11 x1 t10 04 concavity (2012)
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Transcript of 11 x1 t10 04 concavity (2012)
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Concavity
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Concavity The second deriviative measures the change in slope with respect to x,
this is known as concavity
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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
![Page 4: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/4.jpg)
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
![Page 5: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/5.jpg)
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
![Page 6: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/6.jpg)
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
![Page 7: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/7.jpg)
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
![Page 8: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/8.jpg)
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
![Page 9: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/9.jpg)
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
![Page 10: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/10.jpg)
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
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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
![Page 12: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/12.jpg)
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
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Turning Points
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Turning Points All turning points are stationary points.
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Turning Points All turning points are stationary points.
point turningminimum ,0 If xf
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Turning Points All turning points are stationary points.
point turningminimum ,0 If xf
point turningmaximum ,0 If xf
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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
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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
![Page 19: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/19.jpg)
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
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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
![Page 21: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/21.jpg)
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
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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
![Page 23: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/23.jpg)
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
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04
216,1when 2
2
dx
ydx
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04
216,1when 2
2
dx
ydx
point turningmaximum a is 1,2-
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04
216,1when 2
2
dx
ydx
point turningmaximum a is 1,2-
04
23
16,
3
1when
2
2
dx
ydx
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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
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Inflection Points
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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.
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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
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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
![Page 32: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/32.jpg)
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
![Page 33: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/33.jpg)
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
![Page 34: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/34.jpg)
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
![Page 35: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/35.jpg)
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
![Page 36: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/36.jpg)
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
![Page 37: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/37.jpg)
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
![Page 38: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/38.jpg)
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
,
![Page 39: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/39.jpg)
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
![Page 40: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/40.jpg)
Alternative Way of Finding
Inflection Points
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Alternative Way of Finding
Inflection Points
0 occur when inflection of points Possible2
2
dx
yd
![Page 42: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/42.jpg)
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
![Page 43: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/43.jpg)
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
![Page 44: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/44.jpg)
24..3
3
dx
ydge
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24..3
3
dx
ydge
024,2
1when
3
3
dx
ydx
![Page 46: 11 x1 t10 04 concavity (2012)](https://reader034.fdocuments.in/reader034/viewer/2022052621/557e78ecd8b42a03668b517c/html5/thumbnails/46.jpg)
24..3
3
dx
ydge
024,2
1when
3
3
dx
ydx
concavityin change a is there
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24..3
3
dx
ydge
024,2
1when
3
3
dx
ydx
inflection ofpoint a is 32
1
,
concavityin change a is there
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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