管教案新第二章

課題§2-1導(dǎo)數(shù)的概念教學(xué)目的使學(xué)生理解導(dǎo)數(shù)的定義，掌握導(dǎo)數(shù)的幾何意義，會(huì)求曲線的切線方程與法線方程，了解函數(shù)可導(dǎo)與連續(xù)的關(guān)系。教學(xué)重點(diǎn)1．導(dǎo)數(shù)的定義；2．用導(dǎo)數(shù)的定義求函數(shù)在某點(diǎn)的導(dǎo)數(shù)；3．導(dǎo)數(shù)的幾何意義。教學(xué)難點(diǎn)1．導(dǎo)數(shù)的定義；2．函數(shù)可導(dǎo)與連續(xù)的關(guān)系。教學(xué)方法講授法課時(shí)2課型新授課周次班級(jí)星期節(jié)次地點(diǎn)教學(xué)后記：教學(xué)步驟及內(nèi)容：復(fù)習(xí)舊課二、兩個(gè)引例引例1自由落體運(yùn)動(dòng)的瞬時(shí)速度。提問：1．自由落體運(yùn)動(dòng)的位移公式；2．自由落體運(yùn)動(dòng)的瞬時(shí)速度公式；3．自由落體運(yùn)動(dòng)的瞬時(shí)速度公式的推導(dǎo)過程（適當(dāng)討論）。由學(xué)生回答可知自由落體運(yùn)動(dòng)的位移公式為SKIPIF1<0，由于物體的位移SKIPIF1<0是隨時(shí)間SKIPIF1<0連續(xù)變化的，因此在很短的時(shí)間間隔SKIPIF1<0內(nèi)（從SKIPIF1<0到SKIPIF1<0）內(nèi)，速度變化不大，可以用平均速度SKIPIF1<0作為SKIPIF1<0時(shí)的瞬時(shí)速度SKIPIF1<0的近似值，即SKIPIF1<0SKIPIF1<0SKIPIF1<0=SKIPIF1<0=SKIPIF1<0顯然，SKIPIF1<0越小，SKIPIF1<0與SKIPIF1<0越接近，當(dāng)SKIPIF1<0無限變小時(shí)，平均速度就無限接近SKIPIF1<0時(shí)的瞬時(shí)速度．由此，令SKIPIF1<0，如果平均速度SKIPIF1<0的極限存在，就把它定義為物體在時(shí)刻SKIPIF1<0的瞬時(shí)速度SKIPIF1<0，即SKIPIF1<0=SKIPIF1<0=SKIPIF1<0總結(jié)規(guī)律：對(duì)于一般的變速直線運(yùn)動(dòng)的瞬時(shí)速度可由以下式子求得：SKIPIF1<0引例2平面曲線的切線斜率提問：1．什么叫做圓的切線？2．一般的平面曲線的切線怎么定義？（適當(dāng)討論）定義設(shè)點(diǎn)SKIPIF1<0是曲線SKIPIF1<0上的一個(gè)定點(diǎn)，在曲線SKIPIF1<0上另取一點(diǎn)SKIPIF1<0，作割線SKIPIF1<0，當(dāng)動(dòng)點(diǎn)SKIPIF1<0沿曲線SKIPIF1<0向點(diǎn)SKIPIF1<0移動(dòng)時(shí)，割線SKIPIF1<0繞點(diǎn)SKIPIF1<0旋轉(zhuǎn)，設(shè)其極限位置為SKIPIF1<0，則直線SKIPIF1<0稱為曲線SKIPIF1<0在點(diǎn)SKIPIF1<0的切線．如右圖所示．設(shè)曲線SKIPIF1<0的方程是SKIPIF1<0，記點(diǎn)SKIPIF1<0的橫坐標(biāo)為SKIPIF1<0，點(diǎn)SKIPIF1<0的橫坐標(biāo)為SKIPIF1<0（SKIPIF1<0可正可負(fù)），SKIPIF1<0平行SKIPIF1<0軸，設(shè)SKIPIF1<0的傾角為SKIPIF1<0，則SKIPIF1<0的斜率為SKIPIF1<0顯然SKIPIF1<0當(dāng)點(diǎn)SKIPIF1<0沿曲線SKIPIF1<0無限趨近于點(diǎn)SKIPIF1<0時(shí)（這時(shí)SKIPIF1<0，SKIPIF1<0也趨近于SKIPIF1<0的傾角SKIPIF1<0，這時(shí)切線SKIPIF1<0的斜率SKIPIF1<0綜上兩個(gè)引例的結(jié)論可知，雖然這兩個(gè)問題所涉及到的背景知識(shí)不同，但是它們可以用相同的方法求得所需結(jié)果，由此引出導(dǎo)數(shù)的定義。三、導(dǎo)數(shù)的定義1．導(dǎo)數(shù)的定義。定義設(shè)函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0的某鄰域內(nèi)有定義，當(dāng)自變量SKIPIF1<0在點(diǎn)SKIPIF1<0處有增量SKIPIF1<0（點(diǎn)SKIPIF1<0仍在該鄰域內(nèi)）時(shí)，相應(yīng)地函數(shù)有增量SKIPIF1<0如果極限SKIPIF1<0存在，則稱函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處可導(dǎo)，并稱此極限值為函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處的導(dǎo)數(shù)．記作SKIPIF1<0，也可記作SKIPIF1<0，SKIPIF1<0或SKIPIF1<0．即SKIPIF1<0=SKIPIF1<0=SKIPIF1<0這時(shí)就稱函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0的導(dǎo)數(shù)存在，或稱函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0可導(dǎo)；如果極限不存在，則稱函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0不可導(dǎo)。2．由導(dǎo)數(shù)的定義求函數(shù)的導(dǎo)數(shù)。設(shè)函數(shù)SKIPIF1<0，求該函數(shù)在SKIPIF1<0處的導(dǎo)數(shù)的步驟：在SKIPIF1<0處給定SKIPIF1<0求增量SKIPIF1<0算比值SKIPIF1<0取極限SKIPIF1<0例1已知函數(shù)SKIPIF1<0，求SKIPIF1<0。解在SKIPIF1<0處給定SKIPIF1<0（1）求增量SKIPIF1<0（2）算比值SKIPIF1<0（3）取極限SKIPIF1<0SKIPIF1<0因此，SKIPIF1<0=23．幾點(diǎn)說明。1）函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處的導(dǎo)數(shù)也稱為函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處對(duì)自變量的變化率。2）當(dāng)極限SKIPIF1<0與SKIPIF1<0存在時(shí)，分別稱它們?yōu)镾KIPIF1<0的左導(dǎo)數(shù)與右導(dǎo)數(shù)，記為SKIPIF1<0與SKIPIF1<0。且SKIPIF1<0存在當(dāng)且僅當(dāng)SKIPIF1<0與SKIPIF1<0都存在且相等。（利用極限存在的充要條件理解）3）函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處的導(dǎo)數(shù)SKIPIF1<0，就是導(dǎo)函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處的函數(shù)值，即SKIPIF1<0=SKIPIF1<0。（通過例1中改變SKIPIF1<0值的改變進(jìn)行說明）4）如果函數(shù)SKIPIF1<0在SKIPIF1<0，SKIPIF1<0內(nèi)每一點(diǎn)SKIPIF1<0處可導(dǎo)，則稱函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0，SKIPIF1<0內(nèi)可導(dǎo)．顯然導(dǎo)數(shù)值SKIPIF1<0也是SKIPIF1<0的函數(shù)，我們稱它為函數(shù)SKIPIF1<0的導(dǎo)函數(shù)，今后在不會(huì)發(fā)生混淆的情況下，也簡(jiǎn)稱導(dǎo)數(shù)．記作SKIPIF1<0，SKIPIF1<0，SKIPIF1<0或SKIPIF1<0，即SKIPIF1<0=SKIPIF1<0討論：函數(shù)SKIPIF1<0的導(dǎo)數(shù)是什么？（結(jié)論：SKIPIF1<0）思考：函數(shù)SKIPIF1<0的導(dǎo)數(shù)是什么？（結(jié)論：SKIPIF1<0）拓展：函數(shù)SKIPIF1<0的導(dǎo)數(shù)是什么？（結(jié)論：SKIPIF1<0）5）如果函數(shù)SKIPIF1<0在SKIPIF1<0，SKIPIF1<0內(nèi)可導(dǎo)，且在SKIPIF1<0點(diǎn)右導(dǎo)數(shù)存在，在SKIPIF1<0點(diǎn)右導(dǎo)數(shù)存在，則稱函數(shù)SKIPIF1<0在閉區(qū)間SKIPIF1<0，SKIPIF1<0上可導(dǎo)。四、導(dǎo)數(shù)的幾何意義由引例2的分析可知導(dǎo)數(shù)的幾何意義為：函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0的導(dǎo)數(shù)SKIPIF1<0表示曲線SKIPIF1<0在點(diǎn)SKIPIF1<0，SKIPIF1<0的切線的斜率。因此有當(dāng)函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處可導(dǎo)時(shí)，曲線SKIPIF1<0在點(diǎn)SKIPIF1<0，SKIPIF1<0的切線方程為SKIPIF1<0曲線SKIPIF1<0在點(diǎn)SKIPIF1<0，SKIPIF1<0的法線方程為SKIPIF1<0如果SKIPIF1<0在點(diǎn)SKIPIF1<0連續(xù)且導(dǎo)數(shù)為無窮大，則曲線在點(diǎn)SKIPIF1<0，SKIPIF1<0的切線方程為SKIPIF1<0；法線方程為SKIPIF1<0例2求曲線SKIPIF1<0在點(diǎn)(1，1)處的切線和法線方程。解因?yàn)镾KIPIF1<0，所以SKIPIF1<0．于是曲線SKIPIF1<0在點(diǎn)(1，1)處的切線方程為SKIPIF1<0即SKIPIF1<0曲線SKIPIF1<0在點(diǎn)(1，1)處的法線方程為SKIPIF1<0即SKIPIF1<0五、可導(dǎo)與連續(xù)的關(guān)系定理如果函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處可導(dǎo)，則SKIPIF1<0在點(diǎn)SKIPIF1<0處必連續(xù)．注：如果函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處連續(xù)，SKIPIF1<0在點(diǎn)SKIPIF1<0處未必可導(dǎo)。四、課堂練習(xí)：1.用冪函數(shù)的導(dǎo)數(shù)公式求下列函數(shù)的導(dǎo)數(shù)：（1）SKIPIF1<0；（2）SKIPIF1<0；2.求下列曲線在指定點(diǎn)處的切線方程：（1）SKIPIF1<0在點(diǎn)（－1，－1）處；五、教學(xué)內(nèi)容小結(jié)主要內(nèi)容：兩個(gè)引例；導(dǎo)數(shù)的定義；導(dǎo)數(shù)的幾何意義；函數(shù)可導(dǎo)與連續(xù)的關(guān)系。重點(diǎn)：1．導(dǎo)數(shù)的定義；2．用導(dǎo)數(shù)的定義求函數(shù)在某點(diǎn)的導(dǎo)數(shù)；3．導(dǎo)數(shù)的幾何意義。難點(diǎn)：1．導(dǎo)數(shù)的定義；2．函數(shù)可導(dǎo)與連續(xù)的關(guān)系。六、課后思考及作業(yè)p314課題§2-2導(dǎo)數(shù)的運(yùn)算教學(xué)目的使學(xué)生熟記與理解導(dǎo)數(shù)的基本公式與四則運(yùn)算求導(dǎo)法則并能熟練應(yīng)用。使學(xué)生掌握復(fù)合函數(shù)的求導(dǎo)法則教學(xué)重點(diǎn)1．導(dǎo)數(shù)的基本公式；2．四則運(yùn)算求導(dǎo)法則；3.復(fù)合函數(shù)的求導(dǎo)法則教學(xué)難點(diǎn)公式的應(yīng)用教學(xué)方法講授法課時(shí)4課型新授課周次班級(jí)星期節(jié)次地點(diǎn)教學(xué)后記：教學(xué)步驟及內(nèi)容：復(fù)習(xí)舊課提問：1．導(dǎo)數(shù)可以由哪一個(gè)極限式子表示？2．根據(jù)導(dǎo)數(shù)的定義求函數(shù)的導(dǎo)數(shù)有哪幾步？3．導(dǎo)函數(shù)與函數(shù)在某點(diǎn)導(dǎo)數(shù)之間有什么關(guān)系？二、新課講解1．羅列導(dǎo)數(shù)基本公式。SKIPIF1<0（SKIPIF1<0為任意常數(shù)）；SKIPIF1<0（SKIPIF1<0為實(shí)數(shù)）；SKIPIF1<0，特別：SKIPIF1<0；SKIPIF1<0，特別：SKIPIF1<0；SKIPIF1<0；SKIPIF1<0；SKIPIF1<0SKIPIF1<0＊SKIPIF1<0＊SKIPIF1<0SKIPIF1<0；SKIPIF1<0；SKIPIF1<0；SKIPIF1<0。注：要求學(xué)生默記約5分鐘。2．分析部分基本公式特征。課堂練習(xí)：在下列空格處填上適當(dāng)?shù)暮瘮?shù)使等式成立：1）SKIPIF1<0=；（答案：0）2）SKIPIF1<0=；（答案：1）3）SKIPIF1<0=；（答案：0）（答案：SKIPIF1<0）4）SKIPIF1<0=；2、導(dǎo)數(shù)的四則運(yùn)算法則函數(shù)的和、差、積、商的求導(dǎo)法則設(shè)SKIPIF1<0可導(dǎo)，則（1）SKIPIF1<0；（2）SKIPIF1<0（SKIPIF1<0是常數(shù)）；（3）SKIPIF1<0；（4）SKIPIF1<0推廣有限個(gè)可導(dǎo)函數(shù)代數(shù)和的導(dǎo)數(shù)等于和個(gè)函數(shù)導(dǎo)數(shù)的代數(shù)和，即SKIPIF1<0SKIPIF1<0SKIPIF1<0推論SKIPIF1<0（SKIPIF1<0為常數(shù)）．例2已知SKIPIF1<0，求SKIPIF1<0。解SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0例3已知SKIPIF1<0，求SKIPIF1<0。解SKIPIF1<0SKIPIF1<0SKIPIF1<0。例4已知SKIPIF1<0，求SKIPIF1<0。解SKIPIF1<0例5求SKIPIF1<0的導(dǎo)數(shù)。解SKIPIF1<0SKIPIF1<0SKIPIF1<0說明：四則運(yùn)算的求導(dǎo)法則除了直接應(yīng)用公式外，有時(shí)需要將表達(dá)適當(dāng)變形后再應(yīng)用公式。3、復(fù)合函數(shù)的求導(dǎo)法則設(shè)SKIPIF1<0，而SKIPIF1<0，則SKIPIF1<0的導(dǎo)數(shù)為SKIPIF1<0或SKIPIF1<0例6設(shè)SKIP