11x1 t09 04 concavity

Concavity

ConcavityThe second deriviative measures the change in slope with respect to x, this is known as concavity

Concavity upconcaveiscurvethe,0 If xf

The second deriviative measures the change in slope with respect to x, this is known as concavity

Concavity upconcaveiscurvethe,0 If xf downconcaveiscurvethe,0 If xf

The second deriviative measures the change in slope with respect to x, this is known as concavity

Concavity upconcaveiscurvethe,0 If xf downconcaveiscurvethe,0 If xf inflectionofpoint possible,0 If xf

The second deriviative measures the change in slope with respect to x, this is known as concavity

Concavity upconcaveiscurvethe,0 If xf

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

downconcaveiscurvethe,0 If xf inflectionofpoint possible,0 If xf

The second deriviative measures the change in slope with respect to x, this is known as concavity

Concavity upconcaveiscurvethe,0 If xf

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

downconcaveiscurvethe,0 If xf inflectionofpoint possible,0 If xf

3103 2 xxdxdy

The second deriviative measures the change in slope with respect to x, this is known as concavity

Concavity upconcaveiscurvethe,0 If xf

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

downconcaveiscurvethe,0 If xf inflectionofpoint possible,0 If xf

3103 2 xxdxdy

The second deriviative measures the change in slope with respect to x, this is known as concavity

1062

2

xdx

yd

Concavity upconcaveiscurvethe,0 If xf

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

downconcaveiscurvethe,0 If xf inflectionofpoint possible,0 If xf

3103 2 xxdxdy

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 Curve 2

2

dx

yd

Concavity upconcaveiscurvethe,0 If xf

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

downconcaveiscurvethe,0 If xf inflectionofpoint possible,0 If xf

3103 2 xxdxdy

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 Curve 2

2

dx

yd

35

0106 i.e.

x

x

Concavity upconcaveiscurvethe,0 If xf

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

downconcaveiscurvethe,0 If xf inflectionofpoint possible,0 If xf

3103 2 xxdxdy

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 Curve 2

2

dx

yd

35

0106 i.e.

x

x

y

x35

Concavity upconcaveiscurvethe,0 If xf

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

downconcaveiscurvethe,0 If xf inflectionofpoint possible,0 If xf

3103 2 xxdxdy

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 Curve 2

2

dx

yd

35

0106 i.e.

x

x

y

x35

Turning Points

Turning PointsAll turning points are stationary points.

Turning PointsAll turning points are stationary points.

pointturningminimum,0 If xf

Turning PointsAll turning points are stationary points.

pointturningminimum,0 If xf pointturningmaximum,0 If xf

Turning PointsAll turning points are stationary points.

pointturningminimum,0 If xf pointturningmaximum,0 If xf

1 of points turning theFind e.g. 23 xxxy

Turning PointsAll turning points are stationary points.

pointturningminimum,0 If xf pointturningmaximum,0 If xf

1 of points turning theFind e.g. 23 xxxy

123 2 xxdxdy

Turning PointsAll turning points are stationary points.

pointturningminimum,0 If xf pointturningmaximum,0 If xf

1 of points turning theFind e.g. 23 xxxy

123 2 xxdxdy

262

2

xdx

yd

Turning PointsAll turning points are stationary points.

pointturningminimum,0 If xf pointturningmaximum,0 If xf

1 of points turning theFind e.g. 23 xxxy

123 2 xxdxdy

262

2

xdx

yd

0 occur when points Stationary dxdy

Turning PointsAll turning points are stationary points.

pointturningminimum,0 If xf pointturningmaximum,0 If xf

1 of points turning theFind e.g. 23 xxxy

123 2 xxdxdy

262

2

xdx

yd

0 occur when points Stationary dxdy

0123i.e. 2 xx

Turning PointsAll turning points are stationary points.

pointturningminimum,0 If xf pointturningmaximum,0 If xf

1 of points turning theFind e.g. 23 xxxy

123 2 xxdxdy

262

2

xdx

yd

0 occur when points Stationary dxdy

0123i.e. 2 xx 0113 xx

Turning PointsAll turning points are stationary points.

pointturningminimum,0 If xf pointturningmaximum,0 If xf

1 of points turning theFind e.g. 23 xxxy

123 2 xxdxdy

262

2

xdx

yd

0 occur when points Stationary dxdy

0123i.e. 2 xx 0113 xx

1or 31

xx

04

216,1when 2

2

dx

ydx

04

216,1when 2

2

dx

ydx

point turningmaximumais1,2-

04

216,1when 2

2

dx

ydx

point turningmaximumais1,2-

04

2316,

31when 2

2

dxydx

04

216,1when 2

2

dx

ydx

point turningmaximumais1,2-

04

2316,

31when 2

2

dxydx

point turningminimum a is 2722,

31

04

216,1when 2

2

dx

ydx

point turningmaximumais1,2-

04

2316,

31when 2

2

dxydx

point turningminimum a is 2722,

31

Inflection Points

04

216,1when 2

2

dx

ydx

point turningmaximumais1,2-

04

2316,

31when 2

2

dxydx

point turningminimum a is 2722,

31

Inflection PointsA point of inflection is where there is a change in concavity, to see if there is a change, check either side of the point.

04

216,1when 2

2

dx

ydx

point turningmaximumais1,2-

04

2316,

31when 2

2

dxydx

point turningminimum a is 2722,

31

Inflection PointsA 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

04

216,1when 2

2

dx

ydx

point turningmaximumais1,2-

04

2316,

31when 2

2

dxydx

point turningminimum a is 2722,

31

Inflection PointsA 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

xxdxdy 1212 2

04

216,1when 2

2

dx

ydx

point turningmaximumais1,2-

04

2316,

31when 2

2

dxydx

point turningminimum a is 2722,

31

Inflection PointsA 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

xxdxdy 1212 2 12242

2

xdx

yd

0 occur when inflection of points Possible 2

2

dx

yd

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd

21

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

21

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) 21

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1)

(-12)

21

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12)

21

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)

21

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere21

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere21

inflection ofpoint a is 321

,

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere

inflection ofpoint a isit then ,0 If 3

3

dx

yd

21

inflection ofpoint a is 321

,

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere

inflection ofpoint a isit then ,0 If 3

3

dx

yd

24.. 3

3

dx

ydge

21

inflection ofpoint a is 321

,

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere

inflection ofpoint a isit then ,0 If 3

3

dx

yd

24.. 3

3

dx

ydge

21

inflection ofpoint a is 321

,

024,21when 3

3

dx

ydx

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere

inflection ofpoint a isit then ,0 If 3

3

dx

yd

24.. 3

3

dx

ydge

21

inflection ofpoint a is 321

,

024,21when 3

3

dx

ydx

inflection ofpoint a is 321

,

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere

inflection ofpoint a isit then ,0 If 3

3

dx

yd

24.. 3

3

dx

ydge

21

inflection ofpoint a is 321

,

024,21when 3

3

dx

ydx

inflection ofpoint a is 321

,

Horizontal Point of Inflection;

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere

inflection ofpoint a isit then ,0 If 3

3

dx

yd

24.. 3

3

dx

ydge

21

inflection ofpoint a is 321

,

024,21when 3

3

dx

ydx

inflection ofpoint a is 321

,

Horizontal Point of Inflection; 0dxdy

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere

inflection ofpoint a isit then ,0 If 3

3

dx

yd

24.. 3

3

dx

ydge

21

inflection ofpoint a is 321

,

024,21when 3

3

dx

ydx

inflection ofpoint a is 321

,

Horizontal Point of Inflection; 0dxdy

02

2

dx

yd

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere

inflection ofpoint a isit then ,0 If 3

3

dx

yd

24.. 3

3

dx

ydge

21

inflection ofpoint a is 321

,

024,21when 3

3

dx

ydx

inflection ofpoint a is 321

,

Horizontal Point of Inflection; 0dxdy

02

2

dx

yd 03

3

dx

yd

0 occur when inflection of points Possible 2

2

dx

yd

01224 .. xei

21

x x

2

2

dxyd 0

21

21

(-1) (0)

(-12) (12)concavityin change aisthere

inflection ofpoint a isit then ,0 If 3

3

dx

yd

24.. 3

3

dx

ydge

21

inflection ofpoint a is 321

,

024,21when 3

3

dx

ydx

inflection ofpoint a is 321

,

Exercise 10E; 1, 2bc, 3, 6ac,7bd, 8, 10, 12, 14, 16, 18

Horizontal Point of Inflection; 0dxdy

02

2

dx

yd 03

3

dx

yd