# 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