1\)不与\(f(x)=x+\dfrac{1}{x}\)相切,所以\(a≠1\),
\(f^{\prime}(a)=\dfrac{a^2-1}{a^2}=\dfrac{b}{a-1}\),
又因为 \(a+\dfrac{1}{a}=b\),解得\(a=-1±\sqrt{2}\),
即\((-1-\sqrt{2},-2\sqrt{2})\),\((-1+\sqrt{2},2\sqrt{2})\),
所以 \(k_{P M} \times k_{P N}=\dfrac{2 \sqrt{2}}{2+\sqrt{2}} \times \dfrac{2 \sqrt{2}}{\sqrt{2}-2}=-4 \neq-1\),故\(A\)错误;
\(k_{N M}=\dfrac{2 \sqrt{2}+2 \sqrt{2}}{2 \sqrt{2}}=2\),
所以直线\(MN\)的方程为\(y=2(x-1)\),即\(2x-y+2=0\),故\(B\)错误;
\(|M N|=\sqrt{(-1+\sqrt{2}+1+\sqrt{2})^2+(2 \sqrt{2}+2 \sqrt{2})^2}=2 \sqrt{10}\),故\(C\)正确;
\(P(1,0)\)到直线\(MN\)的距离为 \(d=\dfrac{|2-0+2|}{\sqrt{4+1}}=\dfrac{4 \sqrt{5}}{5}\),
所以\(△PMN\)的面积为\(\dfrac{1}{2}|M N| d=\dfrac{1}{2} \times 2 \sqrt{10} \times \dfrac{4 \sqrt{5}}{5}=4 \sqrt{2}\),故\(D\)错误.
故选:\(C\).
 

【巩固练习】

1.已知曲线\(y=\ln x\)的切线过原点,则此切线的斜率为(  )
 A.\(e\) \(\qquad \qquad \qquad \qquad\) B.\(-e\) \(\qquad \qquad \qquad \qquad\) C.\(\dfrac{1}{e}\) \(\qquad \qquad \qquad \qquad\) D.\(-\dfrac{1}{e}\)
 

2.过直线\(y=x-1\)上一点\(P\)可以作曲线\(f(x)=x-\ln ⁡x\)的两条切线,则点\(P\)横坐标\(t\)的取值范围为(  )
 A.\(0<t<1\) \(\qquad \qquad\) B.\(1<t<e\) \(\qquad \qquad\) C.\(0<t<e\) \(\qquad \qquad\) D.\(\dfrac{1}{e}<t<1\)
 

3.(多选)若过点\((1,a)\)可以作出曲线\(y=(x-1)e^x\)的切线\(l\),且\(l\)最多有\(n\)条,\(n∈N^*\),则(  )
 A.\(a≤0\) \(\qquad \qquad \qquad \qquad \qquad \qquad \qquad\) B.当\(n=2\)时,\(a\)值唯一
 C.当\(n=1\)时, \(a<-\dfrac{4}{e}\) \(\qquad \qquad \qquad \qquad\) D.\(na\)的值可以取到\(-4\)
 

参考答案

  1. 答案 \(C\)
    解析 设切点坐标为\((a,\ln a)\),
    \(\because y=\ln x\),\(\therefore y'=\dfrac{1}{x}\),切线的斜率是\(\dfrac{1}{a}\),
    切线的方程为\(y-\ln a=\dfrac{1}{a}(x-a)\),
    将\((0,0)\)代入可得\(\ln a=1\),\(\therefore a=e\),
    \(\therefore\)切线的斜率是\(\dfrac{1}{a}=\dfrac{1}{e}\);
    故选:\(C\).

  2. 答案 \(C\)
    解析 设切点为\((m,m-\ln ⁡m)\),\(m>0\),
    由\(f(x)=x-\ln ⁡x\)的导数为\(f' (x)=1-\dfrac{1}{x}\),
    可得切线的斜率为 \(1-\dfrac{1}{m}\),
    又\(P(t,t-1)\),可得 \(\dfrac{m-\ln m-t+1}{m-t}=1-\dfrac{1}{m}\),
    化为\(t=2m-m\ln m\),
    设\(g(x)=2x-x\ln ⁡x\),
    可得\(g' (x)=2-(1+\ln ⁡x)=1-\ln ⁡x\),
    当\(x>e\)时,\(g' (x)<0\),\(g(x)\)递减;
    当\(0<x<e\)时,\(g' (x)>0\),\(g(x)\)递增.
    可得\(g(x)\)在\(x=e\)处取得最大值\(e\),
    可得当\(0<t<e\)时,方程\(t=2m-m\ln m\)有两解,
    故选:\(C\).

  3. 答案 \(ABD\)
    解析 \(y'=e^x+(x-1)e^x=xe^x\),
    设切线\(l\)的切点为\((x_0,y_0 )\),则切线\(l\)的斜率为\(k=x_0 e^{x_0}\),
    又\(y_0=(x_0-1) e^{x_0}\),
    故由点斜式方程可知切线l的方程为\(y-(x_0-1) e^{x_0}=x_0 e^{x_0} (x-x_0 )\),
    \(\therefore x_0 e^{x_0}=\dfrac{\left(x_0-1\right) e^{x_0-a}}{x_0-1}=e^{x_0}-\dfrac{a}{x_0-1}\),
    则\(a=-e^{x_0} (x_0-1)^2\),
    对于\(A\),由于\(a=-e^{x_0} (x_0-1)^2\),\(e^{x_0}>0,(x_0-1)^2⩾0\),
    故\(a≤0\),选项\(A\)正确;
    对于\(B\),令\(g(x)=-e^x (x-1)^2\),则\(g' (x)=-e^x (x-1)(x+1)\),
    易知\(g(x)\)在\((-∞,-1)\),\((1,+∞)\)上单调递减,在\((-1,1)\)上单调递增,
    且 \(g(-1)=-\dfrac{4}{e}\),\(g(1)=0\),当\(x→-∞\),时,\(g(x)→0\),
    当\(x→+∞\)时,\(g(x)→-∞\),
    作出函数\(g(x)\)的草图如下,
    image.png
    当\(n=2\)时,即\(x_0\)的值有两个,由图象可知,
    当且仅当 \(a=g(-1)=-\dfrac{4}{e}\)时,\(x_0\)的值有两个,选项\(B\)正确;
    对于\(C\),由图象可知,当\(a=0\)或\(a<-\dfrac{4}{e}\)时,\(x_0\)的值唯一,此时\(n=1\),选项\(C\)错误;
    对于\(D\),由图象可知,\(n=1\)或\(2\)或\(3\),若\(na=-4\),则当\(n=1\)时,\(a=-4\),
    由选项\(C\)可知,此时\(a=0\)或\(a<-\dfrac{4}{e}\),而 \(-4<-\dfrac{4}{e}\),
    故\(na\)可能取到\(-4\),选项\(D\)正确.
    故选:\(ABD\).
     

【题型3】两曲线的公切线

【典题1】 若直线\(y=kx+b\)是曲线 \(y=e^{x-2}\)的切线,也是曲线\(y=e^x-1\)的切线,则\(b=\)\(\underline{\quad \quad}\).
解析 设直线\(y=kx+b\)是曲线 \(y=e^{x-2}\)和\(y=e^x-1\)的切点分别为 \(\left(x_1, e^{x_1-2}\right)\)和 \(\left(x_2, e^{x_2}-1\right)\),
则切线分别为 \(y-e^{x_1-2}=e^{x_1-2}\left(x-x_1\right)\), \(y-e^{x_2}+1=e^{x_2}\left(x-x_2\right)\),
化简得: \(y=e^{x_1-2} x+e^{x_1-2}-x_1 e^{x_1-2}\), \(y=e^{x_2} x+e^{x_2}-1-x_2 e^{x_2}\),
依题意有: \(\left\{\begin{array}{l} e^{x_1-2}=e^{x_2} \\ e^{x_1-2}-x_1 e^{x_1-2}=e^{x_2}-1-x_2 e^{x_2} \end{array}\right.\),
\(\therefore x_1-2=x_2\),\(x_2=-\ln ⁡2\),
则 \(b=e^{x_2}-1-x_2 e^{x_2}=\dfrac{1}{2} \ln 2-\dfrac{1}{2}\).
故答案为:\(\dfrac{1}{2} \ln 2-\dfrac{1}{2}\) .
 

【典题2】 若曲线\(C_1:y=ax^2 (a>0)\)与曲线\(C_2:y=e^x\)存在公共切线,则\(a\)的取值范围为(  )
 A. \(\left[\dfrac{e^2}{8},+\infty\right)\) \(\qquad \qquad\) B. \(\left(0, \dfrac{e^2}{8}\right]\) \(\qquad \qquad\)C. \(\left[\dfrac{e^2}{4},+\infty\right)\) \(\qquad \qquad\) D. \(\left(0, \dfrac{e^2}{4}\right]\)
解析 由\(y=ax^2 (a>0)\),得\(y'=2ax\),
由\(y=e^x\),得\(y'=e^x\),
\(\because\)曲线\(C_1:y=ax^2 (a>0)\)与曲线\(C_2:y=e^x\)存在公共切线,则
设公切线与曲线\(C_1\)切于点\((x_1,ax_1^2)\),与曲线\(C_2\)切于点\((x_2,e^{x_2})\),
则\(2 a x_1=e^{x_2}=\dfrac{e^{x_2}-a x_1{ }^2}{x_2-x_1}\),
将\(e^{x_2}=2 a x_1\)代入 \(2 a x_1=\dfrac{e^{x_2}-a x_1{ }^2}{x_2-x_1}\),可得\(2x_2=x_1+2\),
\(\therefore a=\dfrac{e^{\frac{x_1}{2}+1}}{2 x_1}\),记 \(f(x)=\dfrac{e^{\frac{x}{2}+1}}{2 x}\),
则\(f^{\prime}(x)=\dfrac{e^{\frac{x}{2}+1}(x-2)}{4 x^2}\),当\(x∈(0,2)\)时,\(f'(x)<0\).
\(\therefore\)当\(x=2\)时, \(f(x)_{\min }=\dfrac{e^2}{4}\).
\(\therefore a\)的范围是\(\left[\dfrac{e^2}{4},+\infty\right)\) .
故选:\(C\).
 

【巩固练习】

1.若曲线\(y=x-\ln ⁡x\)与曲线\(y=ax^3+x+1\)在公共点处有相同的切线,则实数\(a\)等于(  )
 A. \(\dfrac{e^2}{3}\) \(\qquad \qquad \qquad \qquad\) B.\(-\dfrac{e^2}{3}\) \(\qquad \qquad \qquad \qquad\) C.\(-\dfrac{e}{3}\) \(\qquad \qquad \qquad \qquad\) D.\(\dfrac{e}{3}\)
 

2.若二次函数\(f(x)=x^2+1\)的图象与曲线\(C:g(x)=ae^x+1(a>0)\)存在公共切线,则实数\(a\)的取值范围为\(\underline{\quad \quad}\) .
 

参考答案

  1. 答案 \(B\)
    解析 设公共点的横坐标为\(t\),由\(y'=1-\dfrac{1}{x}\),以及\(y'=3ax^2+1\),
    则\(\left\{\begin{array}{l} t-\ln t=a t^3+t+1 \\ 1-\dfrac{1}{t}=3 a t^2+1 \end{array}\right.\),由第二个方程得\(t^3=-\dfrac{1}{3 a}\),
    ①再代入第一个方程\(\ln ⁡t=-\dfrac{2}{3}\),结合①解得\(a=-\dfrac{e^2}{3}\).
    故选:\(B\).

  2. 答案 \(\left(0, \dfrac{4}{e^2}\right]\)
    解析 \(f(x)=x^2+1\)的导数为\(f'(x)=2x\),
    \(g(x)=ae^x+1\)的导数为\(g'(x)=ae^x\),
    设公切线与\(f(x)=x^2+1\)的图象切于点\((x_1,x_1^2+1)\),
    与曲线\(C:g(x)=ae^x+1\)切于点 \(\left(x_2, a e^{x^2+1}\right)\),
    \(\therefore 2 x_1=a e^{x 2}=\dfrac{a e^{x_2}+1-\left(x_1{ }^2+1\right)}{x_2-x_1}=\dfrac{a e^{x_2}-x_1{ }^2}{x_2-x_1}\),
    化简可得\(2 x_1=\dfrac{2 x_1-x_1^2}{x_2-x_1}\),得\(x_1=0\)或\(2x_2=x_1+2\),
    \(\because 2x_1=ae^{x_2}\),且\(a>0\),\(\therefore x_1>0\),
    则\(2x_2=x_1+2>2\),即\(x_2>1\),
    由\(2 x_1=a e^{x_2}\),得\(a=\dfrac{2 x_1}{e^{x_2}}=\dfrac{4\left(x_2-1\right)}{e^{x_2}}\),
    设\(h(x)=\dfrac{4(x-1)}{e^x}(x>1)\),则\(h^{\prime}(x)=\dfrac{4(2-x)}{e^x}\),
    \(\therefore h(x)\)在\((1,2)\)上递增,在\((2,+∞)\)上递减,
    \(\therefore h(x)_{\max }=h(2)=\dfrac{4}{e^2}\),
    \(\therefore\)实数\(a\)的取值范围为\(\left(0, \dfrac{4}{e^2}\right]\),
    故答案为:\(\left(0, \dfrac{4}{e^2}\right]\).
     

【题型4】综合运用

【典题1】 对任意的\(x>0\), 总有\(f(x)=a-x-|\lg ⁡x|≤0\), 则\(a\)的取值范围是\(\underline{\quad \quad}\).
解析 原问题即\(|\lg ⁡x |≥-x+a\)在区间\((0,+∞)\)上恒成立, 考查临界情况,
即函数\(g(x)=|\lg ⁡x|\)与\(h(x)=-x+a\)相切时的情形,很明显切点横坐标位于区间\((0,1)\)内,
此时,\(g(x)=-\lg ⁡x\), \(g^{\prime}(x)=\dfrac{1}{x \ln 10}\),
由\(g' (x)=-1\)可得: \(x=-\dfrac{1}{\ln 10}=-\lg e\),
则切点坐标为:\((-\lg ⁡e,-\lg ⁡(\lg ⁡e))\),
切线方程为:\(y+\lg ⁡(\lg ⁡e)=x+\lg ⁡e\),
令\(x=0\)可得纵截距为:\(\lg ⁡e-\lg ⁡(\lg ⁡e)\),
结合如图所示的函数图象可得则\(a\)的取值范围是\((-∞,\lg ⁡e-\lg ⁡(\lg ⁡e)]\).
image.png
 

【典题2】 直线\(y=m\)分别与曲线\(y=2(x+1)\), 与\(y=x+\ln ⁡x\)交于点\(A\),\(B\),则\(|AB|\)的最小值为\(\underline{\quad \quad}\).
解析 作出曲线\(y=2(x+1)\), 与\(y=x+\ln ⁡x\)草图如下
image.png
过\(B\)作\(BC⊥AC\), \(|A B|=\dfrac{|B C|}{\sin 60^{\circ}}=\dfrac{|B C|}{\dfrac{\sqrt{3}}{2}}\),
要使\(|AB|\)取到最小值, 只需\(|BC|\)取到最小值即可,
为此对\(y=x+\ln ⁡x\)进行求导得\(y'=1+\dfrac{1}{x}\),
令\(y'=2\), 解得\(x=1\), 代入\(y=x+\ln ⁡x\),知\(y=1\),
所以当\(|BC|\)取到最小值时,\(m=1\),
易知 \(|A B|=1-\left(-\dfrac{1}{2}\right)=\dfrac{3}{2}\).
 

【巩固练习】

1.已知\(M(1,0)\),\(N\)是曲线\(y=e^x\)上一点,则\(|MN|\)的最小值为(  )
 A.\(1\) \(\qquad \qquad \qquad \qquad\) B.\(\sqrt{2}\) \(\qquad \qquad \qquad \qquad\) C.\(e\) \(\qquad \qquad \qquad \qquad\) D. \(\sqrt{e^4+1}\)
 

2.已知定义在 \(\left[\dfrac{1}{\pi}, \pi\right]\)上的函数\(f(x)\),满足 \(f(x)=f\left(\dfrac{1}{x}\right)\),且当\(x∈[1,π]\)时\(f(x)=\ln ⁡x\),若函数\(g(x)=f(x)-ax\)在 \(\left[\dfrac{1}{\pi}, \pi\right]\)上有唯一的零点,则实数\(a\)的取值范围是\(\underline{\quad \quad}\).
 

参考答案

  1. 答案 \(B\)
    解析 \(y=e^x\)的导数为\(y'=e^x\).
    设\(N(m,e^m)\),可得过\(N\)的切线的斜率为\(e^m\).
    当\(MN\)垂直于切线时,\(|MN|\)取得最小值,
    可得 \(\dfrac{e^m}{m-1}=-\dfrac{1}{e^m}\),则 \(e^{2 m}+m=1\).
    因为\(f(x)=e^{2x}+x\)单调递增,且\(f(0)=1\),所以\(m=0\).
    所以\(|MN|\)的最小值为 \(\sqrt{1^2+1^2}=\sqrt{2}\).
    故选:\(B\).

  2. 答案 \(\left(\dfrac{1}{e}, \pi \ln \pi\right] \cup\{0\}\).
    解析 \(\because f(x)=f\left(\dfrac{1}{x}\right)\),\(x∈[1,π]\)时\(f(x)=\ln ⁡x\),
    \(\therefore x \in\left(-\dfrac{1}{\pi}, 1\right]\)时,\(\dfrac{1}{x}∈[1,π]\), \(f\left(\dfrac{1}{x}\right)=\ln \dfrac{1}{x}=f(x)\),
    \(f(x)=-\ln ⁡x\),\(g(x)\)零点, 就是\(y=f(x)\))与\(y=ax\)的交点,
    画出两函数图象,如图,由图知, \(k_{O A}=\pi \ln \pi\),
    过原点与\(y=\ln ⁡x\)相切的直线斜率为\(\dfrac{1}{e}\),
    所有直线与曲线有一个交点的\(a\)的范围是\(\left(\dfrac{1}{e}, \pi \ln \pi\right] \cup\{0\}\).
    image.png
     

分层练习

【A组---基础题】

1.若函数\(y=f(x)\)的图象上存在两点,使得函数的图象在这两点处的切线互相垂直,则称\(y=f(x)\)具有\(T\)性质.下列函数中具有\(T\)性质的是(  )
 A. \(y=\sin ⁡x\) \(\qquad \qquad \qquad \qquad\) B. \(y=\ln ⁡x\) \(\qquad \qquad \qquad \qquad\) C. \(y=e^x\) \(\qquad \qquad \qquad \qquad\) D.\(y=x^3\)
 

2.若过点\(P(-1,m)\)可以作三条直线与曲线\(C:y=xe^x\)相切,则\(m\)的取值范围是(  )
 A.\(\left(-\dfrac{3}{e^2},+\infty\right)\) \(\qquad \qquad\) B.\(\left(-\dfrac{1}{e}, 0\right)\) \(\qquad \qquad\) C.\((0,+∞)\) \(\qquad \qquad\) D. \(\left(-\dfrac{3}{e^2},-\dfrac{1}{e^2}\right)\)
 

3.曲线\(y=e^x\)上的点到直线\(x-y-3=0\)的距离的最小值为(  )
 A.\(\sqrt{2}\) \(\qquad \qquad \qquad \qquad\) B.\(2\) \(\qquad \qquad \qquad \qquad\) C.\(2\sqrt{2}\) \(\qquad \qquad \qquad \qquad\) D.\(4\)
 

4.已知函数\(f(x)=\sin ⁡x-\cos ⁡x\), 直线L过原点且与曲线\(y=f(x)\)相切, 其切点的横坐标从小到大依次排列为 \(x_1\),\(x_2\),\(x_3\),⋯,\(x_n\),⋯, 则下列说法正确的是 (  )
 A. \(|f(x_n )|=1\) \(\qquad \qquad \qquad \qquad \qquad \qquad\) B.数列 \(\left\{x_n\right\}\)为等差数列
 C. \(x_n=\tan \left(x_n+\dfrac{\pi}{4}\right)\) \(\qquad \qquad \qquad \qquad\) D. \(\left[f\left(x_n\right)\right]^2=\dfrac{2 x_n^2}{x_n^2+1}\)
 

5.过平面内一点\(P\)作曲线\(y=|\ln ⁡x|\)两条互相垂直的切线\(l_1\),\(l_2\),切点为\(P_1\),\(P_2\) (\(P_1\),\(P_2\) 不重合),设直线\(l_1\),\(l_2\)分别与\(y\)轴交于点\(A\),\(B\),则下列结论正确的个数是(  )
①\(P_1 P_2\)两点的横坐标之积为定值; ②直线\(P_1 P_2\)的斜率为定值;
③线段\(AB\)的长度为定值; ④三角形\(ABP\)面积的取值范围为\((0,1]\).
 A.\(1\) \(\qquad \qquad \qquad \qquad\) B.\(2\) \(\qquad \qquad \qquad \qquad\) C.\(3\) \(\qquad \qquad \qquad \qquad\) D.\(4\)
 

6.已知函数\(f(x)=\ln ⁡x-x+t\),直线\(l:y=-\dfrac{1}{2} x+\ln ⁡2+2\),点\(P(x_0,f(x_0 ))\)在函数\(y=f(x)\)图像上,则以下说法正确的是(  )
 A.若直线\(l\)是曲线\(y=f(x)\)的切线,则\(t=-3\)
 B.若直线\(l\)与曲线\(y=f(x)\)无公共点,则\(t>-3\)
 C.若\(t=-2\),则点\(P\)到直线\(l\)的最短距离为 \(\sqrt{5}\)
 D.若\(t=-2\),当点\(P\)到直线\(l\)的距离最短时,\(x_0=2\)
 

7.(多选)若过点\(P(-1,t)\)最多可以作出\(n(n∈N^* )\)条直线与函数 \(f(x)=\dfrac{x+1}{e^x}\)的图像相切,则(  )
 A.\(th\)可以等于\(2022\) \(\qquad \qquad \qquad \qquad\) B.\(n\)不可以等于\(3\)
 C.\(te+n>3\) \(\qquad \qquad \qquad \qquad\) D.\(n=1\)时, \(t \in\{0\} \cup\left(\dfrac{4}{e},+\infty\right)\)
 

8.已知曲线\(y=x^2+\dfrac{5}{4}\)在点\(\left(\dfrac{1}{2}, \dfrac{3}{2}\right)\)处的切线为\(l\),数列\(\left\{a_n\right\}\)的首项为\(1\),点\(\left(a_n, a_{n+1}\right)\)\((n∈N^* )\)为切线\(l\)上一点,则数列\(\left\{a_n\right\}\)的前n项和为 \(\underline{\quad \quad}\) .
 

9.若直线\(y=kx+b\)是曲线\(y=\ln ⁡x+2\)的切线,也是曲线\(y=\ln ⁡(x+2)\)的切线,则\(b=\)\(\underline{\quad \quad}\).
 

10.曲线\(y=\ln ⁡(2x-1)\)上的点到直线\(2x-y+8=0\)的最短距离是\(\underline{\quad \quad}\) .
 

11.设函数\(y=2x^2-2(0⩽x⩽1)\)的图象为曲线\(C\),\(R(x_0,y_0 )\)为曲线\(C\)上任意一点过点\(R\)的直线\(PQ\)与曲线\(C\)相切,且与\(x\)轴交于点\(P\),与\(y\)轴交于点\(Q\).当三角形\(POQ\)的面积取得最小值时,\(x_0\)的值为 \(\underline{\quad \quad}\) .
 

参考答案

  1. 答案 \(A\)
    解析 当\(y=\sin ⁡x\)时,\(y'=\cos ⁡x\),有\(\cos ⁡0⋅\cos ⁡π=-1\),
    所以在函数\(y=\sin ⁡x\)图象存在两点\(x=0\),\(x=π\)使条件成立,
    故\(A\)正确;
    函数\(y=\ln ⁡x\),\(y=e^x\),\(y=x^3\)的导数值均非负,不符合题意, 故选\(A\).

  2. 答案 \(D\)
    解析 设切点为\((x_0,y_0)\),过点\(P\)的切线方程为\(y=(x_0+1)e^{x_0} (x-x_0)+x_0 e^{x_0}\),
    代入点\(P\)坐标化简为\(m=(-x_0^2 -x_0-1)e^{x_0}\),即这个方程有三个不等根即可,
    令\(f(x)=(-x^2_0 -x_0-1)e^{x_0}\),求导得到\(f' (x)=(-x-1)(x+2) e^x\),
    函数在\((-∞,-2)\)上单调递减,在\((-2,-1)\)上单调递增,在\((-1,+∞)\)
    上单调递减,
    故得到\(f(-2)<m<f(-1)\),即\(\left(-\dfrac{3}{e^2},-\dfrac{1}{e^2}\right)\),
    故选:\(D\).

  3. 答案 \(C\)
    解析 由\(y=e^x\),得\(y'=e^x\),
    设曲线在\(P(x_0,e^{x_0} )\)处的切线与直线\(x-y-3=0\)平行,
    则\(e^{x_0}=1\),所以\(x_0=0\),切点\(P(0,1)\),
    所以与已知直线平行且与曲线相切的直线为\(x-y+1=0\),
    所以曲线\(y=e^x\)上的点到直线\(x-y-3=0\)的距离的最小值为 \(d=\dfrac{|0-1-3|}{\sqrt{2}}=\dfrac{4}{\sqrt{2}}=2\sqrt{2}\).
    故选:\(C\).

  4. 答案 \(D\)
    解析 设直线\(L\)的方程为\(y=k_n x\),切点为\((x_n,y_n )\),
    由导数的几何意义可得 \(\left\{\begin{array}{l} y_n=\sin x_n-\cos x_n \\ y_n=k_n x_n \\ \cos x_n+\sin x_n=k_n \end{array}\right.\),
    \(\therefore x_n=\dfrac{y_n}{k_n}=\dfrac{\sin x_n-\cos x_n}{\sin x_n+\cos x_n}=\dfrac{\tan x_n-1}{1+\tan x_n}=\tan \left(x_n-\dfrac{\pi}{4}\right)\).故\(C\)错误.
    作出\(y=x\)与\(y=\tan \left(x-\dfrac{\pi}{4}\right)\)的函数图象如图所示:
    image.png
    由图象可知\(\left\{x_n\right\}\)不是等差数列.故\(B\)错误.
    由\(\left\{\begin{array}{l} y_n=\sin x_n-\cos x_n \\ y_n=k_n x_n \\ \cos x_n+\sin x_n=k_n \end{array}\right.\),可得\(\left\{\begin{array}{l} y_n^2=1-2 \sin x_n \cos x_n \\ y_n^2=k_n^2 x_n^2 \\ k_n^2=1+2 \sin x_n \cos x_n \end{array}\right.\),
    \(\therefore y_n^2+k_n^2=2\),\(\therefore k_n^2 x_n^2+k_n^2=2\),
    \(\therefore k_n^2=\dfrac{2}{x_n^2+1}\),
    \(\therefore\left[f\left(x_n\right)\right]^2=y_n^2=2-k_n^2=\dfrac{2 x_n^2}{x_n^2+1}\),故\(A\)错误,\(D\)正确.
    故选:\(D\).

  5. 答案 \(C\)
    解析 作出曲线\(y=|\ln ⁡x|\)的图象,可知,
    过平面内一点\(P\)作曲线\(y=|\ln ⁡x|\)两条互相垂直的切线\(l_1\),\(l_2\),
    切点为\(P_1\),\(P_2\) (\(P_1\),\(P_2\) 不重合),
    则切点\(P_1\)的横坐标在\(x_2∈(0,1)\),\(P_2\)的横坐标在\(x_1∈(1,+∞)\),
    当\(x∈(0,1)\),\(y=-\ln ⁡x\),\(y'=-\dfrac{1}{x}\),\(\therefore k_2=-\dfrac{1}{x_2}\),
    当\(x∈(1,+∞)\),\(y=\ln ⁡x\),\(y'=\dfrac{1}{x}\),\(\therefore k_1=\dfrac{1}{x_1}\),
    \(\therefore -\dfrac{1}{x_2} ×\dfrac{1}{x_1} =-1\),\(\therefore x_1 x_2=1\),故①正确;
    直线\(P_1 P_2\)的斜率为\(\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-\ln x_2-\ln x_1}{x_2-x_1}=-\dfrac{\ln x_1 x_2}{x_2-x_1}=0\),故\(B\)正确;
    过\(P_1\)的切线方程为\(y-y_1=\dfrac{1}{x_1}\left(x-x_1\right)\),
    令\(x=0\),\(\therefore y=y_1-1\),即点\(A(0,y_1-1)\),
    即同理可得\(B(0,y_2+1)\),
    \(\therefore |AB|=|y_2-y_1+2|=|-\ln ⁡x_2-\ln ⁡x_1+2|=2\),故③正确;
    由切线\(l_1\),\(l_2\)联立解得交点\(P\)的横坐标为\(x_P=\dfrac{2 x_1 x_2}{x_1+x_2} \leqslant \dfrac{2}{2 \sqrt{x_1 x_2}}=1\),
    又因为\(P_1\),\(P_2\)不重合,故等号不成立,
    \(\therefore P\)的横坐标\(x_P∈(0,1)\),故\(S_{\triangle A B P}=\dfrac{1}{2}|A B| \cdot\left|x_P\right| \in(0,1)\),故④错误.
    故选:\(C\).
    image.png

  6. 答案 \(D\)
    解析 \(f' (x)=\dfrac{1}{x}-1\),
    \(A\)选项:因为直线为\(f(x)\)的切线,故\(k=-\dfrac{1}{2}\),
    故\(\dfrac{1}{x}-1=-\dfrac{1}{2}\),解得\(x=2\),
    又\(f(2)=\ln ⁡2-2+t\),
    故将\((2,\ln ⁡2-2+t)\)代入直线计算可得\(t=3\),故\(A\)选项错误;
    \(B\)选项:因为\(f' (x)=\dfrac{1}{x}-1\),
    令\(f' (x)=\dfrac{1}{x}-1>0\),解得\(0<x<1\),
    故\(f(x)\)在\((0,1)\)递增,\((1,+∞)\)递减,
    故\(f(x)\)的最大值为\(f(1)=t\),
    又因为直线\(y=-\dfrac{1}{2} x+\ln ⁡2+2\)在\(R\)上单调递减,
    当\(x=1\)时,\(y=\ln ⁡2+\dfrac{3}{2}\),
    故要使得直线与\(f(x)\)无公共点,仅需\(t<\ln ⁡2+\dfrac{3}{2}\),故\(B\)选项错误;
    \(C\)选项:当\(t=-2\)时,\(f(x)=\ln ⁡x-x-2\),要使得到直线距离最短,
    即\(f' (x)=\dfrac{1}{x}-1=-\dfrac{1}{2}\),解得\(x=2\),
    故\(P(2,\ln ⁡2-4)\),
    故\(P\)点到直线的距离 \(d=\dfrac{\left|-\dfrac{1}{2} \times 2-\ln 2+4+\ln 2+2\right|}{\sqrt{\dfrac{1}{4}+1}}=2 \sqrt{5}\),
    故\(C\)选项错误;
    \(D\)选项:由\(C\)选项可知距离最小时,\(P\)横坐标为\(2\),故\(D\)选项正确.
    故选:\(D\).

  7. 答案 \(AD\)
    解析 设过点\(P(-1,t)\)的直线与函数\(f(x)=\dfrac{x+1}{e^x}\)的图像相切时的切点为\((a,b)\),
    则\(b=\dfrac{a+1}{e^a}\) ,
    因为\(f(x)=\dfrac{x+1}{e^x}\), \(f^{\prime}(x)=\dfrac{e^x-(x+1) e^x}{e^{2 x}}=-\dfrac{x}{e^x}\) ,
    所以切线方程为\(y-\dfrac{a+1}{e^a}=-\dfrac{a}{e^a}(x-a)\),
    又\(P(-1,t)\)在切线上,
    所以\(t-\dfrac{a+1}{e^a}=-\dfrac{a}{e^a}(-1-a)\),整理得\(t=\dfrac{(a+1)^2}{e^a}\),
    令\(g(a)=\dfrac{(a+1)^2}{e^a}\),
    则过点\(P(-1,t)\)的直线与函数\(f(x)=\dfrac{x+1}{e^x}\)的图像相切的切线条数
    即为直线\(y=t\)与曲线\(g(a)=\dfrac{(a+1)^2}{e^a}\)的图象的公共点的个数,
    因为\(g^{\prime}(a)=\dfrac{2(a+1) e^a-(a+1)^2 e^a}{e^{2 a}}=\dfrac{-(a+1)(a-1)}{e^a}\),
    令\(g' (a)=0\),得\(a=±1\),
    所以,当\(a<-1\)时,\(g' (a)<0\),\(g(a)\)单调递减,
    当\(-1<a<1\)时,\(g' (a)>0\),\(g(a)\)单调递增,
    当\(a>1\)时,\(g' (a)<0\),\(g(a)\)单调递减,
    因为\(g(-1)=0\), \(g(1)=\dfrac{4}{e}\),\(g(0)=1\),
    所以函数\(g(a)\)的图像大致如图:
    image.png
    由图可知当\(t<0\)时,直线\(y=t\)与曲线\(g(a)=\dfrac{(a+1)^2}{e^a}\)的图像没有公共点,即\(n=0\),
    当\(t=0\)或 \(t>\dfrac{4}{e}\)时,直线\(y=t\)与曲线\(g(a)=\dfrac{(a+1)^2}{e^a}\)的图像有\(1\)个公共点,即\(n=1\),
    当 \(t=\dfrac{4}{e}\)时,直线\(y=t\)与曲线\(g(a)=\dfrac{(a+1)^2}{e^a}\)的图像有\(2\)个公共点,即\(n=2\),
    当\(0<t<\dfrac{4}{e}\)时,直线\(y=t\)与曲线\(g(a)=\dfrac{(a+1)^2}{e^a}\)的图像有\(3\)个公共点,即\(n=3\),
    对于\(A\),当 \(t=2022>\dfrac{4}{e}\),此时\(n=1\),则\(tn=2022\)符合题意,故\(A\)正确;
    对于\(B\),当\(0<t<\dfrac{4}{e}\)时,\(n=3\),故\(B\)错误;
    对于\(C\),当\(t=0\)时,\(n=1\),则\(te+n=1<3\),故\(C\)错误;
    对于\(D\),当\(t=0\)或 \(t>\dfrac{4}{e}\)时,\(n=1\),
    则当\(n=1\)时, \(t \in\{0\} \cup\left(\dfrac{4}{e},+\infty\right)\),故\(D\)正确.
    故选:\(AD\).

  8. 答案 \(\dfrac{n(n+1)}{2}\)
    解析 由\(y=x^2+\dfrac{5}{4}\),得\(y'=2x\),则 \(y^{\prime} |_{x=\dfrac{1}{2}}=1\),
    \(\therefore\)切线 \(l: y-\dfrac{3}{2}=1 \times\left(x-\dfrac{1}{2}\right)\),即\(y=x+1\),
    \(\because\)点 \(\left(a_n, a_{n+1}\right)\)\((n∈N^* )\)为切线\(l\)上一点,
    \(\therefore a_{n+1}=a_n+1\),
    则数列\(\left\{a_n\right\}\)是首项为\(1\),公差为\(1\)的等差数列,
    \(\therefore\)数列\(\left\{a_n\right\}\)的前\(n\)项和为\(S_n=n+\dfrac{n(n-1)}{2} \times 1=\dfrac{n(n+1)}{2}\).

  9. 答案 \(1\)
    解析 设\(y=kx+b\)与\(y=\ln ⁡x+2\)和\(y=\ln ⁡(x+2)\)的切点分别为\((x_1,kx_1+b)\)、\((x_2,kx_2+b)\);
    由导数的几何意义可得 \(k=\dfrac{1}{r}=\dfrac{1}{x+1}\),得\(x_1=x_2+2\),
    切线方程分别为\(y-\left(\ln x_1+2\right)=\dfrac{1}{x_1}\left(x-x_1\right)\),即为 \(y=\dfrac{x}{x_1}+\ln x_1+1\),
    或 \(y-\ln \left(x_2+2\right)=\dfrac{1}{x_2+2}\left(x-x_2\right)\),即为 \(y=\dfrac{x}{x_1}+\dfrac{2-x_1}{x_1}+\ln x_1\),
    \(\therefore \dfrac{2-x_1}{x_1}=1\),解得\(x_1=1\),\(\therefore b=1\).
    故答案为\(1\).

  10. 答案 \(2 \sqrt{5}\)
    解析 \(\because\)曲线\(y=\ln (2x-1)\),\(\therefore y^{\prime}=\dfrac{2}{2 x-1}\),
    分析知直线\(2x-y+8=0\)与曲线\(y=\ln(2x-1)\)相切的点到直线\(2x-y+8=0\)的距离最短,
    \(y^{\prime}=\dfrac{2}{2 x-1}=2\), 解得\(x=1\),把\(x=1\)代入\(y=\ln ⁡(2x-1)\),
    \(\therefore y=0\),\(\therefore\)点\((1,0)\)到直线\(2x-y+8=0\)的距离最短,
    \(\therefore d=\dfrac{|2+8|}{\sqrt{4+1}}=2 \sqrt{5}\).

  11. 答案 \(\dfrac{\sqrt{3}}{3}\)
    解析 由\(y=2x^2-2\),得\(y'=4x\),
    则切线\(PQ\)的方程为\(y-(2x_0^2-2)=4x_0 (x-x_0 )\),
    化简得\(4x_0 x-y-2(x_0^2+1)=0\),
    所以 \(|O P|=\dfrac{x_0^2+1}{2 x_0},|O Q|=2\left(x_0^2+1\right)\),
    所以三角形\(POQ\)的面积为\(\dfrac{1}{2} \cdot 2\left(x_0^2+1\right) \cdot \dfrac{x_0^2+1}{2 x_0}=\dfrac{\left(x_0^2+1\right)^2}{2 x_0}=\dfrac{1}{2}\left(x_0^3+2 x_0+\dfrac{1}{x_0}\right)\),
    令\(v=x_0^3+2x_0+\dfrac{1}{x_0}\) ,
    \(u=v^{\prime}=\left(x_0^3+2 x_0+\dfrac{1}{x_0}\right)^{\prime}=3 x_0^2+2-\dfrac{1}{x_0^2}=\dfrac{3 x_0^4+2 x_0^2-1}{x_0^2}\),
    令\(x_0^2=t∈[0,1]\),则 \(u=\dfrac{3 t^2+2 t-1}{t}\).
    令\(u=0\),解得\(t=\dfrac{1}{3}\)或\(-1\) (舍去),
    所以当\(t \in\left[0, \dfrac{1}{3}\right)\)时,\(u<0\), \(v=x_0^3+2 x_0+\dfrac{1}{x_0}\)单调递减;
    当\(t \in\left(\dfrac{1}{3}, 1\right]\)时,\(u>0\),\(v=x_0^3+2x_0+\dfrac{1}{x_0}\) 单调递增,
    又\(x_0∈[0,1]\),
    所以当且仅当\(x_0=\dfrac{\sqrt{3}}{3}\)时,三角形\(POQ\)的面积取得最小值.
     

【B组---提高题】

1.若直线\(y=k_1 (x+1)-1\)与曲线\(y=e^x\)相切,直线\(y=k_2 (x+1)-1\)与曲线\(y=\ln ⁡x\)相切.则\(k_1 k_2\)的值为(  )
 A.\(\dfrac{1}{2}\) \(\qquad \qquad \qquad \qquad\) B.\(1\) \(\qquad \qquad \qquad \qquad\) C.\(e\)\(\qquad \qquad \qquad \qquad\) D.\(e^2\)
 
2.若曲线\(C_1:y=ax^2 (a>0)\)与曲线\(C_2:y=\ln x\)有唯一的公共点,则实数\(a\)的值为\(\underline{\quad \quad}\).
 

3.已知\(a\),\(b\),\(c∈R\), 且满足\(b^2+c^2=1\), 如果存在两条相互垂直的直线与函数\(f(x)=ax+b\cos ⁡x+ c\sin ⁡x\)的图象都相切,则\(a+\sqrt{2} b+\sqrt{3} c\)的取值范围是\(\underline{\quad \quad}\) .
 
 

参考答案

  1. 答案 \(B\)
    解析 \(y=e^x\)的导数为\(y'=e^x\),\(y=\ln ⁡x\)的导数为\(y'=\dfrac{1}{x}\),
    设与曲线\(y=e^x\)相切的切点为\((m,n)\),
    直线\(y=k_2 (x+1)-1\)与曲线\(y=\ln ⁡x\)相切的切点为\((s,t)\),
    所以\(k_1=e^m\), \(k_2=\dfrac{1}{s},\),即\(m=\ln ⁡k_1\), \(s=\dfrac{1}{k_2}\),
    \(n=k_1=k_1 (1+\ln ⁡k_1 )-1\),即 \(\ln k_1=\dfrac{1}{k_1}\),
    又 \(t=\ln s=-\ln k_2=k_2\left(1+\dfrac{1}{k_2}\right)-1\),
    即\(-\ln k_2=k_2\),可得 \(e^{k_2}=\dfrac{1}{k_2}\) ,
    考虑\(k_1\)\(\ln ⁡x=\dfrac{1}{x}\)为方程的根,\(k_2\)为方程\(e^x=\dfrac{1}{x}\)的根,
    分别画出\(y=e^x\),\(y=\ln ⁡x\)和\(y=\dfrac{1}{x}\),\(y=x\)的图像,
    可得\(y=e^x\)和\(y=\dfrac{1}{x}\)的交点与\(y=\ln ⁡x\)和\(y=\dfrac{1}{x}\)的交点关于直线\(yx\)对称,
    则\(k_1=\dfrac{1}{k_2}\),即\(k_1 k_2=1\).
    故选:\(B\).
    image.png

  2. 答案 \(\dfrac{1}{2e}\)
    解析 由\(y=ax^2\),得\(y'=2ax\),
    由\(y=\ln x\),得\(y^{\prime}=\dfrac{1}{x}\),
    曲线\(y=ax^2 (a>0)\)与曲线\(y=\ln x\)有唯一的公共点,
    则该公共点为两曲线公切线的切点,设为\((s,t)\),
    则\(\left\{\begin{array}{l} 2 a s=\dfrac{1}{s} \\ t=a s^2 \\ t=\ln s \end{array}\right.\),解得\(a=\dfrac{1}{2 e}\).
    故答案为:\(\dfrac{1}{2e}\).

  3. 答案 \([-\sqrt{5},\sqrt{5}]\)
    解析 因为\(b^2+c^2=1\), 故可设\(b=\cos ⁡θ\),\(c=\sin ⁡θ\),\(θ∈[0,2π)\),
    \(\because f(x)=ax+b\cos ⁡x+c\sin ⁡x\),
    \(\therefore f' (x)=a-b\sin ⁡x+c\cos ⁡x=a-\cos ⁡θ\sin ⁡x+\sin ⁡θ\cos ⁡x=a-\sin ⁡(x-θ)\),
    \(\therefore a-1≤f' (x)≤a+1\) 且\(a-1\),\(a+1\)异号,
    \(\because\)存在两条相互垂直的直线与函数\(f(x)\)的图象都相切,
    \(\therefore\)存在\(x_1\),\(x_2\), 使得\(f' (x_1 ) f' (x_2 )=-1\),
    只需\(|a-1||a+1|=|a^2-1|≥1\), 即\(a^2-1≤-1\),
    \(\therefore a^2≤0\),\(\therefore a=0\),
    \(\therefore a+\sqrt{2} b+\sqrt{3} c=\sqrt{2} b+\sqrt{3} c=\sqrt{2} \cos \theta+\sqrt{3} \sin \theta=\sqrt{5} \sin (\theta+\varphi)\),
    其中 \(\tan \varphi=\dfrac{\sqrt{2}}{\sqrt{3}}=\dfrac{\sqrt{6}}{3}\),
    \(\therefore-\sqrt{5} \leq a+\sqrt{2} b+\sqrt{3} c \leq \sqrt{5}\).
     

【C组---拓展题】

1.若直线\(y=k_1 x+b_1\)与直线\(y=k_2 x+b_2 (k_1≠k_2 )\)是曲线\(y=\ln ⁡x\)的两条切线,也是曲线\(y=e^x\)的两条切线,则\(k_1 k_2+b_1+b_2\)的值为\(\underline{\quad \quad}\) .
 

2.若实数\(a\),\(b\),\(c\), \(d\)满足\(|b+a^2-4\ln ⁡a|+|2c-d+2|=0\), 则\((a-c)^2+(b-d)^2\)的最小值为\(\underline{\quad \quad}\).
 

参考答案

  1. 答案 \(-1\)
    解析 由\(y=e^x\)和\(y=\ln ⁡x\)互为反函数可知,
    两条公切线\(y=k_1 x+b_1\)和\(y=k_2 x+b_2\)也互为反函数,
    即\(x=\dfrac{1}{k_1} y-\dfrac{b_1}{k_1}\)满足 \(\dfrac{1}{k_1}=k_2\), \(-\dfrac{b_1}{k_1}=b_2\),
    即\(k_1 k_2=1\), \(b_2=-\dfrac{b_1}{k_1}\) ,
    设直线\(y=k_1 x+b_1\)与\(y=e^x\)和\(y=\ln ⁡x\)分别切于点\(\left(x_1, e^{x_1}\right)\)和\((x_2,\ln ⁡x_2 )\),
    可得切线方程为\(y-e^{x_1}=e^{x_1}\left(x-x_1\right)\)和 \(y-\ln x_2=\dfrac{1}{x_2}\left(x-x_2\right)\),
    整理得:\(y=e^{x_1} x+e^{x_1}-x e^{x_1}\)和 \(y=\dfrac{1}{x_2} x-1+\ln x_2\),
    则\(k_1=e^{x_1}=\dfrac{1}{x_2}\) , \(b_1=e^{x_1}\left(1-x_1\right)=-1+\ln x_2\),
    由\(e^{x_i}=\dfrac{1}{x_2}\) ,得 \(x_1=\ln \dfrac{1}{x_2}=-\ln x_2\),且 \(b_1=\dfrac{1}{x_2}\left(1-x_1\right)=-1+\ln x_2\),
    则\(b_1=\dfrac{1}{x_2}\left(1-x_1\right)=-1-x_1\),
    所以\(x_2=\dfrac{1-x_1}{-1-x_1}\),
    所以 \(k_1 k_2+b_1+b_2=1+b_1-\dfrac{b_1}{k_1}=1+b_1\left(1-\dfrac{1}{k_1}\right)=1+b_1\left(1-x_2\right)\)
    \(=1+\left(-1-x_1\right)\left(1-\dfrac{1-x_1}{-1-x_1}\right)=1+\left(-1-x_1\right)-\left(1-x_1\right)=-1\).

  2. 答案 \(5\)
    解析 \(\because |b+a^2-4\ln ⁡a|+|2c-d+2|=0\),
    \(\therefore b+a^2-4\ln ⁡a=0\),\(2c-d+2=0\).
    将\(b+a^2-4\ln ⁡a=0\)看成\(y+x^2-4\ln ⁡x=0\),
    即曲线\(y=-x^2+4\ln ⁡x\),
    将\(2c-d+2=0\)看成\(2x-y+2=0\), 即直线\(y=2x+2\),
    \((a-c)^2+(b-d)^2\)表示曲线\(y=-x^2+4\ln ⁡x\)上的点与直线\(y=2x+2\)上的点间的距离的平方,
    作与直线\(y=2x+2\)平行的曲线的切线,
    由\(y=-x^2+4\ln ⁡x\), 得 \(y^{\prime}=-2 x+\dfrac{4}{x}\),
    令 \(y^{\prime}=-2 x+\dfrac{4}{x}=2\), 得\(x^2+x-2=0\),解得\(x=1\)或\(x=-2\)(舍去)
    所以切点为\((1,-1)\),
    故点\((1,-1)\)到直线\(2x-y+2=0\)的距离为 \(d=\dfrac{|2 \times 1-(-1)+2|}{\sqrt{5}}=\sqrt{5}\),
    故曲线上的点到直线的最小距离为\(\sqrt{5}\),
    \(\therefore (a-c)^2+(b-d)^2\)的最小值为\(5\).
     

标签:专题,数的几何,导数,ln,dfrac,sqrt,right,qquad,left
From: https://www.cnblogs.com/zhgmaths/p/16960687.html

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