If we now go up from the point hit the curve and then construct another tangent (as in the diagram b, below) then, a similar argument, gives a better approximate root at (nearer than Therefore we would c b). . get = c b – . So if we continued this process we would get = and generally + 1 = d c xn xn – – Sometimes the process fails for some curves or starting points. What happens to the tangent if we try to the process here? An example of the type question which may be seen: f (x) = + 8x 19. x3 – Obtain an approximation to the real root of f(x) = 0 by performing two applications of the procedure to f(x), using = 2 as the first approximation. x Give your answer to 3 decimal places. OPPORTUNITIES FOR REASONING/PROBLEM SOLVING Try different methods to find the roots of the same function. Which is the most efficient method or leads to the more accurate approximation? Consider, for example, iteration vs Newton–Raphson. COMMON MISCONCEPTIONS/EXAMINER REPORT QUOTES Marks are often lost for sign errors and other numerical slips. Students must show full working leading to the correct answer for full marks. Giving a correct answer either without working or following wrong working will result in zero marks. NOTES Graph drawing packages are an essential way to at the curve and the potential position of the roots ‘look’ depending on the first approximation of the root. There will be a rich source of questions from the legacy FP1 papers as this topic was part of that specification. Functions used will be consistent with the differentiation unit, e.g. e2x, etc. Teaching time 9d. Problem solving (9.4) 2 hours OBJECTIVES By the end of the sub-unit, students should: be able to use numerical methods to solve problems in context. TEACHING POINTS
If we now go up from the point b, hit the curve and then construct another tangent (as in the diagram below) then, a similar argument, gives a better approximate root at c (nearer than b). Therefore we would get c = b – . So if we continued this process we would get d = c – and generally xn + 1 = xn – . Sometimes the process fails for some curves or starting points. What happens to the tangent if we try to the process here? An example of the type question which may be seen: f (x) = x3 + 8x – 19. Obtain an approximation to the real root of f(x) = 0 by performing two applications of the procedure to f(x), using x = 2 as the first approximation. Give your answer to 3 decimal places. OPPORTUNITIES FOR REASONING/PROBLEM SOLVING Try different methods to find the roots of the same function. Which is the most efficient method or leads to the more accurate approximation? Consider, for example, iteration vs Newton–Raphson. COMMON MISCONCEPTIONS/EXAMINER REPORT QUOTES Marks are often lost for sign errors and other numerical slips. Students must show full working leading to the correct answer for full marks. Giving a correct answer either without working or following wrong working will result in zero marks. NOTES Graph drawing packages are an essential way to ‘look’ at the curve and the potential position of the roots depending on the first approximation of the root. There will be a rich source of questions from the legacy FP1 papers as this topic was part of that specification. Functions used will be consistent with the differentiation unit, e.g. e2x, etc. 9d. Problem solving (9.4) Teaching time 2 hours OBJECTIVES By the end of the sub-unit, students should: be able to use numerical methods to solve problems in context. TEACHING POINTS
A_level_Maths_2_years_SoW.docx