Question

Make a sketch of the population function (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.

Short answer

Based on these general observations, the population function's sketch will start at a point on the positive y-axis corresponding to \(P_0\) and will either have an upward slope if the growth rate function is positive or a downward slope if the growth rate function is negative. Without specific information about the growth rate function, we cannot draw the exact population function graph.

Step-by-step solution

Identify the growth rate functions

The exercise does not specify the growth rate functions, as it asks to sketch the "following growth rate functions." Since the problem does not provide any specific growth rates, let's consider a general case. Let \(r(t)\) be the growth rate function with respect to time \(t\) (we don't have a specific function since it's not provided in the exercise).

Integrate the growth rate function

To find the population function, we need to integrate the growth rate function with respect to \(t\). Let \(P(t)\) be the population function. Then, \(P(t) = \int r(t) dt + C\), where \(C\) is the constant of integration.

Consider the initial population

The exercise states that the population at time \(t=0\) begins at some positive value. Let this initial population be \(P_0\). We can now use this information to find the constant of integration \(C\). At \(t=0\), \(P(0) = P_0\). So, \(P_0 = \int r(0) dt + C\). To find \(C\), we need to evaluate the integral at \(t=0\). However, since no detailed information about the growth rate function \(r(t)\) is provided, we can not evaluate this integral. Therefore, we'll represent the population function as: \(P(t) = \int r(t) dt + P_0\).

Sketch the population function

With the given information, it's impossible to sketch the exact population function since we don't have a specific growth rate function \(r(t)\). However, we can conclude the general form of the function: If the growth rate function is positive (i.e., the population is increasing with time), the integral would be positive, and hence, the population function would have an upward slope. If the growth rate function is negative (i.e., the population is decreasing with time), the integral would be negative, and hence, the population function would have a downward slope. In both cases, the initial population will be at a positive value of \(P_0\).