函數(shù)y=ln5x+√(41x^2+72)的性質(zhì)及其圖像

函數(shù)y=ln(5x+eq\r(41x2+72))的圖像性質(zhì)主要內(nèi)容：本文主要介紹函數(shù)y=ln(5x+eq\r(41x2+72))的定義域、單調(diào)性、極限等性質(zhì)，并通過(guò)導(dǎo)數(shù)知識(shí)計(jì)算函數(shù)的單調(diào)區(qū)間，簡(jiǎn)要畫(huà)出函數(shù)的圖像示意圖?！?函數(shù)的定義域根據(jù)函數(shù)特征，對(duì)于自然對(duì)數(shù)函數(shù)有5x+eq\r(41x2+72)＞0，(1)當(dāng)x≥0時(shí)，不等式恒成立；(2)當(dāng)x＜0時(shí)，不等式變形有：eq\r(41x2+72)＞-5x，對(duì)不等式兩邊平方，41x2+72>52x2，即-16x2<72，此時(shí)不等式恒成立。綜上所述，該函數(shù)的定義域?yàn)椋?-∞，+∞)?！?函數(shù)的單調(diào)性因?yàn)閥=ln(5x+eq\r(41x2+72))，對(duì)x求導(dǎo)有：所以：eq\f(dy,dx)=eq\f(5+eq\f(41x,eq\r(41x2+72)),5x+eq\r(41x2+72))=eq\f(5eq\r(41x2+72)+41x,eq\r(41x2+72)*(5x+eq\r(41x2+72)))。令5eq\r(41x2+72)+41x=0，則x=-eq\f(15\r(82),82)≈-1.656。此時(shí)函數(shù)的單調(diào)性及單調(diào)區(qū)間如下：（1）當(dāng)x∈(-∞，-eq\f(15\r(82),82))時(shí)，eq\f(dy,dx)<0，此時(shí)函數(shù)為減函數(shù)；（2）當(dāng)x∈[-eq\f(15\r(82),82)，+∞)時(shí)，eq\f(dy,dx)＞0，此時(shí)函數(shù)為增函數(shù)?！?函數(shù)的極限eq\s(lim,x→-∞)ln(5x+eq\r(41x2+72))=+∞，eq\s(lim,x→0)ln(5x+eq\r(41x2+72))=lneq\r(72)。eq\s(lim,x→+∞)ln(5x+eq\r(41x2+72))=+∞?！?函數(shù)的五點(diǎn)圖x-0.33-0.83-1.24-1.66-2.48-3.31-4.145x-1.65-4.15-6.20-8.30-12.40-16.55-20.70eq\r(41x2+72)8.7410.0111.6213.6018.0022.8327.83y1.961.771.691.671.721.841.96※.函數(shù)的圖像示意圖y=ln(5x+eq\r(41x2+72)) y(-4.14,1.96),(-0.33,1.96)(-0.83,1.77)(-3.31,1.84)(-1.2