# 11x1 t12 04 concavity (2010)

Post on 18-Jul-2015

255 views

Category:

## Documents

Tags:

• #### curve22dxyd350106 i

Embed Size (px)

TRANSCRIPT

• 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

Recommended