1. A step-by-step process of trial and improvement
Iteration
A step-by-step process of trial and improvement

2. y = x3 – 2x2 + 2x – 3 Remember zeros occur when y = 0 x -coordinate
y- coordinate
Zero between 1
– 2
Closer to 1∙81 than 1∙82 since 0∙0025 closer to x-axis
Closer to 1∙9 than 1∙7 since 0∙439 closer to x-axis
Closer to 1∙8 than 1∙9 since 0∙048 closer to x-axis 2 1
1 & 2
1∙7
-0∙467
1∙7 & 2
1∙9
0∙439
1∙7 & 1∙9
1∙8
-0∙048
1∙8 & 1∙9
1∙82
0∙0438
1∙8 & 1∙82
1∙81
-0∙0025
1∙81 & 1∙82

3. y = 5 + 2x – x3 Remember zeros occur when y = 0 x -coordinate
y -coordinate
Zero between
Closer to 2∙1 than 2∙07 since 0∙061 closer to x-axis than 0∙270
Much closer to 2 than 3 since 1 closer to x-axis
Closer to 2∙1 than 2 since 0∙061 closer to x-axis than 1
Closer to 2∙09 than 2∙1 since 0∙056 closer to x-axis than 0∙061 2 1 3
– 16
2 & 3
2∙1
-0∙061
2 & 2∙1
2∙07
0∙270
2∙07 & 2∙1
2∙09
0∙056
2∙09 & 2∙1

4. Show that there is a root of the equation
y = x3 – 4x – 7 between x = 2 and x = 3 and
find this root correct to 1 decimal place.
2∙6
2. Show that the equation y = 3 – x – x3 has a root
between x = 1 and x = 2 and find it correct to
1 decimal place.
1∙2

5. 3. The diagram shows the function y = 5 – 2x – x4.
One root lies in the interval [-2,-1] while the other root lies in
the interval [1,2]. Find both roots correct to 1 decimal place.
y = 5 – 2x – x4 -2 -1 1 y x
-1∙7
1∙3

6. 4. The diagram shows the graph of the function
y = x3 – 6x2 + x + 3.
Find each of the roots correct to 1 decimal place.
y = x3 – 6x2 + x + 3. y x
-0∙6
0∙9
5∙7

7. 5. The graph of y = 10 – x3 is shown in the diagram.
By finding point A obtain an approximation for 3√ 10. y
y = 10 – x3  x A
2∙15

8. 6. Show there is a root of the equation y = x3 + x – 19 between x = 2 and x = 3 and find this root correct to 1 decimal place.
7. Show that the equation y = 12 + x – 3x3 has a root between
x = 1 and x = 2 and find it correct to 1 decimal place.

9. 8. The diagram shows the function y = 7 + 2x – x4.
One root lies in the interval [-2,-1] while the other root lies in
the interval [1,2]. Find both roots correct to 1 decimal place.

10. 9. The diagram shows the graph of the function y = x3 + x2 - x - 9.
Find each of the roots correct to 1 decimal place.

11. 10. The graph of y = 18 – x3 is shown in the diagram.
By finding point A obtain an approximation for 3√ 18.