1. Sign Change Re-cap Show that each of the following have a solution for x: 1.The intersection of (between 1.8 and 1.9) and 2.The intersection of(between 3.6 and 3.7) and 3.The equation(between 0.4 and 0.5)

2. Rearranging equations Find as many ways as you can of rearranging to make x the subject:

3. Iteration. Use iteration to approximate solutions to difficult equations. Begin to understand convergence and staircase and cobweb diagrams. starter

4. The 1 st approximate solution lies anywhere between the bounds. The next stage is to improve this estimate. Rearrange the equation to the form. You may spot lots of ways of doing this. e.g. e.g. For the equation :

5. You may spot lots of ways of doing this. e.g. Rearrange the equation to the form. The 1 st approximate solution lies anywhere between the bounds. The next stage is to improve this estimate. (i) Square: or (ii) Rearrange: Cube root: or (iii) Rearrange: Divide by : e.g. For the equation :

6. to iterate means to repeat Let’s take the 2 nd arrangement: Our 1 st estimate of  we will call x 0. We substitute x 0 into the r.h.s. of the formula and the result gives the new estimate x 1. We now have We will then keep repeating the process so we write the formula as This is called an iterative formula. ( Some people start with x 1 which is just as good. )

7. Starting with we get Because we are going to repeat the calculation, we use the ANS function on the calculator. So, Type and press ENTER Type the r.h.s. of the equation, replacing x with ANS, using the ANS button, giving Press ENTER and you get ( 6 d.p. ) Pressing ENTER again replaces with and gives the next estimate and so on. ( 6 d.p. ) Casio emulator

8. If we continue to iterate we eventually get ( to 6 d.p. ) Although I’ve only written down 6 decimal places, the calculator is using the greatest possible accuracy. We get of our answer. Since the answer is correct to 6 decimal places, the exact value of must be within Error Bounds Tip: The index equals the number of d.ps. in the answer.

9. Exercise 1. (a) Show that the equation has a solution  between 2 and 3. (b) Use the iterative formula with to find the solution, giving your answer correct to 4 d.p.

10. Solutions 1. (a) Show that the equation has a solution between 2 and 3. Solution: (a) Let Change of sign (continuous function) (b) Use the iterative formula with to find the solution, giving your answer correct to 4 d.p. ( 4 d.p. )

11. Domino trail In table groups Convergence diagrams

12. Some arrangements of an equation give formulae which do not give a solution. We earlier met 3 arrangements of ( to 6 d.p. ) We used (ii) with to find the solution Now try (i) with We get and after a while the sequence just oscillates between 1 and 0. This iterative sequence does not converge. (i) (ii) (iii)

13. We get Now try the formula with The iteration then fails because we are trying to square root a negative number. Some arrangements of an equation give an iterative sequence which converges to a solution; others do not converge.

16. Exercise B Page 136 Convergence diagrams

17. Iteration You may spot lots of ways of doing this. e.g. Rearrange the equation to the form. The 1 st approximate solution lies anywhere between the bounds. The next stage is to improve this estimate. (i) Square: (ii) Rearrange: Cube root: (iii) Rearrange: Divide by : e.g. For the equation :

18. Iteration to iterate means to repeat Let’s take the 2 nd arrangement: Our 1 st estimate of  we will call x 0. We substitute x 0 into the r.h.s. of the formula and the result gives the new estimate x 1. We now have We will then keep repeating the process so we write the formula as This is called an iterative formula. ( Some people start with x 1 which is just as good. )

19. Iteration SUMMARY To find an approximation to a solution ( or root ) of an equation:  Find a 1 st approximation, often by finding integer bounds for the solution. Let this be x 0.  Rearrange the equation into the form  Write the arrangement as an iterative formula:  Key x 0 into a calculator and ENTER.  Key the r.h.s. of the formula into the calculator, replacing x with ANS.  Press ENTER as many times as required to get the solution to the specified accuracy.

20. Iteration Some arrangements of an equation give formulae which do not give a solution. We earlier met 3 arrangements of ( to 6 d.p. ) We used (ii) with to find the solution Trying (i) with gives and after a while the sequence just oscillates between 1 and 0. The iterative sequence does not converge.

21. Iteration gives Trying the arrangement with The iteration then fails because we are trying to square root a negative number. Some arrangements of an equation give an iterative sequence which converges to a solution; others do not converge.

22. Iteration Solution: (b) gives Let It takes about 7 iterations to reach (4 d.p.)