Flashcards
2022-02-07
What is the first step in using the decimal search method??
Rearranging to get 0 on one side.
How can you find a good estimate to start with for decimal search??
Draw a sketch.
What should you say when an identifying a root using decimal search??
“Gee wiz! The function $f(x)$ is continuous in the interval $[a, b]$ and there is a change of sign, so a root lies near $x$”.
How would you show that $1.398$ is the best approximation to 3 d.p. when using decimal search??
Substitute in the upper and lower bounds $1.3975$ and $1.3985$ and show there is a change of sign.
What’s a common criticism of decimal search??
- Slow
- Can misclassify asymptotes as roots
2022-02-08
How can you solve $f(x) = a$ using fixed-point iteration??
Rearrange into some $x = g(x)$ and see if repeating the expression converges.
What diagram is this??
A staircase diagram.
What diagram is this??
A cobweb diagram.
How do you draw a staircase or cobweb diagram??
Draw the rearranged function along with the line $y = x$.
What is the condition for $f(x)$ to have an attracting fixed point at $x = a$??
$$ |f'(a) < 1| $$
What is the condition for $f(x)$ to have a repelling fixed point at $x = a$??
$$ |f'(a) > 1| $$
2022-02-10
What is the Newton-Rhapson formula for $x_{n+1}$ in terms of $x_n$??
$$ x_{n+1} = x - \frac{f(x_n)}{f'(x_n)} $$
When doesn’t the Newton-Rhapson method work at $x_n$??
When $x_n$ is a stationary point.
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Metadata
date: 2022-02-07 15:56
tags:
- '@?maths'
- '@?year-2'
- '@?public'
- '@?school'
title: Maths - Numerical Methods
