人教A版高中數(shù)學(xué)（必修第一冊）同步講義 4.5 函數(shù)與方程（原卷版）

第第頁第05講4.5.1函數(shù)的零點(diǎn)與方程的解課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①了解函數(shù)的零點(diǎn)與方程的解的關(guān)系，并能結(jié)合函數(shù)的圖象判定函數(shù)的零點(diǎn)。②能根據(jù)函數(shù)零點(diǎn)存在性定理對函數(shù)零點(diǎn)存在進(jìn)行判定，同時(shí)能處理與函數(shù)零點(diǎn)問題相結(jié)合的求參數(shù)及綜合類的問題。通過本節(jié)課的學(xué)習(xí)，要求能判定函數(shù)零點(diǎn)的存在，同時(shí)能解決與函數(shù)零點(diǎn)相結(jié)合的綜合問題知識點(diǎn)01：函數(shù)零點(diǎn)的概念1、函數(shù)零點(diǎn)的概念對于一般函數(shù)SKIPIF1<0，我們把使SKIPIF1<0的實(shí)數(shù)SKIPIF1<0叫做函數(shù)SKIPIF1<0的零點(diǎn).幾何定義:函數(shù)SKIPIF1<0的零點(diǎn)就是方程SKIPIF1<0的實(shí)數(shù)解,也就是函數(shù)SKIPIF1<0的圖象與SKIPIF1<0軸的公共點(diǎn)的橫坐標(biāo).

這樣：方程SKIPIF1<0有實(shí)數(shù)解SKIPIF1<0函數(shù)SKIPIF1<0有零點(diǎn)SKIPIF1<0函數(shù)SKIPIF1<0的圖象與SKIPIF1<0軸有公共點(diǎn)2、已學(xué)基本初等函數(shù)的零點(diǎn)①一次函數(shù)SKIPIF1<0只有一個(gè)零點(diǎn)SKIPIF1<0；②反比例函數(shù)SKIPIF1<0沒有零點(diǎn)；③指數(shù)函數(shù)SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）沒有零點(diǎn)；④對數(shù)函數(shù)SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）只有一個(gè)零點(diǎn)1；⑤冪函數(shù)SKIPIF1<0當(dāng)SKIPIF1<0時(shí)，有一個(gè)零點(diǎn)0；當(dāng)SKIPIF1<0時(shí)，無零點(diǎn)。知識點(diǎn)02：函數(shù)零點(diǎn)存在定理及其應(yīng)用1、函數(shù)零點(diǎn)存在定理如果函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的圖象是一條連續(xù)不斷的曲線，且有SKIPIF1<0，那么函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)至少有一個(gè)零點(diǎn)，即存在SKIPIF1<0，使得SKIPIF1<0，這個(gè)SKIPIF1<0也就是方程SKIPIF1<0的解.說明：定理要求具備兩個(gè)條件:①函數(shù)在區(qū)間SKIPIF1<0上的圖象是連續(xù)不斷的;②SKIPIF1<0.兩個(gè)條件缺一不可.2、函數(shù)零點(diǎn)的求法①代數(shù)法：根據(jù)零點(diǎn)定義，求出方程SKIPIF1<0的實(shí)數(shù)解;②數(shù)形結(jié)合法：作出函數(shù)圖象，利用函數(shù)性質(zhì)求解【即學(xué)即練1】（2023春·四川廣安·高一?？茧A段練習(xí)）函數(shù)SKIPIF1<0的零點(diǎn)為．3、函數(shù)零點(diǎn)個(gè)數(shù)的判斷①利用代數(shù)法，求出所有零點(diǎn)；②數(shù)形結(jié)合，通過作圖，找出圖象與SKIPIF1<0軸交點(diǎn)的個(gè)數(shù)；③數(shù)形結(jié)合，通過分離，將原函數(shù)拆分成兩個(gè)函數(shù)，找到兩個(gè)函數(shù)圖象交點(diǎn)的個(gè)數(shù)；④函數(shù)零點(diǎn)唯一：函數(shù)存在零點(diǎn)+函數(shù)單調(diào).知識點(diǎn)03：二次函數(shù)的零點(diǎn)問題一元二次方程SKIPIF1<0的實(shí)數(shù)根也稱為函數(shù)SKIPIF1<0的零點(diǎn).當(dāng)SKIPIF1<0時(shí)，一元二次方程SKIPIF1<0的實(shí)數(shù)根、二次函數(shù)SKIPIF1<0的零點(diǎn)之間的關(guān)系如下表所示：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0的實(shí)數(shù)根SKIPIF1<0（其中SKIPIF1<0）SKIPIF1<0方程無實(shí)數(shù)根SKIPIF1<0的圖象SKIPIF1<0的零點(diǎn)SKIPIF1<0SKIPIF1<0函數(shù)無零點(diǎn)【即學(xué)即練2】（2023·高一課時(shí)練習(xí)）若函數(shù)SKIPIF1<0的一個(gè)零點(diǎn)是1，則它的另一個(gè)零點(diǎn)是．題型01求函數(shù)的零點(diǎn)【典例1】函數(shù)SKIPIF1<0的零點(diǎn)為．【典例2】已知函數(shù)SKIPIF1<0，則函數(shù)SKIPIF1<0的零點(diǎn)為.【變式1】函數(shù)SKIPIF1<0的零點(diǎn)是【變式2】求下列函數(shù)的零點(diǎn)．(1)SKIPIF1<0；(2)SKIPIF1<0.題型02函數(shù)零點(diǎn)個(gè)數(shù)的判斷【典例1】函數(shù)SKIPIF1<0的零點(diǎn)個(gè)數(shù)為（）A．1B．2C．1或2D．0【典例2】方程SKIPIF1<0的實(shí)數(shù)解的個(gè)數(shù)是（

）A．0B．1C．2D．3【典例3】已知SKIPIF1<0，方程SKIPIF1<0的實(shí)根個(gè)數(shù)為．【典例4】若函數(shù)SKIPIF1<0，則函數(shù)SKIPIF1<0的零點(diǎn)個(gè)數(shù)是.【變式1】函數(shù)SKIPIF1<0的零點(diǎn)的個(gè)數(shù)是（

）A．0B．1C．2D．無數(shù)個(gè)【變式2】已知函數(shù)SKIPIF1<0.(1)作出函數(shù)SKIPIF1<0的圖象；(2)就a的取值范圍討論函數(shù)SKIPIF1<0的零點(diǎn)的個(gè)數(shù)．【變式3】若SKIPIF1<0的值域?yàn)镾KIPIF1<0，則SKIPIF1<0至多有個(gè)零點(diǎn).【變式4】函數(shù)SKIPIF1<0的圖象與函數(shù)SKIPIF1<0的圖象的交點(diǎn)個(gè)數(shù)為個(gè)．題型03判斷函數(shù)零點(diǎn)所在的區(qū)間【典例1】若SKIPIF1<0是方程SKIPIF1<0的解，則SKIPIF1<0（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】設(shè)SKIPIF1<0為方程SKIPIF1<0的解，若SKIPIF1<0，則SKIPIF1<0的值為.【變式1】函數(shù)SKIPIF1<0的零點(diǎn)所在區(qū)間是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式2】函數(shù)SKIPIF1<0的零點(diǎn)為SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，則k的值為（

）A．1B．2C．0D．3題型04已知零點(diǎn)個(gè)數(shù)求參數(shù)的取值范圍【典例1】若方程SKIPIF1<0有兩個(gè)不同的實(shí)數(shù)根，則實(shí)數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】設(shè)SKIPIF1<0表示m，n中的較小數(shù)．若函數(shù)SKIPIF1<0至少有3個(gè)零點(diǎn)，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例3】若函數(shù)SKIPIF1<0有2個(gè)零點(diǎn)，求實(shí)數(shù)a的取值范圍．【典例4】已知函數(shù)SKIPIF1<0是偶函數(shù).當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0.(1)求函數(shù)SKIPIF1<0在SKIPIF1<0上的解析式；(2)若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)，求實(shí)數(shù)a的取值范圍；(3)已知SKIPIF1<0，試討論SKIPIF1<0的零點(diǎn)個(gè)數(shù)，并求對應(yīng)的m的取值范圍.【變式1】設(shè)SKIPIF1<0，函數(shù)SKIPIF1<0若SKIPIF1<0恰有一個(gè)零點(diǎn)，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式2】（多選）已知函數(shù)SKIPIF1<0，若關(guān)于x的方程SKIPIF1<0恰有兩個(gè)互異的實(shí)數(shù)解，則實(shí)數(shù)a的值可以是（

）A．0B．1C．SKIPIF1<0D．2【變式3】若函數(shù)SKIPIF1<0在SKIPIF1<0內(nèi)有且只有一個(gè)零點(diǎn)，則SKIPIF1<0的取值集合是.【變式4】已知SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)．(1)求SKIPIF1<0的值；(2)畫出SKIPIF1<0的圖象，并指出其單調(diào)減區(qū)間；(3)若關(guān)于SKIPIF1<0的方程SKIPIF1<0有2個(gè)不相等的實(shí)數(shù)根，求實(shí)數(shù)SKIPIF1<0的取值范圍．題型05已知零點(diǎn)所在區(qū)間求參數(shù)的取值范圍【典例1】函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上存在零點(diǎn)，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】（多選）函數(shù)SKIPIF1<0的一個(gè)零點(diǎn)在區(qū)間SKIPIF1<0內(nèi)，則實(shí)數(shù)a的可能取值是（

）A．0B．1C．2D．3【典例3】設(shè)SKIPIF1<0為實(shí)數(shù)，函數(shù)SKIPIF1<0在SKIPIF1<0上有零點(diǎn)，則實(shí)數(shù)SKIPIF1<0的取值范圍為．【變式1】若函數(shù)SKIPIF1<0在SKIPIF1<0內(nèi)恰有一個(gè)零點(diǎn)，則a的取值范圍（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式2】）若關(guān)于SKIPIF1<0的方程SKIPIF1<0在SKIPIF1<0上有解，則實(shí)數(shù)SKIPIF1<0的取值范圍是．【變式3】若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)有零點(diǎn)，則實(shí)數(shù)SKIPIF1<0的取值范圍是.題型06二次函數(shù)的零點(diǎn)問題【典例1】已知函數(shù)SKIPIF1<0的零點(diǎn)為SKIPIF1<0，滿足SKIPIF1<0，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】方程SKIPIF1<0的一根大于1，一根小于1，則實(shí)數(shù)SKIPIF1<0的取值范圍是.【典例3】（1）判斷二次函數(shù)SKIPIF1<0在SKIPIF1<0內(nèi)是否存在零點(diǎn)；（2）若二次函數(shù)SKIPIF1<0SKIPIF1<0的兩個(gè)零點(diǎn)均為正數(shù)，求實(shí)數(shù)SKIPIF1<0的取值范圍．【變式1】已知關(guān)于SKIPIF1<0的方程SKIPIF1<0，SKIPIF1<0存在兩個(gè)不同的實(shí)根，則實(shí)數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式2】若關(guān)于SKIPIF1<0的方程SKIPIF1<0在SKIPIF1<0內(nèi)有解，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式3】已知函數(shù)SKIPIF1<0SKIPIF1<0.(1)若該函數(shù)有兩個(gè)不相等的正零點(diǎn)，求SKIPIF1<0的取值范圍；(2)若該函數(shù)有兩個(gè)零點(diǎn)，一個(gè)大于1，另一個(gè)小于1，求SKIPIF1<0的取值范圍．題型07函數(shù)與方程綜合【典例1】已知函數(shù)SKIPIF1<0，常數(shù)SKIPIF1<0.(1)若SKIPIF1<0是奇函數(shù)，求SKIPIF1<0的值；(2)若SKIPIF1<0，SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)有且僅有一個(gè)零點(diǎn)，求實(shí)數(shù)SKIPIF1<0的取值范圍.【典例2】已知函數(shù)SKIPIF1<0(1)證明：函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減；(2)討論關(guān)于x的方程SKIPIF1<0的實(shí)數(shù)解的個(gè)數(shù)．【變式1】已知函數(shù)SKIPIF1<0為奇函數(shù).(1)求實(shí)數(shù)a的值；(2)若方程SKIPIF1<0在區(qū)間SKIPIF1<0上無解，求實(shí)數(shù)m的取值范圍.【變式2】已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)，且當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，函數(shù)SKIPIF1<0在SKIPIF1<0軸左側(cè)的圖象如圖所示．(1)求函數(shù)SKIPIF1<0的解析式；(2)若關(guān)于SKIPIF1<0的方程SKIPIF1<0有SKIPIF1<0個(gè)不相等的實(shí)數(shù)根，求實(shí)數(shù)SKIPIF1<0的取值范圍．A夯實(shí)基礎(chǔ)一、單選題1．已知函數(shù)SKIPIF1<0，則SKIPIF1<0的零點(diǎn)所在的區(qū)間為（

）.A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<02．已知方程SKIPIF1<0的解在SKIPIF1<0內(nèi)，則SKIPIF1<0（

）A．3B．2C．1D．03．已知函數(shù)SKIPIF1<0的兩個(gè)零點(diǎn)都大于2，則實(shí)數(shù)m的取值范圍是（）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<04．已知二次函數(shù)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)的零點(diǎn)情況是（

）A．有兩個(gè)零點(diǎn)B．有唯一零點(diǎn)C．沒有零點(diǎn)D．不確定5．已知函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的零點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<06．設(shè)實(shí)數(shù)a為常數(shù)，則函數(shù)SKIPIF1<0存在零點(diǎn)的充分必要條件是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<07．已知函數(shù)SKIPIF1<0若函數(shù)SKIPIF1<0有五個(gè)零點(diǎn)，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<08．享有“數(shù)學(xué)王子”稱號的德國數(shù)學(xué)家高斯，是近代數(shù)學(xué)奠基者之一，SKIPIF1<0被稱為“高斯函數(shù)”，其中SKIPIF1<0表示不超過SKIPIF1<0的最大整數(shù)，例如：SKIPIF1<0，設(shè)SKIPIF1<0為函數(shù)SKIPIF1<0的零點(diǎn)，則SKIPIF1<0（

）A．3B．4C．5D．6二、多選題9．函數(shù)SKIPIF1<0的零點(diǎn)可以是（）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<010．設(shè)SKIPIF1<0為定義在R上的奇函數(shù)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0為常數(shù)），則（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．函數(shù)SKIPIF1<0僅有一個(gè)零點(diǎn)三、填空題11．已知函數(shù)SKIPIF1<0在SKIPIF1<0上有零點(diǎn)，則實(shí)數(shù)a的取值范圍是．12．若SKIPIF1<0是方程SKIPIF1<0的解，則SKIPIF1<0在區(qū)間內(nèi)(填序號).①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0；④SKIPIF1<0.四、解答題13．已知一次函數(shù)SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0．（1）求這個(gè)函數(shù)的解析式；（2）若函數(shù)SKIPIF1<0，求函數(shù)SKIPIF1<0的零點(diǎn)．14．已知函數(shù)SKIPIF1<0，且SKIPIF1<0的圖象經(jīng)過點(diǎn)SKIPIF1<0.(1)求SKIPIF1<0的值；(2)求SKIPIF1<0在區(qū)間SKIPIF1<0上的最小值；(3)若SKIPIF1<0，求證：SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)存在零點(diǎn).B能力提升1．已知函數(shù)SKIPIF1<0，若SKIPIF1<0且SKIPIF1<0，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<02．已知函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0．若SKIPIF1<0恰有2個(gè)零點(diǎn)，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<03．函數(shù)SKIPIF1<0滿足：①SKIPIF1<0在SKIPIF1<0內(nèi)是單調(diào)遞增函數(shù)；②SKIPIF1<0在SKIPIF1<0上的值域?yàn)镾KIPIF1<0，則稱區(qū)間SKIPIF1<0為SKIPIF1<0的SKIPIF1<0級“調(diào)和區(qū)間”.若函數(shù)SKIPIF1<0存在SKIPIF1<0級“調(diào)和區(qū)間”，則SKIPIF1<0的取值范圍是.4．對于函數(shù)SKIPIF1<0，若存在SKIPIF1<0，使得SKIPIF1<0，則稱SKIPIF1<0為函數(shù)SKIPIF1<0的“不動(dòng)點(diǎn)”.若存在SKIPIF1<0，使得SKIPIF1<0，則稱SKIPIF1<0為函數(shù)SKIPIF1<0的“穩(wěn)定點(diǎn)”.記函數(shù)SKIPIF1<0的“不動(dòng)點(diǎn)”和“穩(wěn)定點(diǎn)”的集合分別為SKIPIF1<0和SKIPIF1<0，即SKIPIF1<0.經(jīng)研究發(fā)現(xiàn)：若函數(shù)SKIPIF1<0為增函數(shù)，則SKIPIF1<0.設(shè)函數(shù)SKIPIF1<0，若存在SKIPIF1<0使SKIPIF1<0成立，則SKIPIF1<0的取值范圍是.C綜合素養(yǎng)1．已知SKIPIF1<0.(1)當(dāng)SKIPIF1<0時(shí)，作出函數(shù)SKIPIF1<0的圖象，若關(guān)于SKIPIF1<0的方程SKIPIF1<0有四個(gè)解，直接寫出SKIPIF1<0的取值范圍；(2)若SKIPIF1<0的定義域和值域均為SKIPIF1<0，求實(shí)數(shù)SKIPIF1<0的值；(3)若SKIPIF1<0是SKIPIF1<0上的嚴(yán)格減函數(shù)，且對任意的SKIPIF1<0，總有SKIPIF1<0成立，求實(shí)數(shù)SKIPIF1<0的取值范圍.2．已知SKIPIF1<0為R上的奇函數(shù)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0．(1)求SKIPIF1<0的值；(2)求SKIPIF1<0的解析式；(3)作出SKIPIF1<0的圖象，并求當(dāng)函數(shù)SKIPIF1<0與函數(shù)SKIPIF1<0圖象恰有三個(gè)不同的交點(diǎn)時(shí)，實(shí)數(shù)m的取值范圍．4.5.2用二分法求方程的近似解課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①理解運(yùn)用二分法逼近方程近似解的數(shù)學(xué)思想。②了解二分法只能用于求變號零點(diǎn)的方法。③借助數(shù)學(xué)工具用二分法求方程的近似解。④能解決與方程近似解有關(guān)的問題。通過本節(jié)課的學(xué)習(xí)，要求會(huì)用二分法進(jìn)行簡單方程近似解的求解，并能根據(jù)題的要求，解決與二分法相關(guān)的參數(shù)問題的處理。知識點(diǎn)01：區(qū)間中點(diǎn)對于區(qū)間SKIPIF1<0，其中點(diǎn)SKIPIF1<0知識點(diǎn)02：二分法1、二分法的概念對于在區(qū)間SKIPIF1<0上圖象連續(xù)不斷且SKIPIF1<0的函數(shù)SKIPIF1<0，通過不斷的把它的零點(diǎn)所在區(qū)間一分為二，使所得區(qū)間的兩個(gè)端點(diǎn)逐步逼近零點(diǎn)，進(jìn)而得到零點(diǎn)近似值的方法叫做二分法（bisection)【即學(xué)即練1】下列圖象中，不能用二分法求函數(shù)零點(diǎn)的是（

）A．

B．

C．

D．

2、用二分法求零點(diǎn)的近似值給定精確度SKIPIF1<0，用二分法求函數(shù)SKIPIF1<0零點(diǎn)SKIPIF1<0的近似值的一般步驟如下：（1）確定零點(diǎn)SKIPIF1<0的初始區(qū)間SKIPIF1<0，驗(yàn)證SKIPIF1<0；（2）求區(qū)間SKIPIF1<0的中點(diǎn)SKIPIF1<0（3）計(jì)算SKIPIF1<0;①若SKIPIF1<0（此時(shí)SKIPIF1<0)，則SKIPIF1<0就是函數(shù)的零點(diǎn)；②若SKIPIF1<0（此時(shí)SKIPIF1<0)，則令SKIPIF1<0；③若SKIPIF1<0（此時(shí)SKIPIF1<0)，則令SKIPIF1<0；（4）判斷是否達(dá)到精確度SKIPIF1<0，若SKIPIF1<0，則得到零點(diǎn)近似值SKIPIF1<0(或SKIPIF1<0)，否則重復(fù)2--4【即學(xué)即練2】已知函數(shù)SKIPIF1<0的表達(dá)式為SKIPIF1<0，用二分法計(jì)算此函數(shù)在區(qū)間SKIPIF1<0上零點(diǎn)的近似值，第一次計(jì)算SKIPIF1<0、SKIPIF1<0的值，第二次計(jì)算SKIPIF1<0的值，第三次計(jì)算SKIPIF1<0的值，則SKIPIF1<0．題型01二分法概念的理解【典例1】用二分法求函數(shù)零點(diǎn)的近似值適合于（

）A．變號零點(diǎn)B．不變號零點(diǎn)C．都適合D．都不適合【典例2】下列函數(shù)圖象與x軸都有公共點(diǎn)，其中不能用二分法求圖中函數(shù)零點(diǎn)近似值的是（

）A．B．C．D．【典例3】（多選）下列函數(shù)中，能用二分法求函數(shù)零點(diǎn)的有（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式1】下列函數(shù)中，不能用二分法求零點(diǎn)的是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式2】下列函數(shù)一定能用“二分法”求其零點(diǎn)的是（

）A．SKIPIF1<0（k，b為常數(shù)，且SKIPIF1<0）B．SKIPIF1<0（a，b，c為常數(shù)，且SKIPIF1<0）C．SKIPIF1<0D．SKIPIF1<0（SKIPIF1<0，k為常數(shù)）【變式3】下列函數(shù)圖象均與SKIPIF1<0軸有交點(diǎn)，其中能用二分法求函數(shù)零點(diǎn)的是題型02確定零點(diǎn)（根）所在區(qū)間【典例1】函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的零點(diǎn)必屬于區(qū)間（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】函數(shù)SKIPIF1<0的零點(diǎn)SKIPIF1<0，對區(qū)間SKIPIF1<0利用兩次“二分法”，可確定SKIPIF1<0所在的區(qū)間為．【典例3】已知定義在SKIPIF1<0上的偶函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，若在SKIPIF1<0內(nèi)關(guān)于SKIPIF1<0的方程SKIPIF1<0恰有3個(gè)不同的實(shí)數(shù)根，則SKIPIF1<0的取值范圍是A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式1】用二分法求方程SKIPIF1<0在SKIPIF1<0內(nèi)的近似解，已知SKIPIF1<0判斷，方程的根應(yīng)落在區(qū)間（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式2】用二分法求函數(shù)SKIPIF1<0的零點(diǎn)可以取的初始區(qū)間是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式3】用二分法求方程SKIPIF1<0近似解時(shí)，所取的第一個(gè)區(qū)間可以是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0題型03用二分法求函數(shù)的零點(diǎn)的近似值【典例1】某同學(xué)在用二分法研究函數(shù)SKIPIF1<0的零點(diǎn)時(shí)，．得到如下函數(shù)值的參考數(shù)據(jù)：x11.251.3751.406251.43751.5SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<00.05670.14600.3284則下列說法正確的是（

）A．1.25是滿足精確度為0.1的近似值B．1.5是滿足精確度為0.1的近似值C．1.4375是滿足精確度為0.05的近似值D．1.375是滿足精確度為0.05的近似值【典例2】（多選）某同學(xué)求函數(shù)SKIPIF1<0的零點(diǎn)時(shí)，用計(jì)算器算得部分函數(shù)值如表所示：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0則方程SKIPIF1<0的近似解（精確度SKIPIF1<0）可取為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例3】用二分法求函數(shù)SKIPIF1<0的一個(gè)零點(diǎn)，其參考數(shù)據(jù)如下：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0據(jù)此數(shù)據(jù)，可知SKIPIF1<0的一個(gè)零點(diǎn)的近似值可取為（誤差不超過0.005）．【變式1】已知函數(shù)SKIPIF1<0的部分函數(shù)值如下表所示：那么函數(shù)SKIPIF1<0的一個(gè)零點(diǎn)的近似值（精確度為0.1）為（

）x10.50.750.6250.5625SKIPIF1<00.6321SKIPIF1<00.27760.0897SKIPIF1<0A．0.55B．0.57C．0.65D．0.7【變式2】）函數(shù)SKIPIF1<0的一個(gè)正數(shù)零點(diǎn)附近的函數(shù)值用二分法逐次計(jì)算，參考數(shù)據(jù)如下：SKIPIF1<0

SKIPIF1<0

SKIPIF1<0SKIPIF1<0

SKIPIF1<0

SKIPIF1<0那么方程的一個(gè)近似解（精確度為0.1）為（

）A．1.5B．1.25C．1.41D．1.44【變式3】用二分法求方程的近似解，求得SKIPIF1<0的部分函數(shù)值數(shù)據(jù)如下表所示：SKIPIF1<0121.51.6251.751.8751.8125SKIPIF1<0-63-2.625-1.459-0.141.34180.5793則當(dāng)精確度為0.1時(shí)，方程SKIPIF1<0的近似解可取為A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0題型04二分法的過程【典例1】用二分法求方程SKIPIF1<0在SKIPIF1<0內(nèi)的近似解，已知SKIPIF1<0判斷，方程的根應(yīng)落在區(qū)間（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】用二分法求函數(shù)SKIPIF1<0在SKIPIF1<0內(nèi)的唯一零點(diǎn)時(shí)，精度為0.001，則經(jīng)過一次二分就結(jié)束計(jì)算的條件是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例3】已知函數(shù)SKIPIF1<0在SKIPIF1<0上有零點(diǎn)，用二分法求零點(diǎn)的近似值（精確度小于0.1）時(shí)，至少需要進(jìn)行次函數(shù)值的計(jì)算.【變式1】已知函數(shù)SKIPIF1<0在SKIPIF1<0內(nèi)有一個(gè)零點(diǎn)，要使零點(diǎn)的近似值的精確度為0.001，若只從二等分區(qū)間的角度來考慮，則對區(qū)間SKIPIF1<0至少需要二等分（

）A．8次B．9次C．10次D．11次【變式2】已知函數(shù)SKIPIF1<0的表達(dá)式為SKIPIF1<0，用二分法計(jì)算此函數(shù)在區(qū)間SKIPIF1<0上零點(diǎn)的近似值，第一次計(jì)算SKIPIF1<0、SKIPIF1<0的值，第二次計(jì)算SKIPIF1<0的值，第三次計(jì)算SKIPIF1<0的值，則SKIPIF1<0．【變式3】）用“二分法”研究函數(shù)SKIPIF1<0的零點(diǎn)時(shí)，第一次計(jì)算SKIPIF1<0，可知必存在零點(diǎn)SKIPIF1<0，則第二次應(yīng)計(jì)算，這時(shí)可以判斷零點(diǎn)SKIPIF1<0．A夯實(shí)基礎(chǔ)一、單選題1．函數(shù)SKIPIF1<0的圖象是連續(xù)不斷的曲線，在用二分法求方程SKIPIF1<0在SKIPIF1<0內(nèi)近似解的過程中可得SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則方程的解所在區(qū)間為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．不能確定2．已知函數(shù)SKIPIF1<0的一個(gè)零點(diǎn)附近的函數(shù)值的參考數(shù)據(jù)如下表：x00.50.531250.56250.6250.751SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<00.0660.2150.5121.099由二分法，方程SKIPIF1<0的近似解（精確度為0.05）可能是（

）A．0.625B．SKIPIF1<0C．0.5625D．0.0663．用二分法求函數(shù)SKIPIF1<0的一個(gè)零點(diǎn)的近似值（誤差不超過SKIPIF1<0）時(shí)，依次計(jì)算得到如下數(shù)據(jù)：SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，關(guān)于下一步的說法正確的是（）A．已經(jīng)達(dá)到對誤差的要求，可以取SKIPIF1<0作為近似值B．已經(jīng)達(dá)到對誤差的要求，可以取SKIPIF1<0作為近似值C．沒有達(dá)到對誤差的要求，應(yīng)該接著計(jì)算SKIPIF1<0D．沒有達(dá)到對誤差的要求，應(yīng)該接著計(jì)算SKIPIF1<04．用二分法研究函數(shù)SKIPIF1<0的零點(diǎn)時(shí)，第一次計(jì)算，得SKIPIF1<0，SKIPIF1<0，第二次應(yīng)計(jì)算SKIPIF1<0，則SKIPIF1<0等于（

）A．1B．SKIPIF1<0C．0.25D．0.755．在用“二分法”求函數(shù)SKIPIF1<0零點(diǎn)近似值時(shí)，若第一次所取區(qū)間為SKIPIF1<0，則第三次所取區(qū)間可能是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<06．下列函數(shù)中，不能用二分法求零點(diǎn)的是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<07．用二分法求函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的零點(diǎn),要求精確度為SKIPIF1<0時(shí),所需二分區(qū)間的次數(shù)最少為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<08．在使用二分法計(jì)算函數(shù)SKIPIF1<0的零點(diǎn)的近似解時(shí)，現(xiàn)已知其所在區(qū)間為SKIPIF1<0，如果要求近似解的精確度為0.1，則接下來至少需要計(jì)算（

）次區(qū)間中點(diǎn)的函數(shù)值．A．2B．3C．4D．5二、多選題9．在用二分法求函數(shù)SKIPIF1<0的一個(gè)正實(shí)數(shù)零點(diǎn)時(shí)，經(jīng)計(jì)算，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則函數(shù)SKIPIF1<0的一個(gè)誤差不超過SKIPIF1<0的正實(shí)數(shù)零點(diǎn)可以為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<010．關(guān)于函數(shù)SKIPIF1<0的零點(diǎn)，下列說法正確的是：（）（參考數(shù)據(jù)：SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0）A．函數(shù)SKIPIF1<0的零點(diǎn)個(gè)數(shù)為1B．函數(shù)SKIPIF1<0的零點(diǎn)個(gè)數(shù)為2C．用二分法求函數(shù)SKIPIF1<0的一個(gè)零點(diǎn)的近似解可取為SKIPIF1<0（精確到SKIPIF1<0）D．用二分法求函數(shù)SKIPIF1<0的一個(gè)零點(diǎn)的近似解可取為SKIPIF1<0（精確到SKIPIF1<0）三、填空題11．函數(shù)SKIPIF1<0的零點(diǎn)SKIPIF1<0，對區(qū)間SKIPIF1<0利用兩次“二分法”，可確定SKIPIF1<0所在的區(qū)間為．12．在用二分法求函數(shù)SKIPIF1<0的零點(diǎn)近似值時(shí)，若第一次所取區(qū)間為SKIPIF1<0，則第三次所取區(qū)間可能是．（寫出一個(gè)符合條件的區(qū)間即可）四、解答題13．利用二分法，求方程SKIPIF1<0的近似解．（精確度為0.1）14．用二分法求SKIPIF1<0在SKIPIF1<0內(nèi)的近似解(精確度為SKIPIF1<0).參考數(shù)據(jù)：x1.1251.251.3751.51.6251.751.8752x2.182.382.592.833.083.363.67

4.5.3函數(shù)模型的應(yīng)用課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①了解函數(shù)模型（如指數(shù)函數(shù)、對數(shù)函數(shù)、冪函數(shù)、分段函數(shù)等在社會(huì)生活中普遍使用的函數(shù)模型）的廣泛應(yīng)用。②在實(shí)際情境中，會(huì)選擇合適的函數(shù)類型刻畫現(xiàn)實(shí)問題的變化規(guī)律。通過本節(jié)課的學(xué)習(xí)，掌握一次函數(shù)、二次函數(shù)、指數(shù)函數(shù)、對數(shù)函數(shù)、冪函數(shù)以及其他函數(shù)模型；會(huì)從實(shí)際問題中抽象出函數(shù)模型，進(jìn)而利用函數(shù)知識求解.高考對函數(shù)應(yīng)用的考查，常與二次函數(shù)、基本不等式等知識交匯．知識點(diǎn)一：常見函數(shù)模型1、一次函數(shù)模型SKIPIF1<0（SKIPIF1<0，SKIPIF1<0為常數(shù)）2、反比例函數(shù)模型SKIPIF1<0（SKIPIF1<0）3、二次函數(shù)模型SKIPIF1<0（SKIPIF1<0）4、指數(shù)函數(shù)模型SKIPIF1<0（SKIPIF1<0且SKIPIF1<0，SKIPIF1<0）5、對數(shù)函數(shù)模型SKIPIF1<0（SKIPIF1<0且SKIPIF1<0，SKIPIF1<0）6、冪函數(shù)模型SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）7、分段函數(shù)模型：兩種或兩種以上上述六種模型的綜合8、對勾函數(shù)模型：SKIPIF1<0題型01指數(shù)、對數(shù)、冪函數(shù)模型的增長差異【典例1】若三個(gè)變量SKIPIF1<0，SKIPIF1<0，SKIPIF1<0隨著變量x的變化情況如下表.x1357911SKIPIF1<0525456585105SKIPIF1<0529245218919685177149SKIPIF1<056.106.616.957.27.6則關(guān)于x分別呈函數(shù)模型：SKIPIF1<0，SKIPIF1<0，SKIPIF1<0變化的變量依次是（

）A．SKIPIF1<0，SKIPIF1<0，SKIPIF1<0B．SKIPIF1<0，SKIPIF1<0，SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0，SKIPIF1<0D．SKIPIF1<0，SKIPIF1<0，SKIPIF1<0【典例2】下列選項(xiàng)是四種生意預(yù)期的收益y關(guān)于時(shí)間x的函數(shù)，從足夠長遠(yuǎn)的角度看，更為有前途的生意是.①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0；④SKIPIF1<0【變式1】下列函數(shù)中，隨著SKIPIF1<0的增大，函數(shù)值的增長速度最快的是(

)A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式2】下列函數(shù)中，隨著SKIPIF1<0的增大，增長速度最快的是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0題型02根據(jù)實(shí)際問題增長率選擇合適的模型【典例1】在一次數(shù)學(xué)實(shí)驗(yàn)中，運(yùn)用計(jì)算器采集到如下一組數(shù)據(jù)：xSKIPIF1<0SKIPIF1<001.02.03.0y0.240.5112.023.988.02則x、y的函數(shù)關(guān)系與下列哪類函數(shù)最接近（其中a、b為待定系數(shù)）？（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】某同學(xué)參加研究性學(xué)習(xí)活動(dòng)，得到如下實(shí)驗(yàn)數(shù)據(jù)：SKIPIF1<0392781SKIPIF1<02SKIPIF1<0SKIPIF1<0SKIPIF1<0以下函數(shù)中最符合變量SKIPIF1<0與SKIPIF1<0的對應(yīng)關(guān)系的是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例3】某旅游風(fēng)景區(qū)發(fā)行的紀(jì)念章即將投放市場，根據(jù)市場調(diào)研情況，預(yù)計(jì)每枚該紀(jì)念章的市場價(jià)SKIPIF1<0（單位：元）與上市時(shí)間SKIPIF1<0（單位：天）的數(shù)據(jù)如下：上市時(shí)間SKIPIF1<0天2620市場價(jià)SKIPIF1<0元10278120為了描述該紀(jì)念章的市場價(jià)SKIPIF1<0與上市時(shí)間SKIPIF1<0的變化關(guān)系，現(xiàn)有以下三種函數(shù)模型供選擇：（1）SKIPIF1<0；（2）SKIPIF1<0；（3）SKIPIF1<0．(1)根據(jù)如表數(shù)據(jù)，請選取一個(gè)恰當(dāng)?shù)暮瘮?shù)模型并說明理由；(2)利用你選取的函數(shù)，求該紀(jì)念章市場價(jià)最低時(shí)的上市天數(shù)及最低的價(jià)格.【變式1】在某種新型材料的研制中，實(shí)驗(yàn)人員獲得了下列一組實(shí)驗(yàn)數(shù)據(jù)．現(xiàn)準(zhǔn)備用下列四個(gè)函數(shù)中的一個(gè)近似地表示這些數(shù)據(jù)的規(guī)律，其中最接近的一個(gè)是（

）SKIPIF1<01.953.003.945.106.12SKIPIF1<00.971.591.982.352.61A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式2】農(nóng)場為了解某農(nóng)作物的產(chǎn)量情況，將近四年的年產(chǎn)量SKIPIF1<0（單位：萬斤）與年份序號x之間的關(guān)系統(tǒng)計(jì)如下：x（第×年）1234SKIPIF1<0（萬斤）4.005.627.008.86若SKIPIF1<0近似符合以下兩種函數(shù)模型之一：①SKIPIF1<0；②SKIPIF1<0．則你認(rèn)為最適合的函數(shù)模型的序號是__________．請簡要說明理由．【變式3】某地西紅柿上市后，通過市場調(diào)查，得到西紅柿種植成本Q（單位：元/10kg）與上市時(shí)間t（單位：天）的數(shù)據(jù)如下表：時(shí)間t79101113種植成本Q1911101119為了描述西紅柿種植成本Q與上市時(shí)間t的變化關(guān)系，現(xiàn)有以下四種函數(shù)模型供選擇：①SKIPIF1<0，②SKIPIF1<0，③SKIPIF1<0，④SKIPIF1<0．(1)選出你認(rèn)為最符合實(shí)際的函數(shù)模型并說明理由，同時(shí)求出相應(yīng)的函數(shù)解析式；(2)在第（1）問的條件下，若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的最大值為110，最小值為10，求實(shí)數(shù)m的最大值．題型03利用二次函數(shù)模型解決實(shí)際問題【典例1】一個(gè)車輛制造廠引進(jìn)了一條摩托車整車裝配流水線，這條流水線生產(chǎn)的摩托車數(shù)量SKIPIF1<0（單位：輛）與創(chuàng)造的價(jià)值SKIPIF1<0（單位：元）之間的關(guān)系為：SKIPIF1<0.如果這家工廠希望在一個(gè)星期內(nèi)利用這條流水線創(chuàng)收60000元以上，請你給出一個(gè)該工廠在這周內(nèi)生成的摩托車數(shù)量的建議，使工廠能夠達(dá)成這個(gè)周創(chuàng)收目標(biāo)，那么你的建議是.【典例2】由于慣性作用，行駛中的汽車在剎車后要滑行一段距離才能停下，這段距離叫做剎車距離．下表是對某種型號汽車剎車性能的測試數(shù)據(jù)．剎車時(shí)車速SKIPIF1<0153040506080剎車距離SKIPIF1<01.236.2011.517.8025.2044.40(1)試選擇合適的函數(shù)模型擬合測試數(shù)據(jù)，并寫出函數(shù)解析式；(2)若車速為SKIPIF1<0，剎車距離為多少？若測得剎車距離為SKIPIF1<0，剎車時(shí)的車速是多少？（可以使用計(jì)算器輔助計(jì)算）【變式1】某商品進(jìn)貨單價(jià)為SKIPIF1<0元，若銷售價(jià)為SKIPIF1<0元，可賣出SKIPIF1<0個(gè)，如果銷售單價(jià)每漲SKIPIF1<0元，銷售量就減少SKIPIF1<0個(gè)，為了獲得最大利潤，則此商品的最佳售價(jià)應(yīng)為．【變式2】為弘揚(yáng)“中國女排精神”，加強(qiáng)青少年體育發(fā)展.學(xué)校在體育課中組織學(xué)生進(jìn)行排球練習(xí)，某同學(xué)以初速度SKIPIF1<0豎直上拋一排球，該排球能夠在拋出點(diǎn)2m以上的位置最多停留時(shí)間為秒（小數(shù)點(diǎn)后保留兩位有效數(shù)字）.（注：若不計(jì)空氣阻力，則豎直上拋的物體距離拋出點(diǎn)的高度SKIPIF1<0與時(shí)間SKIPIF1<0滿足關(guān)系式SKIPIF1<0，其中SKIPIF1<0．）

題型04分段函數(shù)模型的應(yīng)用【典例1】華為消費(fèi)者業(yè)務(wù)產(chǎn)品全面覆蓋手機(jī)、移動(dòng)寬帶終端、終端云等，憑借自身的全球化網(wǎng)絡(luò)優(yōu)勢、全球化運(yùn)營能力，致力于將最新的科技帶給消費(fèi)者，讓世界各地享受到技術(shù)進(jìn)步的喜悅，以行踐言，實(shí)現(xiàn)夢想.已知華為公司生產(chǎn)mate系列的某款手機(jī)的年固定成本為200萬元，每生產(chǎn)1只還需另投入80元.設(shè)華為公司一年內(nèi)共生產(chǎn)該款手機(jī)x萬只并全部銷售完，每萬只的銷售收入為SKIPIF1<0萬元，且SKIPIF1<0(1)寫出年利潤SKIPIF1<0（萬元）關(guān)于年產(chǎn)量SKIPIF1<0（萬只）的函數(shù)解