s":[{"id":"7f51e964-a9e7-4ed3-91d1-272427bacbca","cardType":"basic","front":"If population $P$ changes with time $t$, which derivative represents its growth rate?","back":"$\\frac{dP}{dt}$","points":1,"mcqOptions":null},{"id":"42a2f32e-ce5f-4579-8a54-93fe1dbd1af2","cardType":"cloze","front":"$$\\frac{dy}{dx}$ is the rate of change of [ ].","back":"$$\\frac{dy}{dx}$ is the rate of change of $y$ with respect to $x$.","points":null,"mcqOptions":null},{"id":"c3256c61-f3d5-4a21-afc2-986d6c73a63b","cardType":"basic","front":"A quantity $y$ increases at constant rate $c$. Construct its differential equation.","back":"$\\frac{dy}{dt}=c$","points":1,"mcqOptions":null},{"id":"6c0e744e-9a07-4882-9595-dd3690c2edcb","cardType":"cloze","front":"If $y$ decreases at constant rate $c>0$, then $\\text{[...]}$.","back":"If $y$ decreases at constant rate $c>0$, then $\\boxed{\\frac{dy}{dt}=-c}$.","points":null,"mcqOptions":null},{"id":"cb7f5daf-66f1-4b66-aa8a-f36bf2e70f10","cardType":"basic","front":"The rate of change of $y$ is proportional to $y$. Construct the model.","back":"$\\frac{dy}{dt}=ky$, where $k$ is constant.","points":1,"mcqOptions":null},{"id":"f63bec5b-b6c4-43e6-9abc-9ff7c656d5c9","cardType":"basic","front":"Why is $\\frac{dy}{dt}=y$ generally an incorrect translation of “the rate is proportional to $y$”?","back":"It omits the constant of proportionality $k$.","points":2,"mcqOptions":null},{"id":"8431f8c8-1196-45e0-86d7-9f7203b7bb25","cardType":"basic","front":"A quantity $N$ decreases at a rate proportional to $N^2$. Construct the model for $k>0$.","back":"$\\frac{dN}{dt}=-kN^2$","points":1,"mcqOptions":null},{"id":"834f2cbb-f5f0-4bb5-b7ec-c1ad44d541a6","cardType":"basic","front":"What derivative gives the gradient of the curve $y=f(x)$?","back":"$\\frac{dy}{dx}$","points":1,"mcqOptions":null},{"id":"080ea3a0-d2fc-40f6-90ab-86ee75f7b2e4","cardType":"cloze","front":"The order of a differential equation is determined by its [ ].","back":"The order of a differential equation is determined by its highest derivative.","points":null,"mcqOptions":null},{"id":"9a4bfbae-c07b-420f-b8cc-25a78df3ace5","cardType":"basic","front":"Which derivative describes the rate of change of a curve's gradient?","back":"$\\frac{d^2y}{dx^2}$","points":1,"mcqOptions":null},{"id":"aa7aab5e-9d59-4bd5-a9f2-fe8158d62413","cardType":"basic","front":"Express velocity $v$ in terms of displacement $s$ and time $t$.","back":"$v=\\frac{ds}{dt}$","points":1,"mcqOptions":null},{"id":"b91d7f92-ffd5-41b0-b7f2-38f570fe15f8","cardType":"basic","front":"Express acceleration $a$ directly in terms of displacement $s$.","back":"$a=\\frac{d^2s}{dt^2}$","points":1,"mcqOptions":null},{"id":"2c52b0a9-68d5-4b82-b3bc-2529d26c584f","cardType":"basic","front":"Acceleration is proportional to displacement and acts towards the origin. Construct the model for $k>0$.","back":"$\\frac{d^2s}{dt^2}=-ks$","points":1,"mcqOptions":null},{"id":"04bfeba5-056e-4054-aadf-7b034a622f07","cardType":"basic","front":"Why does $\\frac{d^2s}{dt^2}=-ks$ model acceleration towards the origin?","back":"Acceleration has the opposite sign to displacement.","points":2,"mcqOptions":null},{"id":"4ea1a2eb-dc00-48f9-92a2-00848905619a","cardType":"basic","front":"A population's growth rate is proportional to both $P$ and remaining capacity $M-P$. Construct the model.","back":"$\\frac{dP}{dt}=kP(M-P)$, where $k>0$.","points":1,"mcqOptions":null},{"id":"89bffc6f-0bed-41ec-9b40-d203115f6ce0","cardType":"cloze","front":"For $\\frac{dP}{dt}=kP(5000-P)$, growth is zero at [ ].","back":"For $\\frac{dP}{dt}=

1.7.20 Constructing differential equations (A-level only) Flashcards

1.7.20 Constructing differential equations (A-level only) Flashcards