蘇教版必修一 2.2.1 第1課時(shí) 函數(shù)的單調(diào)性 學(xué)案 (1).doc

22函數(shù)的簡單性質(zhì)22.1函數(shù)的單調(diào)性第1課時(shí)函數(shù)的單調(diào)性1理解并掌握單調(diào)增(減)函數(shù)的定義及其幾何意義(重點(diǎn))2會用單調(diào)性的定義證明函數(shù)的單調(diào)性(重點(diǎn)、難點(diǎn))3會求函數(shù)的單調(diào)區(qū)間(重點(diǎn)、難點(diǎn))基礎(chǔ)初探教材整理1單調(diào)性的定義閱讀教材p37，完成下列問題1定義一般地，設(shè)函數(shù)yf (x)的定義域?yàn)閍，區(qū)間ia.如果對于區(qū)間i內(nèi)的任意兩個(gè)值x1，x2，當(dāng)x1x2時(shí)，都有f (x1)f (x2)，那么就說yf (x)在區(qū)間i上是單調(diào)增函數(shù)，i稱為yf (x)的單調(diào)增區(qū)間如果對于區(qū)間i內(nèi)的任意兩個(gè)值x1，x2，當(dāng)x1f (x2)，那么就說yf (x)在區(qū)間i上是單調(diào)減函數(shù)，i稱為yf (x)的單調(diào)減區(qū)間2函數(shù)單調(diào)性與單調(diào)區(qū)間如果函數(shù)yf (x)在區(qū)間i上是單調(diào)增函數(shù)或單調(diào)減函數(shù)，那么就說函數(shù)yf (x)在區(qū)間i上具有單調(diào)性單調(diào)增區(qū)間和單調(diào)減區(qū)間統(tǒng)稱為單調(diào)區(qū)間1判斷(正確的打“” ，錯(cuò)誤的打“”)(1)所有函數(shù)在定義域上都具有單調(diào)性()(2)若函數(shù)yf (x)在定義域上有f (1)f (2)，則函數(shù)yf (x)是增函數(shù)()(3)若函數(shù)f (x)在實(shí)數(shù)集r上是增函數(shù)，則有f (1)f (4)()(4)若函數(shù)yf (x)在區(qū)間1,3上是減函數(shù)，則函數(shù)f (x)的單調(diào)區(qū)間是1,3()【解析】(1)y2在定義域上無單調(diào)性；(2)只根據(jù)f (1)f (2)，無法確定f (x)的單調(diào)性；(3)由f (x)在r上遞增，可以得出f (1)f (4)；(4)一個(gè)函數(shù)的增區(qū)間也是單調(diào)區(qū)間【答案】(1)(2)(3)(4)2下列說法正確的是_(填序號)定義在(a，b)上的函數(shù)f (x)，若存在x1x2，使f (x1)f (x2)，那么f (x)在(a，b)上為增函數(shù)；定義在(a，b)上的函數(shù)f (x)，若有無窮多對x1，x2(a，b)，使得當(dāng)x1x2時(shí)，有f (x1)f (x2)，那么f (x)在(a，b)上為增函數(shù)；若f (x)在區(qū)間i1上為增函數(shù)，在區(qū)間i2上也為增函數(shù)，那么f (x)在i1i2上也一定為增函數(shù)；若f (x)在區(qū)間i上為增函數(shù)，且f (x1)f (x2)(x1，x2i)，那么x10時(shí)，函數(shù)在r上單調(diào)遞_，當(dāng)k0時(shí)，函數(shù)在(，0)，(0，)上單調(diào)遞_，當(dāng)k0時(shí)，函數(shù)在上單調(diào)遞_，在上單調(diào)遞_，當(dāng)a0時(shí)，函數(shù)在上單調(diào)遞_，在上單調(diào)遞_ 【答案】(1)增減(2)減增(3)減增增減小組合作型利用函數(shù)圖象求單調(diào)區(qū)間作出下列函數(shù)的圖象，并寫出單調(diào)區(qū)間(1)yx24；(2)y；(3)f (x)【精彩點(diǎn)撥】在圖象上看從左向右上升的部分即遞增，從左向右下降的部分即遞減【自主解答】三個(gè)函數(shù)圖象如圖(1)(2)(3)(1)(2)(3)(1)yx24的單調(diào)遞減區(qū)間為(，0)，遞增區(qū)間為(0，)(2)y的單調(diào)增區(qū)間為(，0)，(0，)，無遞減區(qū)間(3)f (x)的單調(diào)增區(qū)間為(，0)，(2，)，遞減區(qū)間為(0,2)1應(yīng)用圖象確定單調(diào)性時(shí)，應(yīng)掌握各種基本函數(shù)的圖象的形狀，并能通過圖象的“上升”或“下降”趨勢來找到函數(shù)的遞增或遞減區(qū)間，但應(yīng)注意端點(diǎn)是否在定義域之內(nèi)2當(dāng)函數(shù)的單調(diào)區(qū)間不唯一時(shí)，中間用“，”隔開，或用“和”連接，但不能用“或”和“”連接再練一題1函數(shù)f (x)x2|x|(xr)的單調(diào)遞增區(qū)間為_. 【解析】(1)f (x)x2|x|圖象如圖所示：f (x)的單調(diào)增區(qū)間為，.【答案】，函數(shù)單調(diào)性的判斷與證明用定義證明函數(shù)f (x)在(1，)上是減函數(shù)【精彩點(diǎn)撥】解答本題可直接利用函數(shù)單調(diào)性的定義來判斷【自主解答】證明：設(shè)x1，x2是區(qū)間(1，)上任意兩個(gè)實(shí)數(shù)，且x1x2，則f (x1)f (x2).1x1x2，x2x10，x110，x210，0，即f (x1)f (x2)，y在(1，)上是減函數(shù)用定義證明(判斷)函數(shù)單調(diào)性的步驟再練一題2證明函數(shù)f (x)在(1，)上單調(diào)遞增【證明】任取x1，x2(1，)，且x11，x1x21，x1x210.又x1x2，x1x20，f (x1)f (x2)，f (x)在(1，)上單調(diào)遞增探究共研型單調(diào)性的應(yīng)用探究1如何利用函數(shù)的單調(diào)性比較兩個(gè)函數(shù)值的大?。俊咎崾尽肯扰袛嗪瘮?shù)f (x)在區(qū)間d上的單調(diào)性，如果函數(shù)f (x)在d上是增函數(shù)，當(dāng)x1x2時(shí)，則f (x1)f (x2)，如果f (x)在d上是減函數(shù)，結(jié)論則相反探究2如果已知函數(shù)的單調(diào)性和函數(shù)值的大小，能否判斷對應(yīng)自變量的大??？【提示】能利用函數(shù)單調(diào)性，將函數(shù)值的大小關(guān)系轉(zhuǎn)化為自變量的大小關(guān)系，即脫去f 符號，轉(zhuǎn)化為自變量的大小關(guān)系已知函數(shù)f (x)是定義在2,2上的增函數(shù)，且f (x2)f (1x)，則x的取值范圍為_【精彩點(diǎn)撥】根據(jù)單調(diào)性可以去掉f ，還應(yīng)考慮定義域【自主解答】f (x)是定義在2,2上的增函數(shù)，且f (x2)f (1x)，x21x，x.又f (x)的定義域?yàn)?,2，0x3，綜上，0x.【答案】1利用函數(shù)單調(diào)性的定義比較大小，一方面是正向應(yīng)用，即若yf (x)在給定區(qū)間上是增函數(shù)，則當(dāng)x1x2時(shí)，f (x1)x2時(shí)，f (x1)f (x2)；另一方面是逆向應(yīng)用，即若yf (x)在給定區(qū)間上是增函數(shù)，則當(dāng)f (x1)f (x2)時(shí)，x1f (x2)時(shí)，x1x2.當(dāng)yf (x)在給定區(qū)間上是減函數(shù)時(shí)，同理可得相應(yīng)結(jié)論2根據(jù)函數(shù)的單調(diào)性研究參數(shù)的取值范圍，往往會根據(jù)函數(shù)在某一區(qū)間上的增減性確定不等式，此時(shí)常需要將含參數(shù)的變量單獨(dú)移到一側(cè)，用變量的范圍推出參數(shù)的范圍再練一題3已知f (x)在r上為減函數(shù)且f (2m)f (9m)，則m的取值范圍是_. 【解析】由題意可得2m9m，m3. 【答案】m31已知函數(shù)f (x)的圖象如圖221所示，則f (x)的單調(diào)減區(qū)間為_圖221【解析】由題圖知，f (x)在上圖象呈下降趨勢，單調(diào)減區(qū)間為.【答案】2下列四個(gè)函數(shù)中，在(0，)上是增函數(shù)的是_(1)f (x)；(2)f (x)x23x；(3)f (x)3x；(4)f (x)|x|.【解析】函數(shù)f (x)的單調(diào)遞增區(qū)間是(，1)，(1，)，顯然在(0，)上是增函數(shù)；函數(shù)f (x)x23x在上單調(diào)遞減，在上單調(diào)遞增；函數(shù)f (x)3x在(0，)上是減函數(shù)；函數(shù)f (x)|x|在(0，)上是減函數(shù)，故(2)(3)(4)錯(cuò)誤【答案】(1)3若函數(shù)f (x)(k2)xb在r上是減函數(shù)，則k的取值范圍為_. 【解析】f (x)(k2)xb在r上是減函數(shù)，k20，k2.【答案】k24已知函數(shù)f (x)則f (x)的單調(diào)增區(qū)間為_【解析】f (x)為分段函數(shù)，當(dāng)x1時(shí)，f (x)單調(diào)遞增，當(dāng)x(1,1)時(shí)，f (x)單調(diào)遞減，當(dāng)x1時(shí)，f (x)單調(diào)遞增【答案】1，)，(，15已知函數(shù)f (x)x2，x1，)(1)判斷函數(shù)f (x)在區(qū)間1，)上的單調(diào)性；(2)解不等式：f f (x1 008)【解】(1)設(shè)1x1x2，f (x1)f (x2