11x1 t12 03 second derivative (2010)

14
The Second Derivative

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Page 1: 11X1 T12 03 second derivative (2010)

The Second Derivative

Page 2: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

Page 3: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

Page 4: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

16 xdx

dy

Page 5: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

16 xdx

dy

62

2

dx

yd

Page 6: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

16 xdx

dy

62

2

dx

yd

32 xyii

Page 7: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

16 xdx

dy

62

2

dx

yd

32 xyii

21

2 3 x

Page 8: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

16 xdx

dy

62

2

dx

yd

32 xyii

21

2 3 x

xxdx

dy23

2

12

12

Page 9: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

16 xdx

dy

62

2

dx

yd

32 xyii

21

2 3 x

xxdx

dy23

2

12

12

2

12 3

xx

Page 10: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

16 xdx

dy

62

2

dx

yd

32 xyii

21

2 3 x

xxdx

dy23

2

12

12

2

12 3

xx

13232

12

122

32

2

2

xxxx

dx

yd

Page 11: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

16 xdx

dy

62

2

dx

yd

32 xyii

21

2 3 x

xxdx

dy23

2

12

12

2

12 3

xx

13232

12

122

32

2

2

xxxx

dx

yd

2

122

322 33

xxx

Page 12: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

16 xdx

dy

62

2

dx

yd

32 xyii

21

2 3 x

xxdx

dy23

2

12

12

2

12 3

xx

13232

12

122

32

2

2

xxxx

dx

yd

2

122

322 33

xxx

33 222

32

xxx

Page 13: 11X1 T12 03 second derivative (2010)

The Second Derivative

2

2

2

2

,,,dx

ydxf

dx

dxfy

23 e.g. 2 xxyi

16 xdx

dy

62

2

dx

yd

32 xyii

21

2 3 x

xxdx

dy23

2

12

12

2

12 3

xx

13232

12

122

32

2

2

xxxx

dx

yd

2

122

322 33

xxx

33 222

32

xxx

33

322

xx

Page 14: 11X1 T12 03 second derivative (2010)

Exercise 10D; 1 to 13, every 2nd in all