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公切線問題的轉(zhuǎn)化路徑(2019課標(biāo)Ⅱ?qū)?shù))

2022-10-19 11:17 作者:數(shù)學(xué)老頑童  | 我要投稿

(2019課標(biāo)Ⅱ,20)已知函數(shù)f%5Cleft(%20x%20%5Cright)%20%3D%5Cln%20%20x-%5Cfrac%7Bx%2B1%7D%7Bx-1%7D.

(1)討論f%5Cleft(%20x%20%5Cright)%20的單調(diào)性,并證明f%5Cleft(%20x%20%5Cright)%20有且僅有兩個零點(diǎn);

(2)設(shè)x_0f%5Cleft(%20x%20%5Cright)%20的一個零點(diǎn),證明曲線y%3D%5Cln%20%20x在點(diǎn)A%5Cleft(%20x_0%2C%5Cln%20%20x_0%20%5Cright)%20處的切線也是曲線y%3D%5Cmathrm%7Be%7D%5Ex的切線.

解:(1)f%5Cleft(%20x%20%5Cright)%20的定義域?yàn)?/p>

%5Ccolor%7Bred%7D%7B%5Cleft(%200%2C1%20%5Cright)%20%5Ccup%20%5Cleft(%201%2C%2B%5Cinfty%20%5Cright)%20%7D,

%5Cbegin%7Baligned%7D%0A%09%5Ccolor%7Bred%7D%7Bf'%5Cleft(%20x%20%5Cright)%20%7D%26%3D%5Cfrac%7B1%7D%7Bx%7D-%5Cfrac%7B1%5Ccdot%20%5Cleft(%20x-1%20%5Cright)%20-%5Cleft(%20x%2B1%20%5Cright)%20%5Ccdot%201%7D%7B%5Cleft(%20x-1%20%5Cright)%20%5E2%7D%5C%5C%0A%09%26%3D%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B2%7D%7B%5Cleft(%20x-1%20%5Cright)%20%5E2%7D%5Ccolor%7Bred%7D%7B%3E0%7D%5C%5C%0A%5Cend%7Baligned%7D

f%5Cleft(%20x%20%5Cright)%20%5Ccolor%7Bred%7D%7B%5Cleft(%200%2C1%20%5Cright)%20%7D%5Ccolor%7Bred%7D%7B%5Cleft(%201%2C%2B%5Cinfty%20%5Cright)%20%7D分別%5Ccolor%7Bred%7D%7B%5Cnearrow%20%7D

又因?yàn)?/p>

%5Ccolor%7Bred%7D%7Bf%5Cleft(%20%5Cfrac%7B1%7D%7B%5Cmathrm%7Be%7D%5E2%7D%20%5Cright)%7D%20%3D%5Cfrac%7B3-%5Cmathrm%7Be%7D%5E2%7D%7B%5Cmathrm%7Be%7D%5E2-1%7D%5Ccolor%7Bred%7D%7B%3C0%7D,

%5Ccolor%7Bred%7D%7Bf%5Cleft(%20%5Cfrac%7B1%7D%7B%5Cmathrm%7Be%7D%7D%20%5Cright)%7D%20%3D%5Cfrac%7B2%7D%7B%5Cmathrm%7Be%7D-1%7D%5Ccolor%7Bred%7D%7B%3E0%7D;

%5Ccolor%7Bred%7D%7Bf%5Cleft(%20%5Cmathrm%7Be%7D%20%5Cright)%7D%20%3D%5Cfrac%7B2%7D%7B1-%5Cmathrm%7Be%7D%7D%5Ccolor%7Bred%7D%7B%3C0%7D,

%5Ccolor%7Bred%7D%7Bf%5Cleft(%20%5Cmathrm%7Be%7D%5E2%20%5Cright)%20%7D%3D%5Cfrac%7B%5Cmathrm%7Be%7D%5E2-3%7D%7B%5Cmathrm%7Be%7D%5E2-1%7D%5Ccolor%7Bred%7D%7B%3E0%7D,

所以f%5Cleft(%20x%20%5Cright)%20僅在%5Cleft(%20%5Cfrac%7B1%7D%7B%5Cmathrm%7Be%7D%5E2%7D%2C%5Cfrac%7B1%7D%7B%5Cmathrm%7Be%7D%7D%20%5Cright)%20%5Cleft(%20%5Cmathrm%7Be%7D%2C%5Cmathrm%7Be%7D%5E2%20%5Cright)%20各存在一個零點(diǎn),即f%5Cleft(%20x%20%5Cright)%20有且僅有兩個零點(diǎn).

(橫屏觀看)

(2)因?yàn)?img type="latex" class="latex" src="https://api.bilibili.com/x/web-frontend/mathjax/tex?formula=x_0" alt="x_0">是f%5Cleft(%20x%20%5Cright)%20的零點(diǎn),

所以%5Ccolor%7Bred%7D%7B%20%5Cln%20%20x_0%3D%5Cfrac%7Bx_0%2B1%7D%7Bx_0-1%7D%7D

%5Cleft(%20%5Cln%20%20x%20%5Cright)%20'%3D%5Cfrac%7B1%7D%7Bx%7D,所以

y%3D%5Cln%20%20x%5Cleft(%20x_0%2C%5Cln%20%20x_0%20%5Cright)%20處的切線斜率為%5Cfrac%7B1%7D%7Bx_0%7D

故該點(diǎn)處的切線方程為

y-%5Cln%20%20x_0%3D%5Cfrac%7B1%7D%7Bx_0%7D%5Cleft(%20x-x_0%20%5Cright)%20,

整理得%5Ccolor%7Bred%7D%7By%3D%5Cfrac%7B1%7D%7Bx_0%7D%5Ccdot%20x%2B%5Cln%20%20x_0-1%7D.

記該直線為%5Ccolor%7Bred%7D%7Bl_1%7D.

設(shè)曲線y%3D%5Cmathrm%7Be%7D%5Exl_1斜率相等的切線為%5Ccolor%7Bred%7D%7Bl_2%7D

設(shè)l_2與曲線y%3D%5Cmathrm%7Be%7D%5Ex相切于%5Cleft(%20x_1%2C%5Cmathrm%7Be%7D%5E%7Bx_1%7D%20%5Cright)%20,

因?yàn)?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=%5Cleft(%20%5Cmathrm%7Be%7D%5Ex%20%5Cright)'%3D%5Cmathrm%7Be%7D%5Ex" alt="%5Cleft(%20%5Cmathrm%7Be%7D%5Ex%20%5Cright)'%3D%5Cmathrm%7Be%7D%5Ex">,

故該點(diǎn)處的切線斜率為%5Cmathrm%7Be%7D%5E%7Bx_1%7D,

因?yàn)?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=%5Ccolor%7Bred%7D%7B%5Cmathrm%7Be%7D%5E%7Bx_1%7D%3D%5Cfrac%7B1%7D%7Bx_0%7D%7D" alt="%5Ccolor%7Bred%7D%7B%5Cmathrm%7Be%7D%5E%7Bx_1%7D%3D%5Cfrac%7B1%7D%7Bx_0%7D%7D">,所以%5Ccolor%7Bred%7D%7Bx_1%3D-%5Cln%20x_0%7D.

所以%5Cleft(%20x_1%2C%5Cmathrm%7Be%7D%5E%7Bx_1%7D%20%5Cright)%20即為%5Ccolor%7Bred%7D%7B%5Cleft(-%5Cln%20x_0%2C%5Cfrac%7B1%7D%7Bx_0%7D%5Cright)%20%7D.

因?yàn)?/p>

%5Cbegin%7Baligned%7D%0A%09y%26%3D%5Cfrac%7B1%7D%7Bx_0%7D%5Ccdot%20%5Cleft(%20%5Ccolor%7Bred%7D%7B-%5Cln%20x_0%7D%20%5Cright)%20%2B%5Cln%20x_0-1%5C%5C%0A%09%26%3D%5Cfrac%7B1%7D%7Bx_0%7D%5Ccdot%20%5Cleft(%20-%5Cfrac%7Bx_0%2B1%7D%7Bx_0-1%7D%20%5Cright)%20%2B%5Cfrac%7Bx_0%2B1%7D%7Bx_0-1%7D-1%5C%5C%0A%09%26%3D%5Ccolor%7Bred%7D%7B%5Cfrac%7B1%7D%7Bx_0%7D%7D%5C%5C%0A%5Cend%7Baligned%7D

所以點(diǎn)%5Cleft(%20-%5Cln%20%20x_0%2C%5Cfrac%7B1%7D%7Bx_0%7D%20%5Cright)%20在直線%5Ccolor%7Bred%7D%7Bl_1%7D.

所以直線l_1與直線l_2重合,

所以l_1也是曲線y%3D%5Cmathrm%7Be%7D%5Ex的切線.


公切線問題的轉(zhuǎn)化路徑(2019課標(biāo)Ⅱ?qū)?shù))的評論 (共 條)

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