高中數(shù)學 第二章 幾個重要的不等式 2.3 數(shù)學歸納法與貝努利不等式 2.3.2 數(shù)學歸納法的應用課件 北師大版選修45

《高中數(shù)學 第二章 幾個重要的不等式 2.3 數(shù)學歸納法與貝努利不等式 2.3.2 數(shù)學歸納法的應用課件 北師大版選修45》由會員分享，可在線閱讀，更多相關《高中數(shù)學 第二章 幾個重要的不等式 2.3 數(shù)學歸納法與貝努利不等式 2.3.2 數(shù)學歸納法的應用課件 北師大版選修45（22頁珍藏版）》請在裝配圖網(wǎng)上搜索。

1、3 3.2 2數(shù)學歸納法的應用目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理1.進一步掌握利用數(shù)學歸納法證明不等式的方法和技巧.2.了解貝努利不等式,并能利用它證明簡單的不等式.目標導航DIANLITOUXI典例透析SUITANG

2、YANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理1.用數(shù)學歸納法證明不等式運用數(shù)學歸納法證明不等式的兩個步驟實際上是分別證明兩個不等式.尤其是第二步:一方面需要我們充分利用歸納假設提供的“便利”,另一方面還需要結合運用比較法、綜合法、分析法、反證法和放縮法等其他不等式的證明方法.名師

3、點撥從“n=k”到“n=k+1”的方法與技巧:在用數(shù)學歸納法證明不等式的問題中,從“n=k”到“n=k+1”的過渡,利用歸納假設是比較困難的一步,它不像用數(shù)學歸納法證明恒等式問題一樣,只需拼湊出所需要的結構來,而證明不等式的第二步中,從“n=k”到“n=k+1”,只用拼湊的方法,有時也行不通,因為對不等式來說,它還涉及“放縮”的問題,它可能需要通過“放大”或“縮小”的過程,才能利用上歸納假設,因此,我們可以利用“比較法”“綜合法”“分析法”等來分析從“n=k”到“n=k+1”的變化,從中找到“放縮尺度”,準確地拼湊出所需要的結構.目標導航DIANLITOUXI典例透析SUITANGYANLIA

4、N隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理【做一做1-1】 設f(k)是定義在正整數(shù)集上的函數(shù),且f(k)滿足:“當f(k)k2成立時,總可推出f(k+1)(k+1)2成立.”那么下列命題總成立的是()A.若f(3)9成立,則當k1時,均有f(k)k2成立B.若f(5)25成立,則當k5時

5、,均有f(k)k2成立C.若f(7)49成立,則當k8時,均有f(k)42,對于任意的k4,總有f(k)k2成立.答案:D目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLI

6、AN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理2.貝努利不等式對任何實數(shù)x-1和任何正整數(shù)n,有(1+x)n1+nx.【做一做2】 設nN+,求證:3n2n.分析:利用貝努利不等式來證明.證明:3n=(1+2)n,根據(jù)貝努利不等式,有(1+2)n1+n2=1+2n.上式右邊舍去1,得(1+2

7、)n2n.3n2n成立.目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理題型一題型二題型三題型一 用數(shù)學歸納法證明不等式 分析:利用數(shù)學歸納法證明. 目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHIS

8、HISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理題型一題型二題型三反思在利用數(shù)學歸納法證明不等式時,要注意對式子的變形,通過放縮、比較、分析、綜合等證明不等式的方法,得出要證明的目標不等式.目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLI

9、TOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理題型一題型二題型三目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHIS

10、HISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理題型一題型二題型三題型二 利用貝努利不等式證明不等式 分析:用求商比較法證明an+14ab2ab+6a2b2=14a2b2=224,g(24)=44-25=224,有f(4)g(24).由此猜想當1n2(nN+)時,f(n)=g(2n).當n3(nN+)時,f(n)g(2n).下面用數(shù)學歸納法證明.(1)當n=3時,由上述計算知猜想成立;目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI

11、典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理題型一題型二題型三(2)假設當n=k(k3,kN+)時,猜想成立,則f(k)g(2k),即(a+b)k-ak-bk4k-2k+1,則當n=k+1時,f(k+1)=(a+b)k+1-ak+1-bk+1=(a+b)(a+b)k-aak-bbk=(a+b)(a+b)k-ak-bk+akb+abk,反思利用數(shù)學歸納法解決探索

12、型不等式問題的思路是:先通過觀察、判斷、猜想得出結論,然后用數(shù)學歸納法證明結論.目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理題型一題型二題型三目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHI

13、SHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理題型一題型二題型三目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUI

14、TANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理12345答案:C 目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理

15、12345答案:B 目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理12345答案:A 目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析S

16、UITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理12345目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理目標導航DIANLITOUXI典例透析SUITANGYANLIAN隨堂演練ZHISHISHULI知識梳理123455已知acdb0,a+b=c+d,n為大于1的正整數(shù),求證:an+bncn+dn.