11X1 T12 04 concavity (2010)

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  • Concavity

  • Concavity The second deriviative measures the change in slope with respect to x,

    this is known as concavity

  • Concavity

    up concave is curve the,0 If xf

    The second deriviative measures the change in slope with respect to x,

    this is known as concavity

  • Concavity

    up concave is curve the,0 If xf

    down concave is curve the,0 If xf

    The second deriviative measures the change in slope with respect to x,

    this is known as concavity

  • Concavity

    up concave is curve the,0 If xf

    down concave is curve the,0 If xf

    inflection ofpoint possible ,0 If xf

    The second deriviative measures the change in slope with respect to x,

    this is known as concavity

  • Concavity

    up concave is curve the,0 If xf

    235sketch derivative second at the lookingBy e.g. 23 xxxy

    down concave is curve the,0 If xf

    inflection ofpoint possible ,0 If xf

    The second deriviative measures the change in slope with respect to x,

    this is known as concavity

  • Concavity

    up concave is curve the,0 If xf

    235sketch derivative second at the lookingBy e.g. 23 xxxy

    down concave is curve the,0 If xf

    inflection ofpoint possible ,0 If xf

    3103 2 xxdx

    dy

    The second deriviative measures the change in slope with respect to x,

    this is known as concavity

  • Concavity

    up concave is curve the,0 If xf

    235sketch derivative second at the lookingBy e.g. 23 xxxy

    down concave is curve the,0 If xf

    inflection ofpoint possible ,0 If xf

    3103 2 xxdx

    dy

    The second deriviative measures the change in slope with respect to x,

    this is known as concavity

    1062

    2

    xdx

    yd

  • Concavity

    up concave is curve the,0 If xf

    235sketch derivative second at the lookingBy e.g. 23 xxxy

    down concave is curve the,0 If xf

    inflection ofpoint possible ,0 If xf

    3103 2 xxdx

    dy

    The second deriviative measures the change in slope with respect to x,

    this is known as concavity

    1062

    2

    xdx

    yd

    0 whenup concave is Curve2

    2

    dx

    yd

  • Concavity

    up concave is curve the,0 If xf

    235sketch derivative second at the lookingBy e.g. 23 xxxy

    down concave is curve the,0 If xf

    inflection ofpoint possible ,0 If xf

    3103 2 xxdx

    dy

    The second deriviative measures the change in slope with respect to x,

    this is known as concavity

    1062

    2

    xdx

    yd

    0 whenup concave is Curve2

    2

    dx

    yd

    3

    5

    0106 i.e.

    x

    x

  • Concavity

    up concave is curve the,0 If xf

    235sketch derivative second at the lookingBy e.g. 23 xxxy

    down concave is curve the,0 If xf

    inflection ofpoint possible ,0 If xf

    3103 2 xxdx

    dy

    The second deriviative measures the change in slope with respect to x,

    this is known as concavity

    1062

    2

    xdx

    yd

    0 whenup concave is Curve2

    2

    dx

    yd

    3

    5

    0106 i.e.

    x

    x

    y

    x

    3

    5

  • Concavity

    up concave is curve the,0 If xf

    235sketch derivative second at the lookingBy e.g. 23 xxxy

    down concave is curve the,0 If xf

    inflection ofpoint possible ,0 If xf

    3103 2 xxdx

    dy

    The second deriviative measures the change in slope with respect to x,

    this is known as concavity

    1062

    2

    xdx

    yd

    0 whenup concave is Curve2

    2

    dx

    yd

    3

    5

    0106 i.e.

    x

    x

    y

    x

    3

    5

  • Turning Points

  • Turning Points All turning points are stationary points.

  • Turning Points All turning points are stationary points.

    point turningminimum ,0 If xf

  • Turning Points All turning points are stationary points.

    point turningminimum ,0 If xf

    point turningmaximum ,0 If xf

  • Turning Points All turning points are stationary points.

    point turningminimum ,0 If xf

    point turningmaximum ,0 If xf

    1 of points turning theFind e.g. 23 xxxy

  • Turning Points All turning points are stationary points.

    point turningminimum ,0 If xf

    point turningmaximum ,0 If xf

    1 of points turning theFind e.g. 23 xxxy

    123 2 xxdx

    dy

  • Turning Points All turning points are stationary points.

    point turningminimum ,0 If xf

    point turningmaximum ,0 If xf

    1 of points turning theFind e.g. 23 xxxy

    123 2 xxdx

    dy

    262

    2

    xdx

    yd

  • Turning Points All turning points are stationary points.

    point turningminimum ,0 If xf

    point turningmaximum ,0 If xf

    1 of points turning theFind e.g. 23 xxxy

    123 2 xxdx

    dy

    262

    2

    xdx

    yd

    0 occur when points Stationary dx

    dy

  • Turning Points All turning points are stationary points.

    point turningminimum ,0 If xf

    point turningmaximum ,0 If xf

    1 of points turning theFind e.g. 23 xxxy

    123 2 xxdx

    dy

    262

    2

    xdx

    yd

    0 occur when points Stationary dx

    dy

    0123 i.e. 2 xx

  • Turning Points All turning points are stationary points.

    point turningminimum ,0 If xf

    point turningmaximum ,0 If xf

    1 of points turning theFind e.g. 23 xxxy

    123 2 xxdx

    dy

    262

    2

    xdx

    yd

    0 occur when points Stationary dx

    dy

    0123 i.e. 2 xx

    0113 xx

  • Turning Points All turning points are stationary points.

    point turningminimum ,0 If xf

    point turningmaximum ,0 If xf

    1 of points turning theFind e.g. 23 xxxy

    123 2 xxdx

    dy

    262

    2

    xdx

    yd

    0 occur when points Stationary dx

    dy

    0123 i.e. 2 xx

    0113 xx

    1or 3

    1 xx

  • 04

    216,1when 2

    2

    dx

    ydx

  • 04

    216,1when 2

    2

    dx

    ydx

    point turningmaximum a is 1,2-

  • 04

    216,1when 2

    2

    dx

    ydx

    point turningmaximum a is 1,2-

    04

    23

    16,

    3

    1when

    2

    2

    dx

    ydx

  • 04

    216,1when 2

    2

    dx

    ydx

    point turningmaximum a is 1,2-

    04

    23

    16,

    3

    1when

    2

    2

    dx

    ydx

    point turningminimum a is 27

    22,

    3

    1

  • Inflection Points

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

    264 of point(s) inflection theFind e.g. 23 xxy

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

    264 of point(s) inflection theFind e.g. 23 xxy

    xxdx

    dy1212 2 1224

    2

    2

    xdx

    yd

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

    264 of point(s) inflection theFind e.g. 23 xxy

    xxdx

    dy1212 2 1224

    2

    2

    xdx

    yd

    0 occur when inflection of points Possible2

    2

    dx

    yd

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

    264 of point(s) inflection theFind e.g. 23 xxy

    xxdx

    dy1212 2 1224

    2

    2

    xdx

    yd

    0 occur when inflection of points Possible2

    2

    dx

    yd

    01224 .. xei

    2

    1x

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

    264 of point(s) inflection theFind e.g. 23 xxy

    xxdx

    dy1212 2 1224

    2

    2

    xdx

    yd

    0 occur when inflection of points Possible2

    2

    dx

    yd

    01224 .. xei

    2

    1x

    x

    2

    2

    dx

    yd0

    2

    1

    2

    1

    2

    1

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

    264 of point(s) inflection theFind e.g. 23 xxy

    xxdx

    dy1212 2 1224

    2

    2

    xdx

    yd

    0 occur when inflection of points Possible2

    2

    dx

    yd

    01224 .. xei

    2

    1x

    x

    2

    2

    dx

    yd0

    2

    1

    2

    1

    (0)

    (12)

    2

    1

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

    264 of point(s) inflection theFind e.g. 23 xxy

    xxdx

    dy1212 2 1224

    2

    2

    xdx

    yd

    0 occur when inflection of points Possible2

    2

    dx

    yd

    01224 .. xei

    2

    1x

    x

    2

    2

    dx

    yd0

    2

    1

    2

    1

    (-1) (0)

    (-12) (12)

    2

    1

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

    264 of point(s) inflection theFind e.g. 23 xxy

    xxdx

    dy1212 2 1224

    2

    2

    xdx

    yd

    0 occur when inflection of points Possible2

    2

    dx

    yd

    01224 .. xei

    2

    1x

    x

    2

    2

    dx

    yd0

    2

    1

    2

    1

    (-1) (0)

    (-12) (12)

    concavityin change a is there

    2

    1

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

    264 of point(s) inflection theFind e.g. 23 xxy

    xxdx

    dy1212 2 1224

    2

    2

    xdx

    yd

    0 occur when inflection of points Possible2

    2

    dx

    yd

    01224 .. xei

    2

    1x

    x

    2

    2

    dx

    yd0

    2

    1

    2

    1

    (-1) (0)

    (-12) (12)

    concavityin change a is there

    2

    1

    inflection ofpoint a is 32

    1

    ,

  • Inflection Points A point of inflection is where there is a change in concavity, to see if

    there is a change, check either side of the point.

    264 of point(s) inflection theFind e.g. 23 xxy

    xxdx

    dy1212 2 1224

    2

    2

    xdx

    yd

    0 occur when inflection of points Possible2

    2

    dx

    yd

    01224 .. xei

    2

    1x

    x

    2

    2

    dx

    yd0

    2

    1

    2

    1

    (-1) (0)

    (-12) (12)

    concavityin change a is there

    2

    1

    inflection ofpoint a is 32

    1

    ,

    Horizontal Point of Inflection; 0dx

    dy0

    2

    2

    dx

    yd0

    3

    3

    dx

    yd

  • Alternative Way of Finding

    Inflection Points

  • Alternative Way of Finding

    Inflection Points

    0 occur when inflection of points Possible2

    2

    dx

    yd

  • Alternative Way of Finding

    Inflection Points

    0 occur when inflection of points Possible2

    2

    dx

    yd

    3 5 7

    3 5 7

    If the first non-zero derivative is of odd order,

    i.e 0 or 0 or 0 etc

    then it is a point of inflection

    d y d y d y

    dx dx dx

  • Alternative Way of Finding

    Inflection Points

    0 occur when inflection of points Possible2

    2

    dx

    yd

    3 5 7

    3 5 7

    If the first non-zero derivative is of odd order,

    i.e 0 or 0 or 0 etc

    then it is a point of inflection

    d y d y d y

    dx dx dx

    4 6 8

    4 6 8

    If the first non-zero derivative is of even order,

    i.e 0 or 0 or 0 etc

    then it is not a point of inflection

    d y d y d y

    dx dx dx

  • 24..3

    3

    dx

    ydge

  • 24..3

    3

    dx

    ydge

    024,2

    1when

    3

    3

    dx

    ydx

  • 24..3

    3

    dx

    ydge

    024,2

    1when

    3

    3

    dx

    ydx

    concavityin change a is there

  • 24..3

    3

    dx

    ydge

    024,2

    1when

    3

    3

    dx

    ydx

    inflection ofpoint a is 32

    1

    ,

    concavityin change a is there

  • 24..3

    3

    dx

    ydge

    024,2

    1when

    3

    3

    dx

    ydx

    inflection ofpoint a is 32

    1

    ,

    Exercise 10E; 1, 2bc, 3, 6ac,

    7bd, 8, 10, 12, 14, 16, 18

    concavityin change a is there