函數(shù)y=ln7x+√(77x^2+5)的性質(zhì)及圖像示意圖

函數(shù)y=ln(7x+eq\r(77x2+5))的圖像性質(zhì)主要內(nèi)容：本文主要介紹函數(shù)y=ln(7x+eq\r(77x2+5))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7x+eq\r(77x2+5)＞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(77x2+5)＞-7x，對(duì)不等式兩邊平方，77x2+5>72x2，即-28x2<5，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(7x+eq\r(77x2+5))，對(duì)x求導(dǎo)有：所以：eq\f(dy,dx)=eq\f(7+eq\f(77x,eq\r(77x2+5)),7x+eq\r(77x2+5))=eq\f(7eq\r(77x2+5)+77x,eq\r(77x2+5)*(7x+eq\r(77x2+5)))。令eq\f(dy,dx)=0，則x=-eq\f(\r(55),22)≈-0.34，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(\r(55),22))時(shí)，eq\f(dy,dx)<0，此時(shí)函數(shù)為減函數(shù)；（2）當(dāng)x∈[-eq\f(\r(55),22)，+∞)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(7x+eq\r(77x2+5))=+∞，eq\s(lim,x→0)ln(7x+eq\r(77x2+5))=lneq\r(5)。eq\s(lim,x→+∞)ln(7x+eq\r(77x2+5))=+∞?！?函數(shù)的五點(diǎn)圖x-0.07-0.17-0.25-0.34-0.51-0.67-0.847x-0.49-1.19-1.75-2.38-3.57-4.69-5.88eq\r(77x2+5)2.322.693.133.735.006.297.70y0.600.410.320.300.360.470.60※.函數(shù)的圖像示意圖y=ln(7x+eq\r(77x2+5)) y(-0.84,0.60),(-0.07,0.60)(-0.17,0.41)(-0.67,0.47)(-0.25,