9709 · 3.6
Numerical solution of equations flashcards
Revision flashcards for Cambridge 9709 Numerical solution of equations (syllabus 3.6). Flip, recall, then mark a real past-paper question.
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What are the two conditions for the change-of-sign rule to confirm a root of $f(x)=0$ in the interval $[a, b]$?
1. The function $f(x)$ must be continuous on the interval $[a, b]$. 2. $f(a)$ and $f(b)$ must have opposite signs, i.e., $f(a) \times f(b) < 0$.
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What is an iterative formula?
A formula of the form $x_{n+1} = g(x_n)$ that generates a sequence of approximations to a root, where the next term is calculated using the previous term.
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How do you show that a root lies between two values, say $x=1$ and $x=2$?
Define $f(x)$ such that the equation is $f(x)=0$. Calculate $f(1)$ and $f(2)$. If one is positive and the other is negative, and $f(x)$ is continuous, a root exists between 1 and 2.
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What is the condition for an iterative process $x_{n+1} = g(x_n)$ to converge to a root $\alpha$?
The condition for convergence is that $|g'(\alpha)| < 1$. In practice, we check that $|g'(x)| < 1$ for all $x$ in an interval containing the root.
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What does a 'cobweb' diagram indicate about convergence?
A cobweb diagram, which spirals inwards, occurs when $-1 < g'(x) < 0$ near the root. The sequence of approximations oscillates about the root as it converges.
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What does a 'staircase' diagram indicate about convergence?
A staircase diagram, which steps towards the root from one side, occurs when $0 < g'(x) < 1$ near the root. The sequence of approximations approaches the root without oscillating.
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What is the Newton-Raphson formula?
The formula is $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$. It uses the tangent to the curve at $x_n$ to find the next approximation.
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When might the Newton-Raphson method fail?
It can fail if the starting value $x_0$ is near a turning point (where $f'(x_n) \approx 0$), or if the starting value is too far from the actual root, potentially leading to a different root or diverging.
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How do you determine the value of a root to '3 decimal places'?
Find an interval $[a, b]$ of width $0.001$ (e.g., $[1.234, 1.235]$) that contains the root. Then, state the root is $1.234$ or $1.235$ by checking which value it is closer to, or more simply, find two values like $1.2345$ and $1.2355$ that round to the same 3 d.p. value and show a sign change between them.
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Common Exam Trap: What must you do when a question involves trigonometric functions?
ALWAYS ensure your calculator is in RADIANS mode, unless the question explicitly specifies degrees. Numerical methods in calculus are based on radian measure.
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How do you show your working for an iterative process in an exam?
Write down the iterative formula, the starting value $x_0$, and then at least two subsequent values ($x_1, x_2, ...$) to at least 2 more decimal places than the final required accuracy. Then state the final answer.