4 Numerical methods in context (A-level only) Flashcards","topicId":"d8cee6b5-29aa-425f-a908-17ea4841489b","topicSlug":"numerical-methods-in-context","cards":[{"id":"43314f7e-e0d8-4164-9123-a827e606d10a","cardType":"basic","front":"If a context requires $A(x)=B(x)$, what function should be used for root-finding?","back":"Use $f(x)=A(x)-B(x)$ and solve $f(x)=0$.","points":1,"mcqOptions":null},{"id":"d2a1c0f0-e0e0-4982-8bd6-c08a28425bb8","cardType":"basic","front":"What does a root of $f(x)=0$ represent on the graph of $y=f(x)$?","back":"An $x$-coordinate where the graph crosses or touches the $x$-axis.","points":1,"mcqOptions":null},{"id":"2ce6a435-6d2a-4975-ac4e-26edcffecd81","cardType":"cloze","front":"For revenue $R(x)$ and cost $C(x)$, the break-even equation is $\\text{[...]}$.","back":"For revenue $R(x)$ and cost $C(x)$, the break-even equation is $\\boxed{R(x)-C(x)=0}$.","points":null,"mcqOptions":null},{"id":"754f5d5a-7c9e-4fd4-b378-ba9264668238","cardType":"basic","front":"Why might a negative mathematical root be rejected in a time model?","back":"The context requires $t\\geq0$.","points":2,"mcqOptions":null},{"id":"c7473a5a-9510-4d03-8599-b64592cce76e","cardType":"cloze","front":"A valid probability must satisfy $\\text{[...]}$.","back":"A valid probability must satisfy $\\boxed{0\\leq p\\leq1}$.","points":null,"mcqOptions":null},{"id":"0d0f867e-9694-485b-9675-d25e4df1966a","cardType":"basic","front":"What restriction usually applies when $x$ counts complete items?","back":"$$x$ must usually be a whole number.","points":1,"mcqOptions":null},{"id":"e0b77fc9-893a-4ee0-8658-9e22bac087f0","cardType":"cloze","front":"For continuous $f$, opposite signs at $a$ and $b$ are expressed as $\\text{[...]}$.","back":"For continuous $f$, opposite signs at $a$ and $b$ are expressed as $\\boxed{f(a)f(b)<0}$.","points":null,"mcqOptions":null},{"id":"0f6e8728-df2c-49b6-a023-e680e62aa66f","cardType":"basic","front":"What does $f(a)f(b)<0$ prove when $f$ is continuous on $[a,b]$?","back":"There is at least one crossing root between $a$ and $b$.","points":1,"mcqOptions":null},{"id":"453092f1-e1be-4823-af47-412ed3088169","cardType":"basic","front":"Why does a change of sign not prove that an interval contains exactly one root?","back":"The function may cross the axis more than once.","points":2,"mcqOptions":null},{"id":"d101f4bc-3c51-4110-bf6b-320274fbeeb8","cardType":"basic","front":"Why is continuity required when using a change of sign to locate a root?","back":"Without continuity, a jump could change the sign without crossing zero.","points":2,"mcqOptions":null},{"id":"2dddb129-36d1-4fc6-a8b3-b75c614ad3ef","cardType":"basic","front":"Why is bisection a reliable root-finding method?","back":"Each step keeps the root trapped in a smaller interval.","points":2,"mcqOptions":null},{"id":"28121948-61b9-4ca0-906e-8312599ebfee","cardType":"cloze","front":"Fixed-point iteration uses $\\text{[...]}$ after rearranging the equation as $x=g(x)$.","back":"Fixed-point iteration uses $\\boxed{x_{n+1}=g(x_n)}$ after rearranging the equation as $x=g(x)$.","points":null,"mcqOptions":null},{"id":"dc91bc73-7d06-42d6-80b4-711d2c47c25b","cardType":"basic","front":"What must happen for a fixed-point iteration to converge?","back":"Its terms must approach a fixed value.","points":1,"mcqOptions":null},{"id":"92d86e9c-1a1f-41e7-9905-30b4ce0df742","cardType":"basic","front":"Why must a fixed-point rearrangement be chosen carefully?","back":"Different rearrangements of the same equation may converge or diverge.","points":2,"mcqOptions":null},{"id":"8932dbfd-3994-48ee-931c-e496c5085ea2","cardType":"cloze","front":"The Newton-Raphson iteration is $$\\text{[...]}$$.","back":"The Newton-Raphson iteration is $$\\boxed{x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}}$$.","points":null,"mcqOptions":null},{"id":"65be7e22-ff7c-417a-8613-d293bc2f60ec","cardType":"basic","front":"When can Newton-Raphson fail

1.12.4 Numerical methods in context (A-level only) Flashcards

1.12.4 Numerical methods in context (A-level only) Flashcards