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Exercise 3C
1 a 3 3 5 0x x− − = Let ( ) 3f 3 5x x x= − −
( )f 2 3= −
( )f 3 13= There is a sign change between f (2) and f (3) so a root of the equation lies in the interval [2, 3]. b
By similar triangles:
1
1
3 132 3
xx−
=−
1 1
1
9 3 13 2616 35
x xx
− = −=
135162.1875
x =
=
( )f 2.1875 1.094...= − There is a sign change, therefore a root lies between x = 2.1875 and x = 3.
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By similar triangles:
2
2
3 132.1875... 1.094...
xx
−=
−
( ) ( )2 2
2
2
1.094... 3 13 2.187514.084... 31.722...
2.250...
x xxx
− = −
==
( )f 2.250... 0.351...= − There is a sign change, therefore a root lies between x = 2.250… and x = 3. By similar triangles:
3
3
3 132.250... 0.351...
xx
−=
−
( ) ( )3 3
3
3
0.351... 3 13 2.25013.351... 30.313...
2.270...
x xxx
− = −
==
( )f 2.270... 0.108...= −
Two successive approximations give x = 2.3, accurate to 1 d.p. 2 a 3 25 8 1 0x x− + = Let ( ) 3 2f 5 8 1x x x= − +
( )f 1 2= −
( )f 2 9= There is a sign change, therefore a root lies between x = 1 and x = 2.
By similar triangles:
1
1
2 91 2x
x−
=−
1 14 2 9 9x x− = −
111 13x =
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1
1311
1.18. .
x =
=
f 1.18 1.920.... . = −
There is a sign change, therefore a root lies between 1.18. .
x = and x = 2.
By similar triangles: 2
2
2 91.920...1.18
. .x
x
−=
−
( )2 2
2
2
1.920... 2 9 1.18
10.920... 14.476...1.325...
. .x x
xx
− = −
==
( )f 1.325... 1.410...= − There is a sign change, therefore a root lies between 1.325...x = and x = 2.
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By similar triangles:
3
3
2 91.325... 1.410...
xx
−=
−
( ) ( )3 3
3
3
1.410... 2 9 1.325...10.410... 14.745...
1.416...
x xxx
− = −
==
( )f 1.416... 0.841...= − There is a sign change, therefore a root lies between 1.416...x = and x = 2.
By similar triangles:
4
4
2 91.416... 0.841...
xx
−=
−
( ) ( )4 4
4
4
0.841... 2 9 1.416...9.841... 14.426...
1.465...
x xxx
− = −
==
( )f 1.465... 0.440...= − There is a sign change, therefore a root lies between 1.465...x = and x = 2.
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2 b By similar triangles:
5
5
2 91.465... 0.440...
xx
−=
−
( ) ( )5 5
5
5
0.440... 2 9 1.465...9.440... 14.066...
1.490...
x xxx
− = −
=
=
( )f 1.490... 0.220...= − Two successive approximations give x = 1.5, accurate to 1 d.p. 3 a 3 33 3 0x x
x x+ = ⇒ − − =
Let ( ) 3f 3x xx
= − −
( )f 3 1= −
( )f 4 0.25= There is a sign change, therefore a root lies between x = 3 and x = 4. b
By similar triangles:
1
1
4 0.253 1x
x−
=−
( )1 1
1
1
4 0.25 31.25 4.75
3.8
x xxx
− = −
==
( )f 3.8 0.010...= There is a sign change, therefore a root lies between x = 3 and x = 3.8.
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By similar triangles:
1
1
3.8 0.010...3 1x
x−
=−
( )1 1
1
1
3.8 0.010... 31.01... 3.831...
3.793
x xxx
− = −
==
Two successive approximations give x = 3.8, accurate to 1 d.p. 4 a 2 cos 1 0x x − = Let ( )f 2 cos 1x x x= −
( )f 1 0.080...=
( )f 1.5 0.787...= − There is a sign change, therefore a root lies between x = 1 and x = 1.5. b
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By similar triangles:
1
1
1.5 0.787...1 0.080...x
x−
=−
( ) ( )1 10.080... 1.5 0.787... 1x x− = −
1
1
0.867... 0.907...1.046...
xx==
( )f 1.046 0.048...= There is a sign change, therefore a root lies between x = 1.046… and x = 1.5.
By similar triangles:
2
2
1.5 0.787...1.046... 0.048...
xx
−=
−
( ) ( )2 20.048... 1.5 0.787... 1.046...x x− = −
20.835... 0.895...x =
2 1.071...x =
( )f 1.071 0.025...= There is a sign change, therefore a root lies between x = 1.071… and x = 1.5.
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By similar triangles:
3
3
1.5 0.787...1.071... 0.025...
xx
−=
−
( ) ( )3 30.025... 1.5 0.787... 1.071...x x− = −
30.812... 0.881...x =
3 1.085...x = Two successive approximations give x = 1.1, accurate to 1 d.p. 5 a 3 22 3 0x x− − = ( ) 3 2f 2 3x x x= − −
( )f 2 3= −
( )f 3 6= There is a sign change between f (2) and f (3), therefore there is a root of the equation in the
interval [2, 3]. 2f ( ) 3 4 (3 4)x x x x x′ = − = −
For 43
x > , ( )f x′ >0 and the function is increasing.
Therefore the root which lies in the interval [2, 3] must be the largest possible root of the equation. b
By similar triangles:
1
1
3 62 3
xx−
=−
( ) ( )1 13 3 6 2x x− = −
19 21x =
1
.2.3x =
.
f 2.3 1.185... = −
There is a sign change, therefore a root lies between x = .
2.3 and x = 3.
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By similar triangles: 2
2
3 6. 1.185...2.3
x
x
−=
−
( )2 2
.1.185... 3 6 2.3x x − = −
27.185... 17.555...x =
2 2.443...x =
( )f 2.443... 0.353...= − There is a sign change, therefore a root lies between x = 2.443... and x = 3.
By similar triangles:
3
3
3 62.443... 0.353...
xx
−=
−
( ) ( )3 30.353... 3 6 2.443...x x− = −
36.353... 15.717x =
3 2.474...x =
( )f 2.474... 0.098...= − There is a sign change, therefore a root lies between x = 2.474… and x = 3.
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By similar triangles:
4
4
3 62.474... 0.098...
xx
−=
−
( ) ( )4 40.098... 3 6 2.474...x x− = −
46.098... 15.140...x =
4 2.482x = Two successive approximations give x = 2.5, accurate to 1 d.p. 6 ( )f 2 3 1xx x= − −
( )f 3 2= −
( )f 4 3=
By similar triangles:
1
1
4 33 2x
x−
=−
( ) ( )1 12 4 3 3x x− = −
15 17x =
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1175
3.4
x =
=