11X1 T10 03 second derivative (2011)

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  • The Second Derivative

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

    16 xdx

    dy

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

    16 xdx

    dy

    62

    2

    dx

    yd

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

    16 xdx

    dy

    62

    2

    dx

    yd

    32 xyii

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

    16 xdx

    dy

    62

    2

    dx

    yd

    32 xyii

    21

    2 3 x

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

    16 xdx

    dy

    62

    2

    dx

    yd

    32 xyii

    21

    2 3 x

    xxdx

    dy23

    2

    12

    12

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

    16 xdx

    dy

    62

    2

    dx

    yd

    32 xyii

    21

    2 3 x

    xxdx

    dy23

    2

    12

    12

    21

    2 3

    xx

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

    16 xdx

    dy

    62

    2

    dx

    yd

    32 xyii

    21

    2 3 x

    xxdx

    dy23

    2

    12

    12

    21

    2 3

    xx

    13232

    12

    122

    32

    2

    2

    xxxx

    dx

    yd

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

    16 xdx

    dy

    62

    2

    dx

    yd

    32 xyii

    21

    2 3 x

    xxdx

    dy23

    2

    12

    12

    21

    2 3

    xx

    13232

    12

    122

    32

    2

    2

    xxxx

    dx

    yd

    21

    22

    322 33

    xxx

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

    16 xdx

    dy

    62

    2

    dx

    yd

    32 xyii

    21

    2 3 x

    xxdx

    dy23

    2

    12

    12

    21

    2 3

    xx

    13232

    12

    122

    32

    2

    2

    xxxx

    dx

    yd

    21

    22

    322 33

    xxx

    33 2223

    2

    xxx

  • The Second Derivative

    2

    2

    2

    2

    ,,,dx

    ydxf

    dx

    dxfy

    23 e.g. 2 xxyi

    16 xdx

    dy

    62

    2

    dx

    yd

    32 xyii

    21

    2 3 x

    xxdx

    dy23

    2

    12

    12

    21

    2 3

    xx

    13232

    12

    122

    32

    2

    2

    xxxx

    dx

    yd

    21

    22

    322 33

    xxx

    33 2223

    2

    xxx

    333

    22

    xx

  • Exercise 10D; 1 to 13, every 2nd in all