:"2559c68c-bc63-4bbb-bd69-44526032a79f","topicSlug":"differentiation-2","cards":[{"id":"c01d3578-32c4-42bd-bfb3-e211c3d97ed6","cardType":"cloze","front":"Using two points, the gradient of a straight line is $\\text{[...]}$.","back":"Using two points, the gradient of a straight line is $\\boxed{\\frac{y_2-y_1}{x_2-x_1}}$.","points":null,"mcqOptions":null},{"id":"b6eaf727-4b7a-4868-92e9-276f92d018eb","cardType":"basic","front":"What rate of change does a chord gradient represent?","back":"The average rate of change between two points.","points":1,"mcqOptions":null},{"id":"59f13126-db56-4fb8-b79f-67846f40967f","cardType":"basic","front":"What rate of change does a tangent gradient represent?","back":"The instantaneous rate of change at a point.","points":1,"mcqOptions":null},{"id":"54116c59-8cdd-433e-a9c6-78b6c399624d","cardType":"basic","front":"What does $f'(a)$ give for the curve $y=f(x)$?","back":"The tangent gradient at $x=a$.","points":1,"mcqOptions":null},{"id":"b748a643-9546-49c9-9fa7-96ff3d60547d","cardType":"cloze","front":"The difference quotient for $f$ is $\\text{[...]}$.","back":"The difference quotient for $f$ is $\\boxed{\\frac{f(x+h)-f(x)}{h}}$.","points":null,"mcqOptions":null},{"id":"60099d95-8d19-4a5a-bde0-c23ad28e54e3","cardType":"cloze","front":"The derivative from first principles is $\\text{[...]}$.","back":"The derivative from first principles is $\\boxed{f'(x)=\\lim_{h\\to 0}\\frac{f(x+h)-f(x)}{h}}$.","points":null,"mcqOptions":null},{"id":"a9a5a07d-cc7a-4a28-bd65-303b964a2dcd","cardType":"basic","front":"Why can $h=0$ not be substituted immediately into the difference quotient?","back":"It would cause division by zero.","points":2,"mcqOptions":null},{"id":"d7e17f72-2dba-40ea-9f7a-1a54315ae6ba","cardType":"basic","front":"As $h\\to0$, what happens geometrically to the chord through $P$ and $Q$?","back":"The chord approaches the tangent at $P$.","points":2,"mcqOptions":null},{"id":"55e6804e-a9e9-4a0d-97c5-60f9acd968b0","cardType":"basic","front":"Using first principles, what is the derivative of $f(x)=x^2$?","back":"$f'(x)=2x$.","points":1,"mcqOptions":null},{"id":"e8cf1303-454a-49b6-a2ee-705cd5ef437e","cardType":"basic","front":"For $y=x^2$, what is the tangent gradient at $x=3$?","back":"$6$.","points":1,"mcqOptions":null},{"id":"129878e7-df2d-431f-9028-a0648f579b7b","cardType":"cloze","front":"The point-gradient equation of a line is $\\text{[...]}$.","back":"The point-gradient equation of a line is $\\boxed{y-y_1=m(x-x_1)}$.","points":null,"mcqOptions":null},{"id":"09864cd4-7b99-40e3-8705-843e273929e9","cardType":"basic","front":"Find the tangent to $y=x^2$ at $x=3$.","back":"$y=6x-9$.","points":2,"mcqOptions":null},{"id":"1b7b80c6-576b-4d98-bd8e-51b04a1a3300","cardType":"basic","front":"What are the units of $\\frac{dy}{dx}$?","back":"Units of $y$ divided by units of $x$.","points":1,"mcqOptions":null},{"id":"86a2b8e7-fe0a-4d94-b284-f7dbcd779a08","cardType":"basic","front":"If $s=t^2+4t$, what is the instantaneous velocity at $t=3$?","back":"$10\\text{ m s}^{-1}$.","points":2,"mcqOptions":null},{"id":"58f1e20a-966d-4134-8b76-18b78603d973","cardType":"basic","front":"Where does the graph of $f'(x)$ lie when $f(x)$ is decreasing?","back":"Below the $x$-axis, since $f'(x)<0$.","points":1,"mcqOptions":null},{"id":"1971772d-3e4f-4d6f-b512-2d9ebdae04cb","cardType":"basic","front":"What happens on the derivative graph where $f$ has a horizontal tangent?","back":"It meets the $x$-axis, since $f'(x)=0$.","points":1,"mcqOptions":null},{"id":"81fad600-b28e-4e8a-9dbb-afdee7652863","cardType":"cloze","front":"The second derivative is written $\\text{[...]}$.","back":"The second derivative is written $\\boxed{f''(x)=\\frac{d^2y}{dx^2}}$.","points":null,"mcqOptions":null},{"id":"9df5f0bf-64c6-423f-b8ee-6fe4fe9eec6c","cardType":"basic","front":"What does $f''(x)>0$ show about the gradient of $f$?","back"

1.7.1 Differentiation Flashcards

1.7.1 Differentiation Flashcards