itle":"1.10.1 Locate roots by change of sign (A-level only) Flashcards","topicId":"99086623-787c-4fea-b934-8cd4adad2f52","topicSlug":"locate-roots-by-change-of-sign","cards":[{"id":"5dc23de5-79c0-4eff-98b3-708d53e1f54d","cardType":"basic","front":"If $f(\\alpha)=0$, where does the graph of $y=f(x)$ meet the $x$-axis?","back":"At $x=\\alpha$.","points":1,"mcqOptions":null},{"id":"9487967d-df64-4c8a-a166-62be3a71ecdf","cardType":"basic","front":"What does $f(x)>0$ show about the graph of $y=f(x)$?","back":"It lies above the $x$-axis.","points":1,"mcqOptions":null},{"id":"3e0a1bde-db14-4374-b092-ecdac18e1336","cardType":"basic","front":"What does $f(x)<0$ show about the graph of $y=f(x)$?","back":"It lies below the $x$-axis.","points":1,"mcqOptions":null},{"id":"963fea55-2326-4ea3-87b5-50520bf3fd65","cardType":"cloze","front":"Opposite endpoint signs are expressed by $\\text{[...]}$.","back":"Opposite endpoint signs are expressed by $\\boxed{f(a)f(b)<0}$.","points":null,"mcqOptions":null},{"id":"f27a0e60-4c00-4429-a010-92cd55a57c9a","cardType":"basic","front":"Why does multiplying endpoint values provide a concise sign-change test?","back":"A positive value times a negative value is negative.","points":2,"mcqOptions":null},{"id":"b37d34c6-7ae8-42e5-bbd3-9f6212964272","cardType":"basic","front":"What property must $f$ have throughout $[a,b]$ for a sign change to guarantee a root?","back":"$$f$ must be continuous on $[a,b]$.","points":1,"mcqOptions":null},{"id":"3f4b95dd-83e4-41a3-8778-46b2d3e43754","cardType":"cloze","front":"If $f$ is continuous on $[a,b]$ and $f(a)f(b)<0$, there is [ ] in $aat least one root in $aevery intermediate value, including zero.","points":2,"mcqOptions":null},{"id":"03fa46a0-d7de-419d-9449-d62802e75b07","cardType":"basic","front":"What does the change-of-sign method determine about a root?","back":"An interval containing the root, not its exact value.","points":1,"mcqOptions":null},{"id":"d86fddba-c4a7-46d8-9da6-5fb3bbd91efe","cardType":"basic","front":"For $f(x)=x^2-5$, what conclusion follows from $f(2)=-1$ and $f(3)=4$?","back":"There is at least one root in $2opposite function signs.","points":1,"mcqOptions":null},{"id":"09439d5f-aac5-40e3-8b42-aeced3e28d82","cardType":"cloze","front":"The width of an interval $[a,b]$ is $\\text{[...]}$.","back":"The width of an interval $[a,b]$ is $\\boxed{b-a}$.","points":null,"mcqOptions":null},{"id":"3d32d216-4cdc-4c1a-8763-85ee9ee025f3","cardType":"basic","front":"For $f(x)=x^3+x-3$, $f(1.21)<0$ and $f(1.22)>0$. Where is a root located?","back":"$$1.21incorrect sign.","points":2,"mcqOptions":null},{"id":"51fae5f6-c605-40ea-b5fb-3c75c50a2cda","cardType":"basic","front":"Can $f(a)f(b)<0$ prove that a continuous function has exactly one root in $(a,b)$?","back":"No - it proves at least one root.","points":1,"mcqOptions":null},{"id":"189425cc-a5bf-4a63-91ed-d2a8f5641b3d","cardType":"basic","front":"What extra property can establish that a root in an interval is unique?","back":"The function is strictly increasing or strictly decreasing throughout.","points":1,"mcqOpt