.9 Modelling using exponential functions Flashcards","topicId":"7ded51f5-4bbe-4122-8189-28f7dab4e43f","topicSlug":"modelling-using-exponential-functions","cards":[{"id":"c0d9e0f8-2704-4879-a15f-1b26510b8d62","cardType":"basic","front":"What pattern over equal time intervals suggests an exponential model?","back":"The same proportional change each interval.","points":1,"mcqOptions":null},{"id":"6fbe5f61-e580-4c30-a6f2-c0a288f828d3","cardType":"basic","front":"What pattern in values at equally spaced times indicates exponential change?","back":"A constant ratio between consecutive values.","points":1,"mcqOptions":null},{"id":"d261751c-29da-4673-951c-d0e308fdbab5","cardType":"basic","front":"What pattern in values at equally spaced times indicates linear change?","back":"A constant difference between consecutive values.","points":1,"mcqOptions":null},{"id":"09ada9fe-dab2-412d-925d-02ac38dfcd6d","cardType":"basic","front":"What multiplier represents growth at decimal rate $r$ per interval?","back":"The multiplier is $1+r$.","points":1,"mcqOptions":null},{"id":"1c9633c5-11c1-4061-b5ce-c50738b7e4a9","cardType":"basic","front":"What multiplier represents decay at decimal rate $r$ per interval?","back":"The multiplier is $1-r$.","points":1,"mcqOptions":null},{"id":"e360a736-b597-4132-8ff0-1e251c3a1f3e","cardType":"basic","front":"What does $b>1$ imply in $P=P_0b^t$?","back":"It represents exponential growth.","points":1,"mcqOptions":null},{"id":"ea212a3d-1bb2-449c-908f-8c30f968d804","cardType":"basic","front":"What does $0<b<1$ imply in $P=P_0b^t$?","back":"It represents exponential decay.","points":1,"mcqOptions":null},{"id":"fa45df32-f5d9-4653-903e-63f8ad0ff48b","cardType":"basic","front":"Why does repeatedly subtracting a percentage of the initial value not model normal percentage decay?","back":"The percentage should apply to the current value each time.","points":2,"mcqOptions":null},{"id":"6e7a3b11-edb1-4d19-9817-ae4629017fd4","cardType":"basic","front":"A culture starts with 500 bacteria and grows by 12% hourly. What is its model?","back":"$N=500(1.12)^t$","points":2,"mcqOptions":null},{"id":"60dcb2cc-6224-4802-8f89-fa05a0133101","cardType":"basic","front":"A £24,000 machine loses 18% annually. What value does the model predict after 5 years?","back":"$24000(0.82)^5\\approx£8{,}898$","points":2,"mcqOptions":null},{"id":"c2391bec-b0fe-4468-9e57-4d858c7d3546","cardType":"basic","front":"Convert $P=700(1.06)^t$ into the form $P=700e^{kt}$.","back":"$P=700e^{0.05827t}$, since $k=\\ln(1.06)$.","points":2,"mcqOptions":null},{"id":"606444e5-e252-4508-8589-ea276f142419","cardType":"basic","front":"Why does the numerical value of $k$ depend on the time unit?","back":"$$k$ measures the continuous proportional change per unit time.","points":2,"mcqOptions":null},{"id":"6ef1292b-38bf-4247-b927-321807f999ed","cardType":"basic","front":"What type of prediction is made within the observed data range?","back":"Interpolation","points":1,"mcqOptions":null},{"id":"6442bca0-2a24-442f-a4af-bc7dfce4546a","cardType":"basic","front":"Why is extrapolation usually less reliable than interpolation?","back":"The model's constant-rate assumption may fail outside the observed range.","points":2,"mcqOptions":null},{"id":"b5264c7b-1267-4f3e-a629-8636ea1ca673","cardType":"basic","front":"Why is long-term exponential population growth often unrealistic?","back":"Food, space or other resources become limited.","points":2,"mcqOptions":null},{"id":"ff20b52f-06d4-49da-8fdd-d411a3796a6c","cardType":"cloze","front":"In $P=P_0b^t$, $\\text{[...]}$ is the value at $t=0$, while $b$ is the multiplier for one time unit.","back":"In $P=P_0b^t$, $\\boxed{P_0}$ is the value at $t=0$, while $b$ is the multiplier for one time unit.","points":null,"mcqOptions":null},{"id":"81482865-8ec3-454f-9122-f7f0ccad5065","ca

1.6.9 Modelling using exponential functions Flashcards

1.6.9 Modelling using exponential functions Flashcards