Standard +0.3 This is a straightforward application of fixed point iteration to solve a transcendental equation. Students need to rearrange the equation into iterative form (e.g., x = ln(1 + e^x)/2) and apply the method until convergence. While it involves exponentials and logarithms, it's a standard textbook exercise requiring only routine application of the iteration technique with no novel insight or complex multi-step reasoning.
State or imply \(e^x + 1 = e^{2x}\), or \(1 + e^{-x} = e^x\), or equivalent
B1
Solve this equation as a quadratic in \(u = e^x\), or in \(e^{-x}\), obtaining one or two roots
M1
Obtain root \(\frac{1}{2}(1 + \sqrt{5})\), or decimal in \([1.61, 1.62]\)
A1
Use correct method for finding \(x\) from a positive root
M1
Obtain \(x = 0.481\) and no other answer
A1
[For the solution 0.481 with no working, award B3 (for 0.48 give B2). However a suitable statement can earn the first B1 in addition, giving a maximum of 4/5 (or 3/5) in such cases.]
OR
Answer
Marks
Guidance
State an appropriate iterative formula, e.g. \(x_{n+1} = \frac{1}{2}\ln(1 + e^{x_n})\) or \(x_{n+1} = \frac{1}{2}\ln(e^{x_n} + e^{2x_n})\)
B1
Use the iterative formula correctly at least once
M1
Obtain final answer 0.481
A1
Show sufficient iterations to justify its accuracy to 3 d.p., or show there is a sign change in the value of a relevant function in the interval \((0.4805, 0.4815)\)
A1
Show that the equation has no other root
A1
[5 marks]
State or imply $e^x + 1 = e^{2x}$, or $1 + e^{-x} = e^x$, or equivalent | B1 |
Solve this equation as a quadratic in $u = e^x$, or in $e^{-x}$, obtaining one or two roots | M1 |
Obtain root $\frac{1}{2}(1 + \sqrt{5})$, or decimal in $[1.61, 1.62]$ | A1 |
Use correct method for finding $x$ from a positive root | M1 |
Obtain $x = 0.481$ and no other answer | A1 | [For the solution 0.481 with no working, award B3 (for 0.48 give B2). However a suitable statement can earn the first B1 in addition, giving a maximum of 4/5 (or 3/5) in such cases.] | [5 marks]
**OR**
State an appropriate iterative formula, e.g. $x_{n+1} = \frac{1}{2}\ln(1 + e^{x_n})$ or $x_{n+1} = \frac{1}{2}\ln(e^{x_n} + e^{2x_n})$ | B1 |
Use the iterative formula correctly at least once | M1 |
Obtain final answer 0.481 | A1 |
Show sufficient iterations to justify its accuracy to 3 d.p., or show there is a sign change in the value of a relevant function in the interval $(0.4805, 0.4815)$ | A1 |
Show that the equation has no other root | A1 | [5 marks]
---