1. Without using gsolv, equation solver, etc. Only calculator operations: x,+,-,^.

2. -1.6 0.6 1.3

3. Solve sinx = 1 – x

4. x-5-4-3-20123456 y x-5-4-3-20123456 y Table Mode

5.  To “show that” 2 given x-values lead to a f(x) sign change between them  Apply the sign change rule to approximate the roots of f(x)=0  Apply the sign change rule to approximate the roots of f(x)=g(x)  Rearrange an equation to make x the subject Topic ObjectivesSkills Practice  Clear explanation of a “show that” question  Calculator skills (evaluating functions)  Checking skills  Algebra (rearrange an equation)  Discussion/teamwork

6. By now we know that it is possible to find the solutions to equations such as... sinx = log(x) by looking at the graphs y=sinx and y=log(x) and seeing where they cross. Equations and Graphs

7. Puzzle Triangle In table groups Equation solver and G-Solv

8. Notice that where the graphs cross the upper graph before the root (solution) becomes the lower afterward (and vice versa for the lower). Change in Biggest

9. Let us call the two functions being equated f(x) and g(x)... The equation to solve is f(x) = g(x) If we input two values for x into f(x) and g(x) and which is the bigger function changes for the different x values there is a solution to the equation between those values. Numerical Test

10. Instead of working with f(x)=g(x) we could work with f(x)- g(x) = 0 the solutions of the equation are now where y=0 We can now identify the existence of a root between two x values if the result of this difference for the x values gives different signs (+ and -). A simpler way to define this... Sign Change Rule

11. 1.Change f(x)=g(x) into f(x)-g(x)=0 (if needed) 2.Substitute lower bound (given) value for x 3.Substitute upper bound (given) value for x 4.There should be a sign change (if not something must have gone wrong; they won’t tell you numbers should indicate a root if they don’t) 5.Make reference to the fact this sign change indicates the presence of a root between the x values. Steps in the sign test

12. Which of these equations have a root between x=3 and x=4? A: log 10 x = 4 – x B: sinx = 3 - 2x C: 2 x = 5x-3 D: cosx = 6 – 2x Answers on whiteboards =0x = 3x = 4Root? logx + x – 4 -0.522880.60206 Yes sinx + 2x – 3 3.141124.243198 No 2 x – 5x + 3 -4 No cosx + 2x - 6 -0.989991.346356 Yes

13. Which of these equations have a root between x=1 and x=2? A: sinx = 3 – x B: log(2x) + 1 = x C: 3 x – 2 = 0 D: cos2x = 1 – x Answers on whiteboards =0x = 1x = 2Root? sinx + x – 3 -1.15853-0.0907 No log(2x) + 1 – x 0.30103-0.39794 Yes 3 x – 2 17 No cos2x - 1 + x -0.416150.346356 Yes

14. Which of these equations have a root between x=1 and x=2? A: x 2 = 5cosx B: 2 x = 4sinx C: 1 / cosx = 4cosx D: 3log(5sinx) = 4 – 2x Answers on whiteboards =0x = 1x = 2Root? x 2 - 5cosx -1.701516.080734 Yes 2 x - 4sinx -1.365880.36281 Yes 1 / cosx - 4cosx -0.31039-0.73841 YES!!! 3log(5sinx) - 4 + 2x -0.127971.973028 Yes

15. There are some times when the sign change occurs but there is no root. 1) The graphs must be continuous (no breaks) in the region. E.g. Between x=-1 and x=1 Limitations

16. There are some times when a root exists but there is no sign change… 2) There may be more than one root in the region E.gBetween x=0 and x=1 Limitations

17. There are some times when a root exists but there is no sign change… 3) There may be a repeated root in the region E.gBetween x=1 and x=2 Limitations

19. Sketch y=sinx and y= 1 – x on the same axes. Using the sketch explain why there is just one root for the equation sinx = 1 - x The root of the equation is  show that  lies between 0 and 1. Final Check

20. Show that f(x)=0 has a solution in the interval 0<x<1 Show that the larger root, p, is such that 4<p<5 Sign Change Rule

21. Rearranging to make x the subject