11x1 t09 04 concavity
TRANSCRIPT
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