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NSWMaths Extension 1Quick questions

Calculus (ME-C1, C2, C3)

Quick questions on Exponential growth and decay: dNdt=kN\frac{dN}{dt} = k N, N=N0ektN = N_0 e^{kt}, doubling and half-life

14short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the model?
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If a quantity N(t)N(t) grows or decays at a rate proportional to itself,
What is general solution?
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$ln⁡∣N∣=kt+C  ⟹  N=N0ekt,\ln |N| = k t + C \implies N = N_0 e^{k t},$
What is finding kk from data?
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Given N(t1)=N1N(t_1) = N_1 and N(0)=N0N(0) = N_0:
What is doubling time and half-life?
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Doubling time for growth: solve 2N0=N0ekTd2 N_0 = N_0 e^{k T_d} to get Td=ln⁡2kT_d = \frac{\ln 2}{k}.
What is continuous compound interest?
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A bank account earning interest at a continuous rate rr satisfies dAdt=rA\frac{dA}{dt} = r A, so A(t)=A0ertA(t) = A_0 e^{r t}. This is the same model.
What is populations with capped growth (logistic)?
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For populations, the model dNdt=kN(M−N)\frac{dN}{dt} = k N (M - N) caps growth at carrying capacity MM. Extension 1 mostly stays with unrestricted growth; logistic growth appears in Extension 2 and in some applications.
What is radioactive decay?
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Carbon-14 has a half-life of 57305730 years. Find kk.
What is mixed growth-decay?
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A drug is absorbed into the bloodstream at a constant rate, and the concentration CC decays exponentially. After 44 hours the concentration is 30%30\% of initial. Find when it falls below 5%5\%.
What is continuous compounding?
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5000investedat5000 invested at 6\%continuous.Whatisthevalueafter continuous. What is the value after 10$ years?
What is working backwards?
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A culture has 400400 after 11 hour and 16001600 after 33 hours. Find N0N_0 and the doubling time.
What is confusing N0N_0 with NN ?
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N0N_0 is the value at t=0t = 0. If the question gives you values at t=2t = 2 and t=5t = 5, neither is N0N_0 unless extracted by working backwards.
What is half-life formula sign?
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Th=−ln⁡2kT_h = -\frac{\ln 2}{k} when k<0k < 0 gives a positive half-life. Using ln⁡2k\frac{\ln 2}{k} directly would give a negative answer.
What is mixing continuous and periodic rates?
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A bank account with 5%5\% annual interest compounded once is not the same as 5%5\% continuous. The continuous-compounding rate is approximately the annual percentage rate for small percentages.
What is approximating too early?
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Keep kk symbolic (k=ln⁡0.810k = \frac{\ln 0.8}{10}) until the final substitution. Truncating to decimals can introduce visible error. :::

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