[Jan 2010] 2. f(x) = + 2x2 3x 11 x3 − − (a) Show that f(x) = 0 can be rearranged as (2) = , x x –2. The equation f(x) = 0 has one positive root α. The iterative formula + 1 = is used to find an approximation to . xn α (b) Taking = 0, find, to 3 decimal places, the values of , and. (3) (c) Show that = 2.057 correct to 3 decimal places. (3) α [June 2012] 2. f(x) = + 3x2 + 4x 12 x3 – (a) Show that the equation f(x) = 0 can be written as = , (3) x x –3. The equation + 3x2 + 4x 12 = 0 has a single root which is between 1 and 2. x3 – (b) Use the iteration formula + 1 = , 0, xn n with = 1 to find, to 2 decimal places, the value of , and . (3) The root of f(x) = 0 is . (c) By choosing a suitable interval, that = 1.272 to 3 decimal places. (3) [June 2011] 2. f (x) = 2 sin (x2) + 2, 0 < 2. x x − (a) Show that f (x) = 0 has a root between = 0.75 and = 0.85. (2) x x The equation f (x) = 0 can be written as = . x (b) Use the iterative formula + 1 = , = 0.8, xn to find the values of , and , giving your answers to 5 decimal places. (3) (c) Show that = 0.80157 is correct to 5 decimal places. (3) [June 2008] 7. f(x) = 3x3 2x 6. – – (a) Show that f (x) = 0 has a root, between = 1.4 and = 1.45. (2) x x α, (b) Show that the equation f (x) = 0 can be written as = , 0. x x (3) (c) Starting with = 1.43, use the iteration
[Jan 2010] 2. f(x) = x3 + 2x2 − 3x − 11 (a) Show that f(x) = 0 can be rearranged as (2) x = , x –2. The equation f(x) = 0 has one positive root α. The iterative formula xn + 1 = is used to find an approximation to α . (b) Taking = 0, find, to 3 decimal places, the values of , and. (3) (c) Show that α = 2.057 correct to 3 decimal places. (3) [June 2012] 2. f(x) = x3 + 3x2 + 4x – 12 (a) Show that the equation f(x) = 0 can be written as x = , x –3. (3) The equation x3 + 3x2 + 4x – 12 = 0 has a single root which is between 1 and 2. (b) Use the iteration formula xn + 1 = , n 0, with = 1 to find, to 2 decimal places, the value of , and . (3) The root of f(x) = 0 is . (c) By choosing a suitable interval, that = 1.272 to 3 decimal places. (3) [June 2011] 2. f (x) = 2 sin (x2) + x − 2, 0 x < 2. (a) Show that f (x) = 0 has a root between x = 0.75 and x = 0.85. (2) The equation f (x) = 0 can be written as x = . (b) Use the iterative formula xn + 1 = , = 0.8, to find the values of , and , giving your answers to 5 decimal places. (3) (c) Show that = 0.80157 is correct to 5 decimal places. (3) [June 2008] 7. f(x) = 3x3 – 2x – 6. (a) Show that f (x) = 0 has a root, α, between x = 1.4 and x = 1.45. (2) (b) Show that the equation f (x) = 0 can be written as x = , x 0. (3) (c) Starting with = 1.43, use the iteration