Flashcards · Maths · Functions, Sequences and Rates of Change
Iteration
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What is iteration?
Repeating a calculation, using each answer as the next input.
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What does \(x_{n+1} = f(x_n)\) mean?
The next value is found by putting the current value into \(f\).
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What is \(x_0\)?
The starting value.
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What is the limit of an iteration?
The value the terms get closer and closer to.
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What is true at the limit?
\(x_{n+1} = x_n\), so \(a = f(a)\).
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How do you find the limit exactly?
Solve \(a = f(a)\).
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What does a change of sign tell you?
A root lies between the two values, if the function is continuous.
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What is the change of sign test for \(x^3 + x - 3\) between 1 and 2?
\(f(1) = -1\) and \(f(2) = 7\), so there is a root.
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Why is the root of \(x^3 + x - 3\) rounded to 1.2?
\(f(1.25) > 0\) and \(f(1.2) < 0\), so it is below 1.25.
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How do you get \(x = 3 - \dfrac{2}{x}\) from \(x^2 - 3x + 2 = 0\)?
Divide by \(x\) and rearrange.
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What should you do with the answer from each step?
Use it as the next input.
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What is \(x_1\) if \(x_{n+1} = 3 - \dfrac{2}{x_n}\) and \(x_0 = 4\)?
2.5.
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Which roots does the iteration \(x_{n+1} = 3 - \dfrac{2}{x_n}\) find?
The roots of \(x^2 - 3x + 2 = 0\), 1 or 2.
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Why do you need a continuous function for a change of sign?
There must be no gaps where the graph jumps over zero.
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Which tier is iteration?
Higher tier on all boards.
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What should you show for each iteration?
Each value, so that the working is clear.