(A-level only) Flashcards","topicId":"ceb39db9-5bd3-44ff-abf3-b20a1f6c769b","topicSlug":"numerical-methods-4","cards":[{"id":"07800fd3-2086-479a-8ddb-5c80841341c0","cardType":"cloze","front":"A root $\\alpha$ of $f(x)=0$ satisfies $\\text{[...]}$.","back":"A root $\\alpha$ of $f(x)=0$ satisfies $\\boxed{f(\\alpha)=0}$.","points":null,"mcqOptions":null},{"id":"25224520-371c-44e4-bd87-d2486aab4be1","cardType":"basic","front":"When does a continuous function have at least one root between $a$ and $b$?","back":"When $f(a)$ and $f(b)$ have opposite signs.","points":1,"mcqOptions":null},{"id":"0a556442-a9be-46d7-ae5e-051ff0936141","cardType":"basic","front":"What does a sign change over an interval fail to prove?","back":"It does not prove there is exactly one root.","points":1,"mcqOptions":null},{"id":"8ab9a0aa-84e1-4018-9075-a71912de7815","cardType":"basic","front":"Why might an even repeated root produce no sign change?","back":"The curve touches the axis without crossing it.","points":2,"mcqOptions":null},{"id":"910738fe-a200-4108-b7b3-425f44d587e2","cardType":"cloze","front":"A fixed-point iteration applies $\\text{[...]}$ repeatedly from $x_0$.","back":"A fixed-point iteration applies $\\boxed{x_{n+1}=g(x_n)}$ repeatedly from $x_0$.","points":null,"mcqOptions":null},{"id":"bab12fb1-4532-42da-a96a-631f3600ce30","cardType":"basic","front":"For $x_{n+1}=\\sqrt{2+x_n}$ and $x_0=1$, calculate $x_1$.","back":"$x_1=\\sqrt{3}\\approx1.732051$.","points":1,"mcqOptions":null},{"id":"55a322ff-9711-4a4f-8267-2ee873152713","cardType":"cloze","front":"If $x_{n+1}=g(x_n)$ converges to $\\alpha$, then $\\text{[...]}$.","back":"If $x_{n+1}=g(x_n)$ converges to $\\alpha$, then $\\boxed{\\alpha=g(\\alpha)}$.","points":null,"mcqOptions":null},{"id":"7f2b479f-3c86-43fb-a977-a8ae2edbfb9e","cardType":"basic","front":"Why is the relevant limit of $x_{n+1}=\\sqrt{2+x_n}$ equal to $2$, not $-1$?","back":"The recurrence produces only non-negative values.","points":2,"mcqOptions":null},{"id":"c4273267-b813-44a2-bcf8-c2f8760b8fcc","cardType":"basic","front":"What do intersections of $y=g(x)$ and $y=x$ represent?","back":"They represent fixed points of $g$.","points":1,"mcqOptions":null},{"id":"4d6a1eed-8e38-41ff-942f-9181f7c61d9a","cardType":"basic","front":"How is one step of a cobweb diagram constructed?","back":"Move vertically to $y=g(x)$, then horizontally to $y=x$.","points":2,"mcqOptions":null},{"id":"d93daafd-2cf4-4193-8d40-6d5e449094cb","cardType":"cloze","front":"Near a fixed point $\\alpha$, iteration usually converges when $\\text{[...]}$.","back":"Near a fixed point $\\alpha$, iteration usually converges when $\\boxed{|g'(\\alpha)|<1}$.","points":null,"mcqOptions":null},{"id":"b069006e-4868-4c4c-a25f-0dfe90022f54","cardType":"basic","front":"Predict the behaviour near $2$ of $x_{n+1}=\\sqrt{x_n+2}$.","back":"It should converge because $|g'(2)|=\\frac14<1$.","points":2,"mcqOptions":null},{"id":"00ef59d7-83fd-4341-9ad7-7d8fd04191cc","cardType":"basic","front":"Predict the behaviour near $2$ of $x_{n+1}=x_n^2-2$.","back":"It should diverge because $|g'(2)|=4>1$.","points":2,"mcqOptions":null},{"id":"9c7bdb7c-8a70-4df4-991d-c93bbb12ee0d","cardType":"cloze","front":"Newton-Raphson uses $\\text{[...]}$ to solve $f(x)=0$.","back":"Newton-Raphson uses $\\boxed{x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}}$ to solve $f(x)=0$.","points":null,"mcqOptions":null},{"id":"5485ad21-dbf4-49f2-a8d0-a6456e5e58db","cardType":"basic","front":"Geometrically, how does Newton-Raphson obtain $x_{n+1}$ from $x_n$?","back":"$$x_{n+1}$ is where the tangent at $(x_n,f(x_n))$ crosses the $x$-axis.","points":2,"mcqOptions":null},{"id":"50301db5-25b8-42a6-9cf9-ae4747c7b0d4","cardType":"basic","front":"How is the Newto

3.8.3 Numerical Methods (A-level only) Flashcards

3.8.3 Numerical Methods (A-level only) Flashcards