2017年1月27日金曜日

学習環境

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第3章(微分係数、導関数)、3(極限)、練習問題1-10.を取り組んでみる。

練習問題1-10.


  1. f'( x )= lim h0 2 ( x+h ) 2 +3( x+h )( 2 x 2 +3x ) h = lim h0 4hx+2 h 2 +3h h = lim h0 ( 4x+2h+3 ) = lim h0 4x+ lim h0 2h+ lim h0 3 =4x+3

  2. f'( x )= lim h0 1 2( x+h )+1 1 2x+1 h = lim h0 2h h( 2x+2h+1 )( 2x+1 ) = lim h0 2 ( 2x+2h+1 )( 2x+1 ) = lim h0 ( 2 ) lim h0 ( 2x+2h+1 )· lim h0 ( 2x+1 ) = 2 ( 2x+1 ) 2

  3. f'( x )= lim h0 x+h x+h+1 x x+1 h = lim h0 ( x+h )( x+1 )x( x+h+1 ) h( x+h+1 )( x+1 ) = lim h0 h h( x+h+1 )( x+1 ) = 1 lim h0 ( x+h+1 )· lim h0 ( x+1 ) = 1 ( x+1 ) 2

  4. f'( x )= lim h0 ( x+h )( x+h+1 )x( x+1 ) h = lim h0 2hx+h h = lim h0 ( 2x+1 ) =2x+1

  5. f'( x )= lim h0 x+h 2( x+h )1 x 2x1 h = lim h0 ( x+h )( 2x1 )x( 2x+2h1 ) h( 2x+2h1 )( 2x1 ) = lim h0 h h( 2x+2h1 )( 2x1 ) = lim h0 ( 1 ) lim h0 ( 2x+2h1 )· lim h0 ( 2x1 ) = 1 ( 2x1 ) 2

  6. f'( x )= lim h0 3 ( x+h ) 3 3 x 3 h = lim h0 3( 3h x 2 +3 h 2 x+ h 3 ) h = lim h0 3·( lim h0 3 x 2 + lim h0 3hx+ lim h0 h 2 ) =3( 3 x 2 ) =9 x 2

  7. f'( x )= lim h0 ( x+h ) 4 x 4 h = lim h0 4h x 3 +6 h 2 x 2 +4 h 3 x+ h 4 h = lim h0 4 x 3 + lim h0 6h x 2 + lim h0 h 2 x+ lim h0 h 3 =4 x 3

  8. f'( x )= lim h0 ( x+h ) 5 x 5 h = lim h0 5h x 4 +10 h 2 x 3 +10 h 3 x 2 +5 h 4 x+ h 5 h = lim h0 5 x 4 + lim h0 10h x 3 + lim h0 10 h 2 x+ lim h0 5 h 3 x+ lim h0 h 4 =5 x 4

  9. f'( x )= lim h0 2 ( x+h ) 3 2 x 3 h = lim h0 2( 3h x 2 +3 h 2 x+ h 3 ) h = lim h0 2·( lim h0 3 x 2 + lim h0 3hx+ lim h0 h 2 ) =6 x 2

  10. f'( x )= lim h0 ( 1 2 ( x+h ) 3 +( x+h ) )( 1 2 x 3 +x ) h = lim h0 1 2 ( 3h x 2 +3 h 2 x+ h 3 )+h h = lim h0 1 2 ·( lim h0 3 x 2 + lim h0 3hx+ lim h0 h 2 )+ lim h0 1 = 3 2 x 2 +1

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