12 x1 t04 07 approximations to roots (2010)

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Approximations To Roots (1) Halving The Interval

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Page 1: 12 X1 T04 07 approximations to roots (2010)

Approximations To Roots(1) Halving The Interval

Page 2: 12 X1 T04 07 approximations to roots (2010)

Approximations To Roots(1) Halving The Interval

y

x

If y = f(x) is a continuous function over the interval , and f(a) and f(b) are opposite in sign,

bxa

xfy

Page 3: 12 X1 T04 07 approximations to roots (2010)

Approximations To Roots(1) Halving The Interval

y

x

If y = f(x) is a continuous function over the interval , and f(a) and f(b) are opposite in sign,

bxa

xfy af

a bf

b

Page 4: 12 X1 T04 07 approximations to roots (2010)

Approximations To Roots(1) Halving The Interval

y

x

If y = f(x) is a continuous function over the interval , and f(a) and f(b) are opposite in sign,

bxa

xfy af

a bf

b

then at least one root of the equation f(x) = 0 lies in the interval bxa

Page 5: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of in the interval01924 xx 31 x

Page 6: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of in the interval01924 xx

01619211 4

f

31 x

1924 xxxf 068

193233 4

f

Page 7: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of in the interval01924 xx

01619211 4

f

22

311

x

31 x

1924 xxxf 068

193233 4

f

01

192222 4

f

1 2 3

Page 8: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of in the interval01924 xx

01619211 4

f

21intervalin liessolution x

22

311

x

31 x

1924 xxxf 068

193233 4

f

01

192222 4

f

1 2 3

Page 9: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of in the interval01924 xx

01619211 4

f

21intervalin liessolution x

22

311

x

31 x

1924 xxxf 068

193233 4

f

01

192222 4

f

1 2 3

5.12

212

x

09.10195.125.15.1 4

f

1 1.5 2

Page 10: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of in the interval01924 xx

01619211 4

f

21intervalin liessolution x

22

311

x

31 x

1924 xxxf 068

193233 4

f

01

192222 4

f

1 2 3

21.5intervalin liessolution x

5.12

212

x

09.10195.125.15.1 4

f

1 1.5 2

Page 11: 12 X1 T04 07 approximations to roots (2010)

75.12

25.13

x

012.61975.1275.175.1 4

f

1.5 1.75 2

Page 12: 12 X1 T04 07 approximations to roots (2010)

21.75intervalin liessolution x

75.12

25.13

x

012.61975.1275.175.1 4

f

1.5 1.75 2

Page 13: 12 X1 T04 07 approximations to roots (2010)

21.75intervalin liessolution x

75.12

25.13

x

012.61975.1275.175.1 4

f

1.5 1.75 2

88.12

275.14

x

075.21988.1288.188.1 4

f

1.75 1.88 2

Page 14: 12 X1 T04 07 approximations to roots (2010)

21.75intervalin liessolution x

75.12

25.13

x

012.61975.1275.175.1 4

f

1.5 1.75 2

21.88intervalin liessolution x

88.12

275.14

x

075.21988.1288.188.1 4

f

1.75 1.88 2

Page 15: 12 X1 T04 07 approximations to roots (2010)

21.75intervalin liessolution x

75.12

25.13

x

012.61975.1275.175.1 4

f

1.5 1.75 2

21.88intervalin liessolution x

88.12

275.14

x

075.21988.1288.188.1 4

f

1.75 1.88 2

94.12

288.15

x

096.01994.1294.194.1 4

f

1.88 1.94 2

Page 16: 12 X1 T04 07 approximations to roots (2010)

21.75intervalin liessolution x

75.12

25.13

x

012.61975.1275.175.1 4

f

1.5 1.75 2

21.88intervalin liessolution x

88.12

275.14

x

075.21988.1288.188.1 4

f

1.75 1.88 2

21.94intervalin liessolution x

94.12

288.15

x

096.01994.1294.194.1 4

f

1.88 1.94 2

Page 17: 12 X1 T04 07 approximations to roots (2010)

97.12

294.16

x

0001.01997.1297.197.1 4

f

1.94 1.97 2

Page 18: 12 X1 T04 07 approximations to roots (2010)

97.11.94intervalin liessolution x

97.12

294.16

x

0001.01997.1297.197.1 4

f

1.94 1.97 2

Page 19: 12 X1 T04 07 approximations to roots (2010)

97.11.94intervalin liessolution x

97.12

294.16

x

0001.01997.1297.197.1 4

f

1.94 1.97 2

96.12

97.194.17

x

032.0

1996.1296.196.1 4

f 1.94 1.96 1.97

Page 20: 12 X1 T04 07 approximations to roots (2010)

97.11.94intervalin liessolution x

97.12

294.16

x

0001.01997.1297.197.1 4

f

1.94 1.97 2

97.11.96intervalin liessolution x

96.12

97.194.17

x

032.0

1996.1296.196.1 4

f 1.94 1.96 1.97

Page 21: 12 X1 T04 07 approximations to roots (2010)

97.11.94intervalin liessolution x

97.12

294.16

x

0001.01997.1297.197.1 4

f

1.94 1.97 2

97.11.96intervalin liessolution x

96.12

97.194.17

x

032.0

1996.1296.196.1 4

f 1.94 1.96 1.97

97.12

97.196.18

x

Page 22: 12 X1 T04 07 approximations to roots (2010)

97.11.94intervalin liessolution x

97.12

294.16

x

0001.01997.1297.197.1 4

f

1.94 1.97 2

97.11.96intervalin liessolution x

96.12

97.194.17

x

032.0

1996.1296.196.1 4

f 1.94 1.96 1.97

1.97isroot for theion approximatan x97.1

297.196.1

8

x

Page 23: 12 X1 T04 07 approximations to roots (2010)

(2) Newton’s Method of Approximation

Page 24: 12 X1 T04 07 approximations to roots (2010)

(2) Newton’s Method of Approximationy

x

Page 25: 12 X1 T04 07 approximations to roots (2010)

(2) Newton’s Method of Approximationy

x

If is a good first approximation to a root of the equation f(x) = 0, then a closer approximation is given by;

0x

0

001 xf

xfxx

Page 26: 12 X1 T04 07 approximations to roots (2010)

(2) Newton’s Method of Approximationy

x

If is a good first approximation to a root of the equation f(x) = 0, then a closer approximation is given by;

0x

0

001 xf

xfxx

by;given are,,,,ionsapproximat Successive 132 nn xxxx n

nnn xf

xfxx

1

Page 27: 12 X1 T04 07 approximations to roots (2010)

(2) Newton’s Method of Approximationy

x

If is a good first approximation to a root of the equation f(x) = 0, then a closer approximation is given by;

0x

0

001 xf

xfxx

by;given are,,,,ionsapproximat Successive 132 nn xxxx n

nnn xf

xfxx

1

NOTE:must be a good first

approximationNewton’s method finds where the tangent at cuts the x axis

0x

0x

Page 28: 12 X1 T04 07 approximations to roots (2010)

(2) Newton’s Method of Approximationy

x

xfy

If is a good first approximation to a root of the equation f(x) = 0, then a closer approximation is given by;

0x

0

001 xf

xfxx

by;given are,,,,ionsapproximat Successive 132 nn xxxx n

nnn xf

xfxx

1

NOTE:must be a good first

approximationNewton’s method finds where the tangent at cuts the x axis

0x

0x

0x

Page 29: 12 X1 T04 07 approximations to roots (2010)

(2) Newton’s Method of Approximationy

x

xfy

If is a good first approximation to a root of the equation f(x) = 0, then a closer approximation is given by;

0x

0

001 xf

xfxx

by;given are,,,,ionsapproximat Successive 132 nn xxxx n

nnn xf

xfxx

1

NOTE:must be a good first

approximationNewton’s method finds where the tangent at cuts the x axis

0x

0x

0x1x

Page 30: 12 X1 T04 07 approximations to roots (2010)

(2) Newton’s Method of Approximationy

x

xfy

If is a good first approximation to a root of the equation f(x) = 0, then a closer approximation is given by;

0x

0

001 xf

xfxx

by;given are,,,,ionsapproximat Successive 132 nn xxxx n

nnn xf

xfxx

1

NOTE:must be a good first

approximationNewton’s method finds where the tangent at cuts the x axis

0x

0x

0x1x

failwillmethodtheaxis || tangenti.e.0If 0

xxf

Page 31: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of 01924 xx

Page 32: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of 01924 xx

24 3 xxf

1924 xxxf

Page 33: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of 01924 xx

24 3 xxf

1924 xxxf

5.10 x 9375.10

195.125.15.1 4

f

5.1525.145.1 3

f

Page 34: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of 01924 xx

24 3 xxf

1924 xxxf

5.10 x 9375.10

195.125.15.1 4

f

5.1525.145.1 3

f

21.25.15

9375.105.1

0

001

xfxfxx

Page 35: 12 X1 T04 07 approximations to roots (2010)

e.g Find an approximation to two decimal places for a root of 01924 xx

24 3 xxf

1924 xxxf

5.10 x 9375.10

195.125.15.1 4

f

5.1525.145.1 3

f

21.25.15

9375.105.1

0

001

xfxfxx

2744.91921.2221.221.2 4

f

1754.45

221.2421.2 3

f

Page 36: 12 X1 T04 07 approximations to roots (2010)

00.21754.45

2744.921.22

x

Page 37: 12 X1 T04 07 approximations to roots (2010)

00.21754.45

2744.921.22

x 1

192222 4

f

35

2242 3

f

Page 38: 12 X1 T04 07 approximations to roots (2010)

00.21754.45

2744.921.22

x 1

192222 4

f

35

2242 3

f

97.135123

x

Page 39: 12 X1 T04 07 approximations to roots (2010)

00.21754.45

2744.921.22

x 1

192222 4

f

35

2242 3

f

97.135123

x 001.0

1997.1297.197.1 4

f

58.32

297.1497.1 3

f

Page 40: 12 X1 T04 07 approximations to roots (2010)

00.21754.45

2744.921.22

x 1

192222 4

f

35

2242 3

f

97.135123

x 001.0

1997.1297.197.1 4

f

58.32

297.1497.1 3

f

97.158.32

001.097.14

x

Page 41: 12 X1 T04 07 approximations to roots (2010)

00.21754.45

2744.921.22

x 1

192222 4

f

35

2242 3

f

97.135123

x 001.0

1997.1297.197.1 4

f

58.32

297.1497.1 3

f

97.158.32

001.097.14

x

rootfor theion approximatbetter ais1.97 x

Page 42: 12 X1 T04 07 approximations to roots (2010)

( ) Use Newton's Method to obtain an approximation to 23 correct to two decimal placesii

Page 43: 12 X1 T04 07 approximations to roots (2010)

( ) Use Newton's Method to obtain an approximation to 23 correct to two decimal placesii

2 23f x x

Page 44: 12 X1 T04 07 approximations to roots (2010)

( ) Use Newton's Method to obtain an approximation to 23 correct to two decimal placesii

2 23f x x

2f x x

Page 45: 12 X1 T04 07 approximations to roots (2010)

( ) Use Newton's Method to obtain an approximation to 23 correct to two decimal placesii

2 23f x x

2f x x

21

11

21

1

232

232

nn n

n

n

n

xx xx

xx

Page 46: 12 X1 T04 07 approximations to roots (2010)

( ) Use Newton's Method to obtain an approximation to 23 correct to two decimal placesii

2 23f x x

2f x x

21

11

21

1

232

232

nn n

n

n

n

xx xx

xx

0 5x

Page 47: 12 X1 T04 07 approximations to roots (2010)

( ) Use Newton's Method to obtain an approximation to 23 correct to two decimal placesii

2 23f x x

2f x x

21

11

21

1

232

232

nn n

n

n

n

xx xx

xx

0 5x

2

15 232 5

x

1 4.8x

Page 48: 12 X1 T04 07 approximations to roots (2010)

( ) Use Newton's Method to obtain an approximation to 23 correct to two decimal placesii

2 23f x x

2f x x

21

11

21

1

232

232

nn n

n

n

n

xx xx

xx

0 5x

2

15 232 5

x

1 4.8x

2

24.8 232 4.8

x

2 4.795833333x

2 4.80 (to 2 dp)x

Page 49: 12 X1 T04 07 approximations to roots (2010)

( ) Use Newton's Method to obtain an approximation to 23 correct to two decimal placesii

2 23f x x

2f x x

21

11

21

1

232

232

nn n

n

n

n

xx xx

xx

0 5x

2

15 232 5

x

1 4.8x

2

24.8 232 4.8

x

2 4.795833333x

2 4.80 (to 2 dp)x

23 4.80 (to 2 dp)

Page 50: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

Page 51: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s MethodApproximations oscillate

Page 52: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

y

x

Approximations oscillate

Page 53: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

y

x

Approximations oscillate

want to find this root

Page 54: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

y

x

Approximations oscillate

1x

want to find this root

2x

Page 55: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

y

x

Approximations oscillate

1x 2x

want to find this root

Page 56: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

y

x

Approximations oscillate

1x 2x

wrong side of stationary pointconverges to wrong root

want to find this root

Page 57: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

y

x

Approximations oscillate

1x 2x

wrong side of stationary pointconverges to wrong rooty

x

want to find this root

Page 58: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

y

x

Approximations oscillate

1x 2x

wrong side of stationary pointconverges to wrong rooty

x

want to find this root

want to find this root

Page 59: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

y

x

Approximations oscillate

1x 2x

wrong side of stationary pointconverges to wrong rooty

x1x 2x

want to find this root

want to find this root

Page 60: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

y

x

Approximations oscillate

1x 2x

wrong side of stationary pointconverges to wrong rooty

x1x 2x

want to find this root

want to find this root

Page 61: 12 X1 T04 07 approximations to roots (2010)

Other Possible Problems with Newton’s Method

y

x

Approximations oscillate

1x 2x

wrong side of stationary pointconverges to wrong rooty

x1x 2x

want to find this root

want to find this root

Exercise 6E; 1, 3ac, 6adf, 8a, 10, 12