導(dǎo)數(shù)定義及其在中學(xué)數(shù)學(xué)中的應(yīng)用畢業(yè)論文

1、本科畢業(yè)論文設(shè)計題 目導(dǎo)數(shù)定義及其在中學(xué)數(shù)學(xué)中的應(yīng)用學(xué) 院 數(shù)學(xué)與統(tǒng)計學(xué)院 專 業(yè) 數(shù)學(xué)與應(yīng)用數(shù)學(xué) 年 級 2021級 學(xué) 號 姓 名 指 導(dǎo) 教 師 成 績 目錄 TOC o 1-3 h z u HYPERLINK l _Toc353648300 摘要1 HYPERLINK l _Toc353648302 1.引言1 HYPERLINK l _Toc353648303 2. 導(dǎo)數(shù)的知識儲藏2 HYPERLINK l _Toc353648304 21 導(dǎo)數(shù)的定義與幾何意義2 HYPERLINK l _Toc353648305 22 依定義求導(dǎo)數(shù)的方法2 HYPERLINK l _Toc3536

2、48306 23 導(dǎo)數(shù)的運算2 HYPERLINK l _Toc353648307 2.31幾種常見函數(shù)的導(dǎo)數(shù)2 HYPERLINK l _Toc353648308 232 導(dǎo)數(shù)的四那么運算法那么2 HYPERLINK l _Toc353648309 3導(dǎo)數(shù)在解題中的應(yīng)用3 HYPERLINK l _Toc353648310 3.1 利用導(dǎo)數(shù)定義巧妙解題3 HYPERLINK l _Toc353648311 3.11 利用導(dǎo)數(shù)的定義求函數(shù)的極限3 HYPERLINK l _Toc353648312 3.12 利用導(dǎo)數(shù)的定義求函數(shù)在某點處的導(dǎo)數(shù)值5 HYPERLINK l _Toc3536483

3、13 3.13 利用導(dǎo)數(shù)定義判斷函數(shù)的單調(diào)性6 HYPERLINK l _Toc353648314 3.14 利用導(dǎo)數(shù)定義判斷函數(shù)切線的斜率6 HYPERLINK l _Toc353648315 3.2導(dǎo)數(shù)與其他知識點的聯(lián)系7 HYPERLINK l _Toc353648316 3.21導(dǎo)數(shù)與函數(shù)7 HYPERLINK l _Toc353648317 3.22 利用切線的幾何意義解決問題9 HYPERLINK l _Toc353648318 3.23 導(dǎo)數(shù)與不等式10 HYPERLINK l _Toc353648319 3.24 導(dǎo)數(shù)與數(shù)列10 HYPERLINK l _Toc35364832

4、0 3.25 導(dǎo)數(shù)的實際應(yīng)用11 HYPERLINK l _Toc353648321 4.總結(jié)125.參考文獻(xiàn).126.致謝.12導(dǎo)數(shù)定義及其在中學(xué)數(shù)學(xué)中的應(yīng)用田茜西南大學(xué)數(shù)學(xué)與統(tǒng)計學(xué)院，重慶 400715 摘要：導(dǎo)數(shù)概念是數(shù)學(xué)分析根本概念，是近代數(shù)學(xué)的重要根底，同時也是聯(lián)系初、高等數(shù)學(xué)的紐帶。導(dǎo)數(shù)在中學(xué)數(shù)學(xué)中的應(yīng)用特別廣泛。本文將結(jié)合高考中導(dǎo)數(shù)相關(guān)題目，主要從導(dǎo)數(shù)的定義、導(dǎo)數(shù)與函數(shù)、導(dǎo)數(shù)與不等式、導(dǎo)數(shù)與數(shù)列以及導(dǎo)數(shù)的實際應(yīng)用幾個方面分析導(dǎo)數(shù)在中學(xué)數(shù)學(xué)中的廣泛應(yīng)用。 關(guān)鍵詞：導(dǎo)數(shù)；定義；函數(shù)；應(yīng)用The definition of derivative and its application

5、in the middle school mathematicsTian XiSchool of Mathematics and Statistics, Southwest University, Chongqing 400715, ChinaAbstract: Derivative concept is the basic concept in mathematical analysis and is the important basis of modern mathematics. It is a link between elementary mathematics and advan

6、ced mathematics. Derivative is widely used in the middle school mathematics. Combined with college entrance examination, this thesis mainly analyzes the application of derivative in the middle school mathematics from the perspectives of the definition of derivative, derivatives and functions, deriva

7、tives and non-equality, derivative and derivative series and the practical applications of derivatives. Key words: derivative； function； definition； application 1.引言自從導(dǎo)數(shù)進(jìn)入高中數(shù)學(xué)教材以后，與導(dǎo)數(shù)有關(guān)的問題就成了歷年高考的熱點正確運用導(dǎo)數(shù)的思想方法與根本理論解決中學(xué)數(shù)學(xué)中的問題，成為了中學(xué)數(shù)學(xué)教師和學(xué)生重點關(guān)注的對象。但是導(dǎo)數(shù)對于學(xué)生而言仍舊是一個難點，因為導(dǎo)數(shù)與其他的知識結(jié)合十分緊密。本文運用導(dǎo)數(shù)的思想方法和根本理論，探討導(dǎo)

8、數(shù)在各個方面的根本應(yīng)用，希望在解決導(dǎo)數(shù)相關(guān)題目時有一定的思路與方法。2. 導(dǎo)數(shù)的知識儲藏21 導(dǎo)數(shù)的定義與幾何意義定義：設(shè)函數(shù)在包含的某個區(qū)間上有定義，如果比值在 趨于0時趨于確定的極限值，那么稱此極限值為函數(shù)在處的導(dǎo)數(shù)或微商，記作 注： 幾何意義：函數(shù)在處的導(dǎo)數(shù)的幾何意義，就是曲線在點處的切線的斜率，即斜率為過點P的切線方程為：22 依定義求導(dǎo)數(shù)的方法1求函數(shù)的改變量2求平均變化率3取極限，得導(dǎo)數(shù) 23 導(dǎo)數(shù)的運算2.31幾種常見函數(shù)的導(dǎo)數(shù)(C為常數(shù))；()；。232 導(dǎo)數(shù)的四那么運算法那么1；2；3； 4；5假設(shè)那么。3導(dǎo)數(shù)在解題中的應(yīng)用3.1 利用導(dǎo)數(shù)定義巧妙解題 導(dǎo)數(shù)的定義，在解題中有

9、廣泛的應(yīng)用，與此同時可以簡化解題的步驟。下面將從導(dǎo)數(shù)的定義在函數(shù)、不等式、曲線斜率以及綜合應(yīng)用四個方面加以闡述。3.11 利用導(dǎo)數(shù)的定義求函數(shù)的極限例1設(shè)，那么等于A B. C. D.分析：此題看似一個求極限的問題，但是根據(jù)導(dǎo)數(shù)的定義其實為在處的導(dǎo)數(shù)。故有兩種解法，一種運用極限知識求解，另一種運用求導(dǎo)方法巧妙求解。解法一：解： C此種解法給出的是利用極限的根本知識求解，當(dāng)時，。解法二：解：由導(dǎo)函數(shù)的定義可知，=， =因此選C解法一與解法二分別從兩個角度來解析此題，一方面是從極限的角度，另一方面利用導(dǎo)數(shù)定義，解法二更為巧妙簡單。例2假設(shè)那么的值為A -3 B.-6 C.-9 D.-12分析：此題

10、同理可以利用導(dǎo)數(shù)定義巧妙求解，但是此處給出的極限式子有別于定義標(biāo)準(zhǔn)形式，需要進(jìn)行適當(dāng)?shù)淖冃巍?解：根據(jù)導(dǎo)數(shù)的定義可知， =4= D此題關(guān)鍵在于將此題給出的極限式子與導(dǎo)數(shù)定義產(chǎn)生聯(lián)系，并明確導(dǎo)數(shù)定義中并非確定的一個數(shù)，任意的數(shù)均可。例3那么的值為 . A.0 B.2 C.3 D.6分析：此題也容易想到運用導(dǎo)數(shù)定義求解，但關(guān)鍵在于將充分運用在極限式子中,將3轉(zhuǎn)換為，將極限式子進(jìn)行變形，具體解法如下。解： 應(yīng)選C.以上三道例題展現(xiàn)了函數(shù)的極限與導(dǎo)數(shù)的定義充分結(jié)合，將極限轉(zhuǎn)換成求函數(shù)的導(dǎo)數(shù)。3.12 利用導(dǎo)數(shù)的定義求函數(shù)在某點處的導(dǎo)數(shù)值例4，用導(dǎo)數(shù)定義求。解法一：利用導(dǎo)數(shù)求導(dǎo)公式可得解法二：導(dǎo)數(shù)定義求

11、解 在用極限來定義導(dǎo)數(shù)主要有導(dǎo)函數(shù)以及在某個點處的導(dǎo)數(shù)值兩個方面，解法二直接運用在的導(dǎo)數(shù)定義來求解比先求的值，再求更加簡單。例5.，求 分析：假設(shè)此題采取利用導(dǎo)數(shù)公式對進(jìn)行求導(dǎo)，過程十分復(fù)雜，并且這個式子還可以無限往后，可以繼續(xù)乘上型的單項式。所以此題才用導(dǎo)數(shù)的定義求解。解：在一般情況下，先求出，在取，但是此題對于的求解比擬繁瑣，直接求得更為簡單。3.13 利用導(dǎo)數(shù)定義判斷函數(shù)的單調(diào)性例6. 在實數(shù)R上的可導(dǎo)函數(shù)對于任意實數(shù) 都有，假設(shè)存在實數(shù)a,b 使且求證：12在是單調(diào)遞增函數(shù)。分析：此題中的沒有給出準(zhǔn)確的表達(dá)式，第二問要證明在R上的單調(diào)性，只能利用導(dǎo)數(shù)的定義。證明：1又，即2=即=，所以

12、在是單調(diào)遞增函數(shù)。3.14 利用導(dǎo)數(shù)定義判斷函數(shù)切線的斜率例7. 設(shè)為可導(dǎo)函數(shù)且滿足，問曲線在點處的切線斜率是否存在？假設(shè)存在求在該點的切線斜率；假設(shè)不存在，請說明理由.解析：為可導(dǎo)函數(shù), . 即在點處存在切線斜率,且在點處切線的斜率為-2.3.2導(dǎo)數(shù)與其他知識點的聯(lián)系 3.21導(dǎo)數(shù)與函數(shù)在中學(xué)數(shù)學(xué)中，導(dǎo)數(shù)與函數(shù)的聯(lián)系主要在于利用導(dǎo)數(shù)對函數(shù)的單調(diào)性、極值、最值、函數(shù)對應(yīng)曲線的切線以及函數(shù)在某個區(qū)間內(nèi)的恒成立問題進(jìn)行研究。往往函數(shù)這幾個方面在解題中是緊密聯(lián)系在一起的，對于，當(dāng)時，在區(qū)間上單調(diào)遞增；當(dāng)時，在區(qū)間上單調(diào)遞減。函數(shù)的恒成立問題一般結(jié)合二次函數(shù)的性質(zhì)與別離參數(shù)的方法求解。例8. 函數(shù)在與

13、時都取得極值1求的值與函數(shù)的單調(diào)區(qū)間2假設(shè)對，不等式恒成立，求c的取值范圍。分析：函數(shù)在某點處取得極值必有在該點處的導(dǎo)數(shù)值必為0，反過來導(dǎo)數(shù)等于0的點并不一定是取得極值的點，還需要檢驗，這是學(xué)生易犯錯誤的地方。利用便可求出的值。解：1 由題意可知，且解得 即或時，；時，單調(diào)遞增區(qū)間，；單調(diào)遞減區(qū)間2由1可得， 時在,上單調(diào)遞增，在遞減 即或第2小題是利用函數(shù)的單調(diào)性解決不等式恒成立問題，在恒成立即大于的最大值即可。這就歸結(jié)于在一個區(qū)間內(nèi)，對于參數(shù)恒成立，即大于等于在區(qū)間上的最大值；恒成立，即小于等于在區(qū)間上的最小值。例9.設(shè)函數(shù) (1)求曲線在點處的切線方程；(2)求函數(shù)的單調(diào)區(qū)間；(3)假設(shè)

14、函數(shù)在區(qū)間內(nèi)單調(diào)遞增，求的取值范圍.分析：導(dǎo)函數(shù)在某點處的值的幾何意義，即函數(shù)在該點處的切線斜率，該點稱之為切點解：1 在點切線方程為：2由1可知：當(dāng)時，單調(diào)遞增 ，單調(diào)遞減；當(dāng)時，單調(diào)遞增 ，單調(diào)遞減.3當(dāng)時，即；當(dāng)時，即.第2、3問重點在于，判斷導(dǎo)數(shù)的正負(fù)情況首先與的正負(fù)有緊密聯(lián)系，故此討論與是必要的.引申：導(dǎo)數(shù)的幾何意義往往和切線產(chǎn)生聯(lián)系，過點處的切線，一種情況點p在曲線上，即為切點。另一種情況點p不在曲線上 ，做題關(guān)鍵設(shè)切點坐標(biāo)，通過斜率建立等量關(guān)系。3.22 利用切線的幾何意義解決問題例10. 點P是曲線上任意一點，求P到直線的距離的最小值。分析：要求曲線上一點到直線的最小距離，實際

15、上P點為曲線的一條與直線平行的切線的切點，故此題的關(guān)鍵為設(shè)切點，利用距離公式。解：設(shè)切點，由題意得： 得或舍 例11. 函數(shù)的圖像如下圖,以下不等關(guān)系正確的選項是( ) A B C D 分析：根據(jù)導(dǎo)數(shù)定義結(jié)合導(dǎo)數(shù)的幾何意義, 表示圖像在處的切線斜率，表示圖像在處的切線斜率，所以表示圖像在A(2, )以及B(3, )兩點的割線斜率。 由圖可見,答案應(yīng)為C。3.23 導(dǎo)數(shù)與不等式例12. 對于R上可導(dǎo)的任意函數(shù)，假設(shè)滿足，那么必有 A BC D分析：與0的大小關(guān)系與有關(guān)，當(dāng)時，；時，。所以的單調(diào)性有兩種情況，一種是在單調(diào)遞減，單調(diào)遞增；另一種特殊的情況這種情況學(xué)生容易忽略。由此可得所以。應(yīng)選C3.

16、24 導(dǎo)數(shù)與數(shù)列例13. 假設(shè)等比數(shù)列的首項為，且，求公比。解：由題意得：設(shè)曲線在點處的切線與軸交點的橫坐標(biāo)為，令，求的值。解：由題意可得，故在點1,1處的切線方程為：，那么即：3.25 導(dǎo)數(shù)的實際應(yīng)用例15. 如圖，將邊長為1的正六邊形鐵皮的六個角各切去一個全等的四邊形，再沿虛線折起，做成一個無蓋的正六棱柱容器。當(dāng)這個正六棱柱容器的底面邊長為_時，其容積最大。分析：設(shè)這個正六棱柱容器的底面邊長為，那么底面面積正六棱柱高。所以正六棱柱體積故此在單調(diào)遞增，在單調(diào)遞減，即在處取得最大值。例16. 某商場銷售某種商品的經(jīng)驗說明，該商品每日的銷售量(單位：千克)與銷售價格(單位：元/千克)滿足函數(shù)關(guān)系

17、式，其中，為常數(shù) 銷售價格為元/千克時，每日可售出該商品千克。(1) 求的值；(2)假設(shè)該商品的本錢為元/千克，試確定銷售價格的值，使商場每日銷售該商品所獲得的利潤最大.解： (1)因為時，所以，.(2)由(1)可知，該商品每日的銷售量。所以商場每日銷售該商品所獲得的利潤從而又因為，于是，當(dāng)時，函數(shù)在區(qū)間上單調(diào)遞增；當(dāng)時，函數(shù)在區(qū)間上單調(diào)遞減；由此可得，是函數(shù)在區(qū)間內(nèi)的極大值點，也是最大值點.所以，當(dāng)時，函數(shù)取得最大值.答：當(dāng)銷售價格為4元/千克時，商場每日銷售該商品所獲得的利潤最大4.總結(jié)導(dǎo)數(shù)已經(jīng)成為了高考的熱門，通過以上對導(dǎo)數(shù)定義及其在函數(shù)、不等式、數(shù)列及其實際應(yīng)用的討論，對于解決有關(guān)導(dǎo)數(shù)

18、問題給與了思路與方法。參考文獻(xiàn)：1 魏霞.導(dǎo)數(shù)在中學(xué)數(shù)學(xué)中的地位和應(yīng)用J.現(xiàn)代閱讀教育版.2021082 劉韌.論導(dǎo)數(shù)在不等式中的應(yīng)用J.現(xiàn)代商貿(mào)工業(yè).2021063 蔣亞紅.再談“切線問題J.考試周刊.2007(04).致謝：在本次論文的撰寫過程中，陳波老師對我的幫助非常大，感謝陳波老師在百忙中抽出時間對我的論文加以修改、指導(dǎo)。我非常佩服陳波老師具有淵博的知識與工作認(rèn)真負(fù)責(zé)的精神。我衷心的希望陳波老師在今后的日子里工作順利以及身體健康。最后我對所有的數(shù)學(xué)與統(tǒng)計學(xué)院的老師們表示感謝，由衷的祝愿我的母校蒸蒸日上。 9JWKffwvG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswS

19、yXTy#&QA9wkxFyeQ!djs#XuyUP2kNXpRWXmA&UE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2z

20、Vkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz8

21、49GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg&6a*CZ7

22、H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ!djs#XuyUP2kNXpRWXmA&UE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxG89AmUE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadN

23、u#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4

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25、8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3t

26、nGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5

27、pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP

28、$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz8

29、49GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8

30、9AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&g

31、TXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9

32、CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjq

33、v$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxG

34、jqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6

35、X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqvadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazad

36、Nu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazad

37、Nu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6

38、YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z

39、89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z

40、89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%MadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn

41、%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*adNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$

42、UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD

43、6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%M

44、z849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&Mu

45、WFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&Mu

46、WFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWw

47、cvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vS

48、TT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5u

49、xGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7

50、JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X

51、4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%MzadNu#KN&MuWFA5ux

52、Y7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%M

53、z849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!

54、z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#Q

55、cUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcv

56、R9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Am

57、YWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYW

58、pazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXR

59、m6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9Cpb

60、K!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqvUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89Am