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 $$P'$$ plotted against the population $$P. $$Here, $$P' is $$constant and positive, meaning the rate of change of the population does not depend on the population size and remains steady over time. Step 2: Translate the constant positive growth rate into a differential equation. Since $$P' is $$constant, we can write $$\displaystyle \frac{dP}{dt} = k$$, where $$k is a $$positive constant representing the constant growth rate. Step 3: Solve the differential equation. Integrate both sides with respect to $$t to $$find $$P(t)$$: $$\displaystyle P(t) = kt + C$$, where $$C is $$the initial population at time $$t=0. $$Step 4: Interpret the solution. The population function $$P(t) is a $$linear function of time with a positive slope $$k$$, indicating the population increases steadily and linearly over time. Step 5: Sketch the population function $$P(t). $$Start at the initial population $$C on $$the vertical axis at $$t=0$$ and draw a straight line with positive slope $$k$$ extending to the right, showing continuous linear growth.

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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Key Concepts

Population Growth Rate Function

Relationship Between Derivative and Function Shape

Initial Conditions in Differential Equations