:"2f7f491c-6644-4716-99c9-1ad9df7d54ea","topicSlug":"differentiation-7","cards":[{"id":"69962e2f-c53a-46fb-81ba-f1e4e05549d8","cardType":"basic","front":"What does $f'(x)>0$ show about $f(x)$?","back":"$$f(x)$ is increasing.","points":1,"mcqOptions":null},{"id":"3ee53fa6-6fce-4937-b818-94e827becc1f","cardType":"basic","front":"What does $f'(x)<0$ show about $f(x)$?","back":"$$f(x)$ is decreasing.","points":1,"mcqOptions":null},{"id":"e5df060d-b99d-4aad-b770-e2d6a2b8794b","cardType":"basic","front":"What does $f'(x)=0$ show about the tangent?","back":"The tangent is horizontal.","points":1,"mcqOptions":null},{"id":"aeec2a55-f4f8-405c-9727-d2c9c837bc8c","cardType":"cloze","front":"For $y=f(x)$, the derivative $\\text{[...]}$ gives the gradient at each point.","back":"For $y=f(x)$, the derivative $\\boxed{f'(x)=\\frac{dy}{dx}}$ gives the gradient at each point.","points":null,"mcqOptions":null},{"id":"f4276e62-94b9-4a45-99f5-9c5b734332d5","cardType":"basic","front":"Which equation locates the stationary points of $f(x)$?","back":"Solve $f'(x)=0$.","points":1,"mcqOptions":null},{"id":"435bc370-754f-47b3-9cd7-03ac88a487ac","cardType":"basic","front":"Why does the sign of $f(x)$ not determine whether the function is increasing?","back":"It only shows whether the graph is above or below the $x$-axis.","points":2,"mcqOptions":null},{"id":"8040d90e-b65f-49cc-8a10-e33a75952738","cardType":"cloze","front":"If $f'(x)$ changes from positive to negative, the stationary point is a [ ].","back":"If $f'(x)$ changes from positive to negative, the stationary point is a local maximum.","points":null,"mcqOptions":null},{"id":"2219c294-5bcb-47db-8304-b7d5e19d88b5","cardType":"cloze","front":"If $f'(x)$ changes from negative to positive, the stationary point is a [ ].","back":"If $f'(x)$ changes from negative to positive, the stationary point is a local minimum.","points":null,"mcqOptions":null},{"id":"4ff0a519-1c8d-4146-a970-0e16db989d68","cardType":"basic","front":"What does a positive derivative on both sides of a stationary point show?","back":"The function is increasing on both sides, so it does not turn.","points":2,"mcqOptions":null},{"id":"a0004929-3f64-443e-81f1-d32a18b2b309","cardType":"basic","front":"Why is one test value from each interval usually sufficient?","back":"A continuous derivative keeps the same sign between its zeros or undefined points.","points":2,"mcqOptions":null},{"id":"d796da13-9660-4872-94fa-f374071125bf","cardType":"basic","front":"Which values must be used to split the domain for a sign analysis?","back":"

Zeros of $f'(x)$
Undefined values in the domain

","points":3,"mcqOptions":null},{"id":"39c95be1-6821-4403-9437-d44fb7e44640","cardType":"basic","front":"Why must an undefined value split an interval?","back":"The function has no permitted input there, even if $f'(x)$ keeps its sign.","points":2,"mcqOptions":null},{"id":"e82be6b8-c9d7-4230-8ee7-26c7bf41000e","cardType":"basic","front":"Why are open endpoints used when stating where $f'(x)>0$?","back":"At stationary endpoints, $f'(x)=0$, not $f'(x)>0$.","points":2,"mcqOptions":null},{"id":"5f607a49-585b-4f13-92cc-70eadf9e8993","cardType":"basic","front":"For $f(x)=x^2+4x+7$, where is the function increasing?","back":"$x>-2$.","points":1,"mcqOptions":null},{"id":"2399a28d-fca8-4075-9601-c7f411bdcd51","cardType":"basic","front":"For $f(x)=x^2+4x+7$, where is the function decreasing?","back":"$x<-2$.","points":1,"mcqOptions":null},{"id":"b77a8c26-fb12-4a9c-8c48-7ee2561e3793","cardType":"cloze","front":"For $f(x)=x^3-3x^2-9x+5$, the factorised derivative is $\\text{[...]}$.","back":"For $f(x)=x^3-3x^2-9x+5$,

1.7.6 Differentiation Flashcards

1.7.6 Differentiation Flashcards