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==微分,増減,極値,グラフ==
~教科書入門レベル/楽しいワーク~
問題1 y=x3−3xの増減,極値を調べて,グラフを描いてください
導関数,増減表
増減を調べる
極値を求める
グラフを描く

問題2 y=2x3−3x2+1の増減,極値を調べて,グラフを描いてください
導関数,増減表
増減を調べる
極値を求める
グラフを描く

問題3 y=x3−x2−x+2の増減,極値を調べて,グラフを描いてください
導関数,増減表
増減を調べる
極値を求める
グラフを描く

問題4 y=−x3+3x2−2の増減,極値を調べて,グラフを描いてください
導関数,増減表
増減を調べる
極値を求める
グラフを描く

問題5 y=x3の増減,極値を調べて,グラフを描いてください
導関数,増減表
増減を調べる
極値を求める
グラフを描く

[参考1]
 「増減を調べよ」という問題に対して,次のどちらの形で答えてもよい.(どちらかが正しくて,どちらかが間違っているのではない)

開区間で答える
書き方
閉区間で答える
書き方(多数派)
問題1の答x<−1, 1<xで増加
−1<x<1で減少
x≦−1, 1≦xで増加
−1≦x≦1で減少
問題2の答x<0, 1<xで増加
0<x<1で減少
x≦0, 1≦xで増加
0≦x≦1で減少
数学Ⅱ
教科書の例
D社, T社S社,K社,T社
a<x<bのように端を含まない区間を開区間といい,
 a≦x≦bのように端を含む区間を閉区間という.
• 数学Ⅱの段階で,踏み込んだ解説が書かれている教科書はないが,結論から言えば「開区間で増加であるならば,端の点を含めた閉区間で増加」「開区間で減少であるならば,端の点を含めた閉区間で減少」であるということがいえる.
 その証明は,数学Ⅲで習う「平均値の定理」でできる.(数学Ⅱではそこまで踏み込まない)
• 教科書各社の「増加区間,減少区間」の書き方を数学ⅡⅢを通して見ると,「閉区間で書く答案が多数派」になっている.
• 上記の問題1~4の解答も,閉区間で書くことができる(その方が多数派).
[参考2]
 「閉区間で書く答案」にすると,y=x3のグラフは,「x≦0, 0≦xで増加」すなわち「xの全区間で増加」と言えます.

 また,いつも弁当で見ている(?),ご飯粒ごはんつぶの下の方の形(4次関数でf '(x)=0であるが,減少から減少への中休みになっている箇所)も含めて,右図の茶色で示した部分は減少区間です.
• 「閉区間で書く答案」では,『境目の値(極大値,極小値となるxの値)』は,こうもり軍団のようなもので,けもの軍団(減少区間)にも鳥軍団(増加区間)にも入っていて『ずるい!』と考える生徒が多いが,実際には極大値,極小値のところは『増加でもあり,減少でもある』ことになります.
 また,y=x3のグラフでは,「x≦0で増加」「0≦xで増加」,結局「全区間で増加」となります.

問題6 y=x33+x2xの増減,極値を調べて,グラフを描いてください
導関数,増減表
増減を調べる
極値を求める
グラフを描く

問題7 y=x44x33x2の増減,極値を調べて,グラフを描いてください
導関数,増減表
増減を調べる
極値を求める
グラフを描く

問題8 y=x44x33x22+xの増減,極値を調べて,グラフを描いてください
導関数,増減表
増減を調べる
極値を求める
グラフを描く

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