Question

9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.

Accepted Answer

Step 1: Understand the graph provided. The graph shows the growth rate function $$P'(t)$$ plotted against the population $$P. $$The curve starts at $$P=0$$ with $$P'=0$$, rises to a maximum positive growth rate at some intermediate population, and then decreases back to $$P'=0 at$$ $$P=K. $$Step 2: Interpret the meaning of $$P' in $$terms of population growth. Since $$P'(t)$$ represents the rate of change of the population $$P$$ with respect to time $$t$$, positive values of $$P'$$ mean the population is increasing, and negative values would mean the population is decreasing. Here, $$P' is $$zero at $$P=0$$ and $$P=K$$, and positive in between. Step 3: Sketch the population function $$P(t)$$ starting from a positive initial population $$P(0). $$Since $$P'(t) is $$positive for $$0 \u003c P \u003c K$$, the population will increase over time until it approaches $$K. At$$ $$P=K$$, the growth rate $$P' is $$zero, so the population stabilizes and stops growing. Step 4: Consider the shape of $$P(t). $$Because $$P'(t)$$ increases then decreases as $$P$$ approaches $$K$$, the population growth rate accelerates initially, then slows down as it nears the carrying capacity $$K. $$This suggests an S-shaped or logistic growth curve for $$P(t). $$Step 5: Conclude that the population $$P(t)$$ will start at a positive value, increase over time with a decreasing growth rate as it approaches $$K$$, and eventually level off at $$P=K$$, representing a stable equilibrium population.

9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.

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Concetti chiave

Relationship Between a Function and Its Derivative

Interpreting the Graph of the Growth Rate Function

Sketching Population Functions from Growth Rates