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

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

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

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

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

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

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

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.

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.

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

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