導(dǎo)數(shù)的幾何意義廣大附中施永紅一．作業(yè)討論并由學(xué)生陳述：問題：1．導(dǎo)數(shù)SKIPIF1<0的本質(zhì)是什么？2．在高臺(tái)跳水運(yùn)動(dòng)中，SKIPIF1<0秒SKIPIF1<0時(shí)運(yùn)動(dòng)員相對于水面的高度是SKIPIF1<0（單位：SKIPIF1<0），求運(yùn)動(dòng)員在SKIPIF1<0時(shí)的瞬時(shí)速度，并解釋此時(shí)的運(yùn)動(dòng)狀態(tài)。3．圓面積SKIPIF1<0是半徑SKIPIF1<0的函數(shù)SKIPIF1<0，求圓在SKIPIF1<0時(shí)的面積瞬時(shí)變化率，并解釋此時(shí)圓面積的瞬時(shí)變化規(guī)律。4．球體積SKIPIF1<0是半徑SKIPIF1<0的函數(shù)SKIPIF1<0，問：球在SKIPIF1<0時(shí)的體積瞬時(shí)變化率，并解釋此時(shí)球體積的瞬時(shí)變化規(guī)律。參考答案：1．導(dǎo)數(shù)SKIPIF1<0的本質(zhì)是函數(shù)SKIPIF1<0在SKIPIF1<0處的瞬時(shí)變化率，即：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0，所以SKIPIF1<0。運(yùn)動(dòng)員在SKIPIF1<0時(shí)的瞬時(shí)速度為SKIPIF1<0。這說明運(yùn)動(dòng)員在SKIPIF1<0附近，正以大約SKIPIF1<0的速率下落。3．SKIPIF1<0＝SKIPIF1<0SKIPIF1<0，所以SKIPIF1<0。這說明圓在SKIPIF1<0的附近，面積正以大約SKIPIF1<0的瞬時(shí)變化率增大。（思考，動(dòng)畫）SKIPIF1<0SKIPIF1<0SKIPIF1<0=SKIPIF1<0，所以SKIPIF1<0。這說明球在SKIPIF1<0的附近，體積正以大約SKIPIF1<0的瞬時(shí)變化率增大。（思考，動(dòng)畫）二．（一）引入：函數(shù)的思想中，數(shù)形結(jié)合是一種重要而且直觀的數(shù)學(xué)思想方法。我們已經(jīng)學(xué)習(xí)了導(dǎo)數(shù)的本質(zhì)及其物理意義或者實(shí)際意義，我們不禁要問：導(dǎo)數(shù)的幾何意義是什么？例如：上述的三個(gè)問題中，SKIPIF1<0,SKIPIF1<0,SKIPIF1<0的幾何意義分別是什么？我們利用數(shù)形結(jié)合，畫出函數(shù)的圖像，看看通過函數(shù)的圖像研究，能有什么新的體會(huì)。（二）轉(zhuǎn)化：為了使研究更具有操作性，我們拋開問題的實(shí)際背景，從研究更簡單的函數(shù)入手，這也是數(shù)學(xué)研究的常用方法。（三）問題與探索：求函數(shù)SKIPIF1<0在SKIPIF1<0處的導(dǎo)數(shù)SKIPIF1<0，說說它的本質(zhì)，并用數(shù)形結(jié)合的方法，探索SKIPIF1<0的幾何意義。SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0。所以SKIPIF1<0SKIPIF1<0的本質(zhì)是函數(shù)SKIPIF1<0在SKIPIF1<0處的瞬時(shí)變化率是SKIPIF1<0，即函數(shù)SKIPIF1<0在SKIPIF1<0的附近正以大約SKIPIF1<0的瞬時(shí)變化率增大。畫出函數(shù)SKIPIF1<0的圖像，從圖像上的哪幾個(gè)點(diǎn)開始研究？APQB1MNBSKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0APQB1MNBSKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0聯(lián)想：定義SKIPIF1<0從點(diǎn)SKIPIF1<0.點(diǎn)SKIPIF1<0入手。那么點(diǎn)SKIPIF1<0,點(diǎn)SKIPIF1<0的幾何意義是什么？請?jiān)趫D形中體現(xiàn)體現(xiàn)出來。SKIPIF1<0的幾何意義是什么？請?jiān)趫D形中體現(xiàn)體現(xiàn)出來。填表，討論，發(fā)言，展示。靜態(tài)動(dòng)態(tài)SKIPIF1<0靜態(tài)有關(guān)量幾何意義ASKIPIF1<0SKIPIF1<0SKIPIF1<0坐標(biāo)的幾何表示BSKIPIF1<0SKIPIF1<0SKIPIF1<0坐標(biāo)的幾何表示點(diǎn)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0橫坐標(biāo)的差（自變量的增量）SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0縱坐標(biāo)的差SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0割線SKIPIF1<0的斜率割線SKIPIF1<0（極限位置）即：切線SKIPIF1<0動(dòng)態(tài)SKIPIF1<0SKIPIF1<0＝切線SKIPIF1<0的斜率切線SKIPIF1<0的斜率（四）動(dòng)畫演示，閱讀課本，小結(jié)：平均變化率SKIPIF1<0在趨向于某一時(shí)刻的變化過程中，自變量的增量SKIPIF1<0越來越小，能越過任意小的間隔，但始終不能為0，不然，我們就得不到平均變化率的值；平均變化率在圖形中的幾何意義是割線AB的斜率。２．當(dāng)SKIPIF1<0趨近于0時(shí)，平均變化率SKIPIF1<0無限趨近于的常數(shù)就是SKIPIF1<0，從圖形上看，當(dāng)SKIPIF1<0趨近于0時(shí)，點(diǎn)B沿曲線趨近于點(diǎn)A，割線繞點(diǎn)A轉(zhuǎn)動(dòng)，它的極限位置為直線AD，這條直線叫做此曲線在點(diǎn)A處的切線。于是當(dāng)SKIPIF1<0時(shí)，割線AB的斜率趨近于過點(diǎn)A的切線的斜率，即：SKIPIF1<0＝切線SKIPIF1<0的斜率還可以利用點(diǎn)斜式求出切線AD的方程：SKIPIF1<0，即：SKIPIF1<0。（展示學(xué)生作品）（五）訓(xùn)練：分別在函數(shù)SKIPIF1<0，SKIPIF1<0和SKIPIF1<0的圖像上，用圖形來體現(xiàn)導(dǎo)數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的幾何意義，并用數(shù)學(xué)語言表述出來。（展示學(xué)生作品）1．SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<02．3．SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0三．學(xué)生繼續(xù)動(dòng)手：（一）在函數(shù)SKIPIF1<0的圖像中，請描述，比較曲線SKIPIF1<0在SKIPIF1<0附近增（減）以及增（減）快慢的情況。在SKIPIF1<0附近呢？（二）請估計(jì)曲線SKIPIF1<0在SKIPIF1<0的瞬時(shí)變化率，把數(shù)據(jù)用表格的形式列出。SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0函數(shù)的瞬時(shí)變化率SKIPIF1<0反思，發(fā)現(xiàn)。導(dǎo)函數(shù)。（探索作業(yè)求SKIPIF1<0SKIPIF1<0，SKIPIF1<0）單調(diào)性：在某個(gè)區(qū)間上，當(dāng)SKIPIF1<0單調(diào)遞增；當(dāng)SKIPIF1<0單調(diào)遞減。求出SKIPIF1<0，驗(yàn)證你的結(jié)論。SKIPIF1<0＝SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0。驗(yàn)證：（1）數(shù)據(jù)；（2）單調(diào)性。思考：1．如圖表示人體血管中的藥物濃度SKIPIF1<0（單位：SKIPIF1<0）隨時(shí)間SKIPIF1<0（單位：SKIPIF1<0）變化的函數(shù)圖像，根據(jù)圖像，估計(jì)SKIPIF1<0時(shí)，血管中藥物濃度的瞬時(shí)變化率，把數(shù)據(jù)用表格的形式列出。有什么發(fā)現(xiàn)？SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<00.20.40.60.8藥物濃度的瞬時(shí)變化率2．下圖是利用信息技術(shù)畫出的函數(shù)SKIPIF1<0的圖像，請根據(jù)圖像，估計(jì)SKIPIF1<0時(shí)，氣球的瞬時(shí)膨脹率。有什么發(fā)現(xiàn)？SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0四．小結(jié)：1．知識(shí)結(jié)構(gòu)：基本初等函數(shù)導(dǎo)數(shù)公式平均速度瞬時(shí)速度（物理意義）及導(dǎo)數(shù)運(yùn)算法則平均變化率瞬時(shí)變化率（導(dǎo)數(shù)本質(zhì)）導(dǎo)數(shù)平均變化率瞬時(shí)變化率（導(dǎo)數(shù)本質(zhì)）導(dǎo)數(shù)與函數(shù)單調(diào)性的關(guān)系導(dǎo)數(shù)與極（最）值的關(guān)系2．?dāng)?shù)學(xué)思想：運(yùn)動(dòng)變化的觀點(diǎn)（以靜態(tài)刻劃動(dòng)態(tài)），數(shù)形結(jié)合（函數(shù)思想）以直代曲（以近似求精確）（微積分思想方法）作業(yè)：習(xí)題3.1A5,6.B2,3.書上。探索作業(yè)1、求SKIPIF1<0SKIPIF1<0，SKIPIF1<02。用導(dǎo)數(shù)定義及單調(diào)性定義解釋：在某個(gè)區(qū)間上，當(dāng)SKIPIF1<0單調(diào)遞增；當(dāng)