HOME THEORY MEDIA MISSION Now that you are juggling formulas like a pro, let me ask you this: How could you combine the growth rate k = n / t and the growth formula $N_t = 2^n * N_0$ from before? a) k = $log_2(N_0 + N_t)$ / t b) k = $log_{10}(N_t - N_0)$ / t c) k = $log_2(N_t / N_0)$ / t d) k = $log_{10}(N_0 * N_t)$ / t VIEW THEORY
Added by David H.
Close
Step 1
Growth rate: $k = n / t$ 2. Growth formula: $N_t = 2^n * N_0$ The goal is to combine these two formulas to express k in terms of $N_t$, $N_0$, and t. Show more…
Show all steps
Your feedback will help us improve your experience
Madhur L and 82 other Intro Stats / AP Statistics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
thoughtful
Madhur L.
(a) Let $\mathcal{N}(t)=\mathcal{N}_{0} e^{k t} .$ Show that $[\mathcal{N}(t+1)-\mathcal{N}(t)] / \mathcal{N}(t)=e^{k}-1 .$ (This is actually done in detail in the text. So, ideally, you should look back only if you get stuck or want to check your answer.) (b) Assume as given the following approximation, which was introduced in Exercise 26 of Section 5.2 $e^{x} \approx x+1 \quad$ provided $x$ is close to zero Use this approximation to explain why $e^{k}-1 \approx k$ provided that $k$ is close to zero. Remark: Combining this result with that in part (a), we conclude that the relative growth rate for the function $\mathcal{N}(t)=\mathcal{N}_{0} e^{k t}$ is approximately equal to the growth constant $k .$ As explained in the text, this is one of the reasons why in applications we've not distinguished between the relative growth rate and the decay constant $k$
Exponential and Logarithmic Functions
Exponential Growth and Decay
N(t) = K / (1 + (K - 1)e^(-rt)) Where r and K are positive constants representing the population growth rate and carrying capacity, respectively. (A) (5) Plot N(t) as a function of time t, t >= 0, for K = 1000, r = 0.02, and K = 2000, r = 0.02. What happens to the population size as time increases? (B) (5) Find the rate of change of the population per unit of time, dN/dt (do not substitute any values for K or r). (C) (10) Note that dN/dt = rN(1 - N/K) (you can check this by substituting N(t) on the right hand side of this equation and simplify until you obtain what you found in part (B)). Plot dN/dt and the per capita rate of growth 1/N * dN/dt, 0 <= N <= 1000, as a function of N, in the same coordinate plane for K = 1000, r = 0.02. What happens to the per capita rate of growth as the population increase?
Adi S.
Recommended Textbooks
Elementary Statistics a Step by Step Approach
The Practice of Statistics for AP
Introductory Statistics
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD