函數(shù)的單調(diào)性課件

11.1函數(shù)(59)11.1函數(shù)(59)21.1函數(shù)(59)21.1函數(shù)(59)1.1.1變量與區(qū)間1.集合具有某種特定性質(zhì)的事物的總體.組成這個(gè)集合的事物稱為該集合的元素.有限集無(wú)限集31.1函數(shù)(59)1.1.1變量與區(qū)間1.集合具有某種特定性質(zhì)的事物的總體數(shù)集分類:N----自然數(shù)集Z----整數(shù)集Q----有理數(shù)集R----實(shí)數(shù)集數(shù)集間的關(guān)系:例如不含任何元素的集合稱為空集.例如,規(guī)定空集為任何集合的子集.41.1函數(shù)(59)數(shù)集分類:N----自然數(shù)集Z----整數(shù)集Q----有理數(shù)2.區(qū)間是指介于某兩個(gè)實(shí)數(shù)之間的全體實(shí)數(shù).這兩個(gè)實(shí)數(shù)叫做區(qū)間的端點(diǎn).稱為開區(qū)間,稱為閉區(qū)間,51.1函數(shù)(59)2.區(qū)間是指介于某兩個(gè)實(shí)數(shù)之間的全體實(shí)數(shù).這兩個(gè)實(shí)數(shù)叫做區(qū)間稱為半閉區(qū)間,稱為半開區(qū)間,有限區(qū)間無(wú)限區(qū)間區(qū)間長(zhǎng)度的定義：兩端點(diǎn)間的距離(線段的長(zhǎng)度)稱為區(qū)間的長(zhǎng)度.61.1函數(shù)(59)稱為半閉區(qū)間,稱為半開區(qū)間,有限區(qū)間無(wú)限區(qū)間區(qū)間長(zhǎng)度的定義：3.鄰域71.1函數(shù)(59)3.鄰域71.1函數(shù)(59)4.常量與變量在某過(guò)程中數(shù)值保持不變的量稱為常量,注意常量與變量是相對(duì)“過(guò)程”而言的.通常用字母a,b,c等表示常量,而數(shù)值變化的量稱為變量.常量與變量的表示方法：用字母x,y,t等表示變量.81.1函數(shù)(59)4.常量與變量在某過(guò)程中數(shù)值保持不變的量稱為常量,注意常量5.絕對(duì)值運(yùn)算性質(zhì):絕對(duì)值不等式:91.1函數(shù)(59)5.絕對(duì)值運(yùn)算性質(zhì):絕對(duì)值不等式:91.1函數(shù)(59)1.1.2函數(shù)的概念例圓內(nèi)接正多邊形的周長(zhǎng)圓內(nèi)接正n邊形Or)101.1函數(shù)(59)1.1.2函數(shù)的概念例圓內(nèi)接正多邊形的周長(zhǎng)圓內(nèi)接正因變量自變量數(shù)集D叫做這個(gè)函數(shù)的定義域111.1函數(shù)(59)因變量自變量數(shù)集D叫做這個(gè)函數(shù)的定義域111.1函數(shù)(5自變量因變量對(duì)應(yīng)法則f函數(shù)的兩要素：定義域與對(duì)應(yīng)法則.約定:定義域是自變量所能取的使算式有意義的一切實(shí)數(shù)值（即自然定義域）.121.1函數(shù)(59)自變量因變量對(duì)應(yīng)法則f函數(shù)的兩要素：定義域與對(duì)應(yīng)法則.約定:定義：如果自變量在定義域內(nèi)任取一個(gè)數(shù)值時(shí)，對(duì)應(yīng)的函數(shù)值總是只有一個(gè)，這種函數(shù)叫做單值函數(shù)，否則叫與多值函數(shù)．131.1函數(shù)(59)定義：如果自變量在定義域內(nèi)任取一個(gè)數(shù)值時(shí)，對(duì)應(yīng)的函數(shù)值總(1)符號(hào)函數(shù)幾個(gè)特殊的函數(shù)舉例1-1xyo141.1函數(shù)(59)(1)符號(hào)函數(shù)幾個(gè)特殊的函數(shù)舉例1-1xyo14(2)取整函數(shù)y=[x][x]表示不超過(guò)的最大整數(shù)12345-2-4-4-3-2-14321-1-3xyo階梯曲線151.1函數(shù)(59)(2)取整函數(shù)y=[x]1234有理數(shù)點(diǎn)無(wú)理數(shù)點(diǎn)?1xyo(3)狄利克雷函數(shù)161.1函數(shù)(59)有理數(shù)點(diǎn)無(wú)理數(shù)點(diǎn)?1xyo(3)狄利克雷函數(shù)161.1(4)取最值函數(shù)yxoyxo171.1函數(shù)(59)(4)取最值函數(shù)yxoyxo171.1函數(shù)(59)在自變量的不同變化范圍中,對(duì)應(yīng)法則用不同的式子來(lái)表示的函數(shù),稱為分段函數(shù)。181.1函數(shù)(59)在自變量的不同變化范圍中,對(duì)應(yīng)法則用不同的式例1脈沖發(fā)生器產(chǎn)生一個(gè)單三角脈沖,其波形如圖所示,寫出電壓U與時(shí)間的函數(shù)關(guān)系式.解單三角脈沖信號(hào)的電壓191.1函數(shù)(59)例1脈沖發(fā)生器產(chǎn)生一個(gè)單三角脈沖,其波形如圖所示,寫出電壓201.1函數(shù)(59)201.1函數(shù)(59)例2解故211.1函數(shù)(59)例2解故211.1函數(shù)(59)1.1.3函數(shù)的特性M-Myxoy=f(x)X有界無(wú)界M-MyxoX1．函數(shù)的有界性:221.1函數(shù)(59)1.1.3函數(shù)的特性M-Myxoy=f(x)X有界無(wú)界M2．函數(shù)的單調(diào)性:xyo231.1函數(shù)(59)2．函數(shù)的單調(diào)性:xyo231.1函數(shù)(59)xyo241.1函數(shù)(59)xyo241.1函數(shù)(59)3．函數(shù)的奇偶性:偶函數(shù)yxox-x251.1函數(shù)(59)3．函數(shù)的奇偶性:偶函數(shù)yxox-x251.1函數(shù)(59奇函數(shù)yxox-x261.1函數(shù)(59)奇函數(shù)yxox-x261.1函數(shù)(59)4．函數(shù)的周期性:（通常周期函數(shù)的周期是指其最小正周期）.271.1函數(shù)(59)4．函數(shù)的周期性:（通常周期函數(shù)的周期是指其最小正周期）.2例3解單值函數(shù),有界函數(shù),偶函數(shù),周期函數(shù)(無(wú)最小正周期)不是單調(diào)函數(shù),281.1函數(shù)(59)例3解單值函數(shù),有界函數(shù),偶函數(shù),周期函數(shù)(無(wú)最小正周期)1.1.4復(fù)合函數(shù)與反函數(shù)1.復(fù)合函數(shù)定義:291.1函數(shù)(59)1.1.4復(fù)合函數(shù)與反函數(shù)1.復(fù)合函數(shù)定義:291注意:1.不是任何兩個(gè)函數(shù)都可以復(fù)合成一個(gè)復(fù)合函數(shù)的;2.復(fù)合函數(shù)可以由兩個(gè)以上的函數(shù)經(jīng)過(guò)復(fù)合構(gòu)成.301.1函數(shù)(59)注意:1.不是任何兩個(gè)函數(shù)都可以復(fù)合成一個(gè)復(fù)合函數(shù)的;2.復(fù)DWDW2.反函數(shù)311.1函數(shù)(59)DWDW2.反函數(shù)311.1函數(shù)(59)

直接函數(shù)與反函數(shù)的圖形關(guān)于直線對(duì)稱。321.1函數(shù)(59)直接函數(shù)與反函數(shù)的圖形關(guān)于直線對(duì)稱。31.1.5初等函數(shù)(1)冪函數(shù)1.基本初等函數(shù)331.1函數(shù)(59)1.1.5初等函數(shù)(1)冪函數(shù)1.基本初等函數(shù)(2)指數(shù)函數(shù)341.1函數(shù)(59)(2)指數(shù)函數(shù)341.1函數(shù)(59)(3)對(duì)數(shù)函數(shù)351.1函數(shù)(59)(3)對(duì)數(shù)函數(shù)351.1函數(shù)(59)(4)三角函數(shù)正弦函數(shù)361.1函數(shù)(59)(4)三角函數(shù)正弦函數(shù)361.1函數(shù)(59)余弦函數(shù)371.1函數(shù)(59)余弦函數(shù)371.1函數(shù)(59)正切函數(shù)381.1函數(shù)(59)正切函數(shù)381.1函數(shù)(59)余切函數(shù)391.1函數(shù)(59)余切函數(shù)391.1函數(shù)(59)正割函數(shù)401.1函數(shù)(59)正割函數(shù)401.1函數(shù)(59)余割函數(shù)411.1函數(shù)(59)余割函數(shù)411.1函數(shù)(59)(5)反三角函數(shù)421.1函數(shù)(59)(5)反三角函數(shù)421.1函數(shù)(59)431.1函數(shù)(59)431.1函數(shù)(59)441.1函數(shù)(59)441.1函數(shù)(59)冪函數(shù),指數(shù)函數(shù),對(duì)數(shù)函數(shù),三角函數(shù)和反三角函數(shù)統(tǒng)稱為基本初等函數(shù).451.1函數(shù)(59)冪函數(shù),指數(shù)函數(shù),對(duì)數(shù)函數(shù),三角函數(shù)和反三角2.初等函數(shù)

由常數(shù)和基本初等函數(shù)經(jīng)過(guò)有限次四則運(yùn)算和有限次的函數(shù)復(fù)合步驟所構(gòu)成的函數(shù),稱為初等函數(shù)。例4設(shè)解461.1函數(shù)(59)2.初等函數(shù)由常數(shù)和基本初等函數(shù)經(jīng)過(guò)有471.1函數(shù)(59)471.1函數(shù)(59)綜上所述481.1函數(shù)(59)綜上所述481.1函數(shù)(59)1.1.6小結(jié)與思考題基本概念集合,區(qū)間,鄰域,常量與變量,絕對(duì)值.函數(shù)的概念函數(shù)的特性有界性,單調(diào)性,奇偶性,周期性.反函數(shù)1.小結(jié)491.1函數(shù)(59)1.1.6小結(jié)與思考題基本概念函數(shù)的概念函數(shù)的特性反函數(shù)函數(shù)的分類函數(shù)初等函數(shù)非初等函數(shù)(分段函數(shù),有無(wú)窮多項(xiàng)的函數(shù)等)代數(shù)函數(shù)超越函數(shù)有理函數(shù)無(wú)理函數(shù)有理整函數(shù)(多項(xiàng)式函數(shù))有理分函數(shù)(分式函數(shù))501.1函數(shù)(59)函數(shù)的分類函數(shù)初等函數(shù)非初等函數(shù)(分段函數(shù),有無(wú)窮多項(xiàng)的函數(shù)2.思考題一511.1函數(shù)(59)2.思考題一511.1函數(shù)(59)思考題一解答設(shè)則故521.1函數(shù)(59)思考題一解答設(shè)則故521.1函數(shù)(59)3.思考題二531.1函數(shù)(59)3.思考題二531.1函數(shù)(59)思考題二解答不能．541.1函數(shù)(59)思考題二解答不能．541.1函數(shù)(59)課堂練習(xí)題551.1函數(shù)(59)課堂練習(xí)題551.1函數(shù)(59)561.1函數(shù)(59)561.1函數(shù)(59)571.1函數(shù)(59)571.1函數(shù)(59)課堂練習(xí)題答案581.1函數(shù)(59)課堂練習(xí)題答案581.1函數(shù)(59)591.1函數(shù)(59)591.1函數(shù)(59)601.1函數(shù)(59)11.1函數(shù)(59)611.1函數(shù)(59)21.1函數(shù)(59)1.1.1變量與區(qū)間1.集合具有某種特定性質(zhì)的事物的總體.組成這個(gè)集合的事物稱為該集合的元素.有限集無(wú)限集621.1函數(shù)(59)1.1.1變量與區(qū)間1.集合具有某種特定性質(zhì)的事物的總體數(shù)集分類:N----自然數(shù)集Z----整數(shù)集Q----有理數(shù)集R----實(shí)數(shù)集數(shù)集間的關(guān)系:例如不含任何元素的集合稱為空集.例如,規(guī)定空集為任何集合的子集.631.1函數(shù)(59)數(shù)集分類:N----自然數(shù)集Z----整數(shù)集Q----有理數(shù)2.區(qū)間是指介于某兩個(gè)實(shí)數(shù)之間的全體實(shí)數(shù).這兩個(gè)實(shí)數(shù)叫做區(qū)間的端點(diǎn).稱為開區(qū)間,稱為閉區(qū)間,641.1函數(shù)(59)2.區(qū)間是指介于某兩個(gè)實(shí)數(shù)之間的全體實(shí)數(shù).這兩個(gè)實(shí)數(shù)叫做區(qū)間稱為半閉區(qū)間,稱為半開區(qū)間,有限區(qū)間無(wú)限區(qū)間區(qū)間長(zhǎng)度的定義：兩端點(diǎn)間的距離(線段的長(zhǎng)度)稱為區(qū)間的長(zhǎng)度.651.1函數(shù)(59)稱為半閉區(qū)間,稱為半開區(qū)間,有限區(qū)間無(wú)限區(qū)間區(qū)間長(zhǎng)度的定義：3.鄰域661.1函數(shù)(59)3.鄰域71.1函數(shù)(59)4.常量與變量在某過(guò)程中數(shù)值保持不變的量稱為常量,注意常量與變量是相對(duì)“過(guò)程”而言的.通常用字母a,b,c等表示常量,而數(shù)值變化的量稱為變量.常量與變量的表示方法：用字母x,y,t等表示變量.671.1函數(shù)(59)4.常量與變量在某過(guò)程中數(shù)值保持不變的量稱為常量,注意常量5.絕對(duì)值運(yùn)算性質(zhì):絕對(duì)值不等式:681.1函數(shù)(59)5.絕對(duì)值運(yùn)算性質(zhì):絕對(duì)值不等式:91.1函數(shù)(59)1.1.2函數(shù)的概念例圓內(nèi)接正多邊形的周長(zhǎng)圓內(nèi)接正n邊形Or)691.1函數(shù)(59)1.1.2函數(shù)的概念例圓內(nèi)接正多邊形的周長(zhǎng)圓內(nèi)接正因變量自變量數(shù)集D叫做這個(gè)函數(shù)的定義域701.1函數(shù)(59)因變量自變量數(shù)集D叫做這個(gè)函數(shù)的定義域111.1函數(shù)(5自變量因變量對(duì)應(yīng)法則f函數(shù)的兩要素：定義域與對(duì)應(yīng)法則.約定:定義域是自變量所能取的使算式有意義的一切實(shí)數(shù)值（即自然定義域）.711.1函數(shù)(59)自變量因變量對(duì)應(yīng)法則f函數(shù)的兩要素：定義域與對(duì)應(yīng)法則.約定:定義：如果自變量在定義域內(nèi)任取一個(gè)數(shù)值時(shí)，對(duì)應(yīng)的函數(shù)值總是只有一個(gè)，這種函數(shù)叫做單值函數(shù)，否則叫與多值函數(shù)．721.1函數(shù)(59)定義：如果自變量在定義域內(nèi)任取一個(gè)數(shù)值時(shí)，對(duì)應(yīng)的函數(shù)值總(1)符號(hào)函數(shù)幾個(gè)特殊的函數(shù)舉例1-1xyo731.1函數(shù)(59)(1)符號(hào)函數(shù)幾個(gè)特殊的函數(shù)舉例1-1xyo14(2)取整函數(shù)y=[x][x]表示不超過(guò)的最大整數(shù)12345-2-4-4-3-2-14321-1-3xyo階梯曲線741.1函數(shù)(59)(2)取整函數(shù)y=[x]1234有理數(shù)點(diǎn)無(wú)理數(shù)點(diǎn)?1xyo(3)狄利克雷函數(shù)751.1函數(shù)(59)有理數(shù)點(diǎn)無(wú)理數(shù)點(diǎn)?1xyo(3)狄利克雷函數(shù)161.1(4)取最值函數(shù)yxoyxo761.1函數(shù)(59)(4)取最值函數(shù)yxoyxo171.1函數(shù)(59)在自變量的不同變化范圍中,對(duì)應(yīng)法則用不同的式子來(lái)表示的函數(shù),稱為分段函數(shù)。771.1函數(shù)(59)在自變量的不同變化范圍中,對(duì)應(yīng)法則用不同的式例1脈沖發(fā)生器產(chǎn)生一個(gè)單三角脈沖,其波形如圖所示,寫出電壓U與時(shí)間的函數(shù)關(guān)系式.解單三角脈沖信號(hào)的電壓781.1函數(shù)(59)例1脈沖發(fā)生器產(chǎn)生一個(gè)單三角脈沖,其波形如圖所示,寫出電壓791.1函數(shù)(59)201.1函數(shù)(59)例2解故801.1函數(shù)(59)例2解故211.1函數(shù)(59)1.1.3函數(shù)的特性M-Myxoy=f(x)X有界無(wú)界M-MyxoX1．函數(shù)的有界性:811.1函數(shù)(59)1.1.3函數(shù)的特性M-Myxoy=f(x)X有界無(wú)界M2．函數(shù)的單調(diào)性:xyo821.1函數(shù)(59)2．函數(shù)的單調(diào)性:xyo231.1函數(shù)(59)xyo831.1函數(shù)(59)xyo241.1函數(shù)(59)3．函數(shù)的奇偶性:偶函數(shù)yxox-x841.1函數(shù)(59)3．函數(shù)的奇偶性:偶函數(shù)yxox-x251.1函數(shù)(59奇函數(shù)yxox-x851.1函數(shù)(59)奇函數(shù)yxox-x261.1函數(shù)(59)4．函數(shù)的周期性:（通常周期函數(shù)的周期是指其最小正周期）.861.1函數(shù)(59)4．函數(shù)的周期性:（通常周期函數(shù)的周期是指其最小正周期）.2例3解單值函數(shù),有界函數(shù),偶函數(shù),周期函數(shù)(無(wú)最小正周期)不是單調(diào)函數(shù),871.1函數(shù)(59)例3解單值函數(shù),有界函數(shù),偶函數(shù),周期函數(shù)(無(wú)最小正周期)1.1.4復(fù)合函數(shù)與反函數(shù)1.復(fù)合函數(shù)定義:881.1函數(shù)(59)1.1.4復(fù)合函數(shù)與反函數(shù)1.復(fù)合函數(shù)定義:291注意:1.不是任何兩個(gè)函數(shù)都可以復(fù)合成一個(gè)復(fù)合函數(shù)的;2.復(fù)合函數(shù)可以由兩個(gè)以上的函數(shù)經(jīng)過(guò)復(fù)合構(gòu)成.891.1函數(shù)(59)注意:1.不是任何兩個(gè)函數(shù)都可以復(fù)合成一個(gè)復(fù)合函數(shù)的;2.復(fù)DWDW2.反函數(shù)901.1函數(shù)(59)DWDW2.反函數(shù)311.1函數(shù)(59)

直接函數(shù)與反函數(shù)的圖形關(guān)于直線對(duì)稱。911.1函數(shù)(59)直接函數(shù)與反函數(shù)的圖形關(guān)于直線對(duì)稱。31.1.5初等函數(shù)(1)冪函數(shù)1.基本初等函數(shù)921.1函數(shù)(59)1.1.5初等函數(shù)(1)冪函數(shù)1.基本初等函數(shù)(2)指數(shù)函數(shù)931.1函數(shù)(59)(2)指數(shù)函數(shù)341.1函數(shù)(59)(3)對(duì)數(shù)函數(shù)941.1函數(shù)(59)(3)對(duì)數(shù)函數(shù)351.1函數(shù)(59)(4)三角函數(shù)正弦函數(shù)951.1函數(shù)(59)(4)三角函數(shù)正弦函數(shù)361.1函數(shù)(59)余弦函數(shù)961.1函數(shù)(59)余弦函數(shù)371.1函數(shù)(59)正切函數(shù)971.1函數(shù)(59)正切函數(shù)381.1函數(shù)(59)余切函數(shù)981.1函數(shù)(59)余切函數(shù)391.1函數(shù)(59)正割函數(shù)991.1函數(shù)(59)正割函數(shù)401.1函數(shù)(59)余割函數(shù)1001.1函數(shù)(59)余割函數(shù)411.1函數(shù)(59)(5)反三角函數(shù)1011.1函數(shù)(59)(5)反三角函數(shù)421.1函數(shù)(59)1021.1函數(shù)(59)431.1函數(shù)(59)1031.1函數(shù)(59)441.1函數(shù)(59)冪函數(shù),指數(shù)函數(shù),對(duì)數(shù)函數(shù),三角函數(shù)和反三角函數(shù)統(tǒng)稱為基本初等函數(shù).1041.1函數(shù)(59)冪函數(shù),指數(shù)函數(shù),對(duì)數(shù)函數(shù),三角函數(shù)和反三角2.初等函數(shù)