由雙曲函數(shù)的切線引出的幾個(gè)常見不等式

2022-10-10 18:50 作者:數(shù)學(xué)老頑童 0人讀過 | 我要投稿

容易求出，雙曲正弦函數(shù)

$%5Csinh%20%20x%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft(%20%5Cmathrm%7Be%7D%5Ex-%5Cmathrm%7Be%7D%5E%7B-x%7D%20%5Cright)%20$

的圖象在點(diǎn) $%5Cleft(%200%2C0%20%5Cright)%20$ 處的切線方程為

$y%3Dx$ .

并且不難推出：

$%5Cbbox%5B%23def%2C5px%2Cborder%3A1px%20solid%5D%7B%0A%5Cbegin%7Balign%7D%0A%5Cforall%20x%5Cin%20%5Cleft(%20-%5Cinfty%20%2C0%20%5Cright)%20%2C%5Cfrac%7B1%7D%7B2%7D%5Cleft(%20%5Cmathrm%7Be%7D%5Ex-%5Cmathrm%7Be%7D%5E%7B-x%7D%20%5Cright)%20%26%3Cx%3B%0A%5C%5C%0A%5Cforall%20x%5Cin%20%5Cleft(%200%2C%2B%5Cinfty%20%5Cright)%20%2C%5Cfrac%7B1%7D%7B2%7D%5Cleft(%20%5Cmathrm%7Be%7D%5Ex-%5Cmathrm%7Be%7D%5E%7B-x%7D%20%5Cright)%20%26%3Ex.%0A%5Cend%7Balign%7D%7D$

在上式中，換元，令 $%5Cmathrm%7Be%7D%5Ex%3Dt$ ，可得：

$%5Cbbox%5B%23def%2C5px%2Cborder%3A1px%20solid%5D%7B%0A%5Cbegin%7Baligned%7D%09%5Cforall%20t%5Cin%20%5Cleft(%200%2C1%20%5Cright)%20%2C%5Cln%20%20t%26%3E%5Cfrac%7B1%7D%7B2%7D%5Cleft(%20t-%5Cfrac%7B1%7D%7Bt%7D%20%5Cright)%20%3B%5C%5C%09%5Cforall%20t%5Cin%20%5Cleft(%201%2C%2B%5Cinfty%20%5Cright)%20%2C%5Cln%20%20t%26%3C%5Cfrac%7B1%7D%7B2%7D%5Cleft(%20t-%5Cfrac%7B1%7D%7Bt%7D%20%5Cright)%20.%5C%5C%5Cend%7Baligned%7D%7D$

再換，使 $t$ 為 $%5Csqrt%7Bt%7D%20$ ，可得

$%5Cbbox%5B%23def%2C5px%2Cborder%3A1px%20solid%5D%7B%0A%5Cbegin%7Baligned%7D%09%5Cforall%20t%5Cin%20%5Cleft(%200%2C1%20%5Cright)%20%2C%5Cln%20%20t%26%3E%5Csqrt%7Bt%7D-%5Cfrac%7B1%7D%7B%5Csqrt%7Bt%7D%7D%3B%5C%5C%09%5Cforall%20t%5Cin%20%5Cleft(%201%2C%2B%5Cinfty%20%5Cright)%20%2C%5Cln%20%20t%26%3C%5Csqrt%7Bt%7D-%5Cfrac%7B1%7D%7B%5Csqrt%7Bt%7D%7D.%5C%5C%5Cend%7Baligned%7D%7D$

容易求出，雙曲正切函數(shù)

$%5Ctanh%20%20x%3D%5Cfrac%7B%5Cmathrm%7Be%7D%5Ex-%5Cmathrm%7Be%7D%5E%7B-x%7D%7D%7B%5Cmathrm%7Be%7D%5Ex%2B%5Cmathrm%7Be%7D%5E%7B-x%7D%7D$

的圖象在點(diǎn) $%5Cleft(%200%2C0%20%5Cright)%20$ 處的切線方程為

$y%3Dx$ .

并且不難推出：

$%5Cbbox%5B%23def%2C5px%2Cborder%3A1px%20solid%5D%7B%0A%5Cbegin%7Baligned%7D%0A%09%5Cforall%20x%5Cin%20%5Cleft(%20-%5Cinfty%20%2C0%20%5Cright)%20%2C%5Cfrac%7B%5Cmathrm%7Be%7D%5Ex-%5Cmathrm%7Be%7D%5E%7B-x%7D%7D%7B%5Cmathrm%7Be%7D%5Ex%2B%5Cmathrm%7Be%7D%5E%7B-x%7D%7D%26%3Ex%3B%5C%5C%0A%09%5Cforall%20x%5Cin%20%5Cleft(%200%2C%2B%5Cinfty%20%5Cright)%20%2C%5Cfrac%7B%5Cmathrm%7Be%7D%5Ex-%5Cmathrm%7Be%7D%5E%7B-x%7D%7D%7B%5Cmathrm%7Be%7D%5Ex%2B%5Cmathrm%7Be%7D%5E%7B-x%7D%7D%26%3Cx.%5C%5C%0A%5Cend%7Baligned%7D%7D$

在上式中，換元，令 $%5Cmathrm%7Be%7D%5Ex%3Dt$ ，可得：

$%5Cbbox%5B%23def%2C5px%2Cborder%3A1px%20solid%5D%7B%0A%5Cbegin%7Baligned%7D%0A%09%5Cforall%20t%5Cin%20%5Cleft(%200%2C1%20%5Cright)%20%2C%5Cln%20%20t%26%3C%5Cfrac%7Bt%5E2-1%7D%7Bt%5E2%2B1%7D%3B%5C%5C%0A%09%5Cforall%20t%5Cin%20%5Cleft(%201%2C%2B%5Cinfty%20%5Cright)%20%2C%5Cln%20%20t%26%3E%5Cfrac%7Bt%5E2-1%7D%7Bt%5E2%2B1%7D.%5C%5C%0A%5Cend%7Baligned%7D%7D$

再換，使 $t%5E2$ 為 $t$ ，使 $t$ 為 $%5Csqrt%7Bt%7D%20$ ，可得：

$%5Cbbox%5B%23def%2C5px%2Cborder%3A1px%20solid%5D%7B%5Cbegin%7Baligned%7D%09%5Cforall%20t%5Cin%20%5Cleft(%200%2C1%20%5Cright)%20%2C%5Cln%20%20t%26%3C%5Cfrac%7B2%5Cleft(%20t-1%20%5Cright)%7D%7Bt%2B1%7D%3B%5C%5C%09%5Cforall%20t%5Cin%20%5Cleft(%201%2C%2B%5Cinfty%20%5Cright)%20%2C%5Cln%20%20t%26%3E%5Cfrac%7B2%5Cleft(%20t-1%20%5Cright)%7D%7Bt%2B1%7D.%5C%5C%5Cend%7Baligned%7D%7D$

標(biāo)簽：

由雙曲函數(shù)的切線引出的幾個(gè)常見不等式的評(píng)論 (共條)

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