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

To find the population function, \(P(t)\), as a function of time, given the growth rate function, \(G(t)\), follow these steps: 1. Write the relationship between the population function and the growth rate function as: \[P(t) = P(0) + \int_0^t G(u) du\] 2. Integrate the given growth rate function, \(G(t)\), with respect to time to find the population function, \(P(t)\). 3. Consider the behavior of the growth rate function when sketching the graph of the population function. If \(G(t)\) is positive, \(P(t)\) increases; if \(G(t)\) is negative, \(P(t)\) decreases. If \(G(t)\) is increasing, \(P(t)\) is concave up; if \(G(t)\) is decreasing, \(P(t)\) is concave down. 4. Sketch the graph of the population function, \(P(t)\), as a function of time, considering the behaviors obtained from the growth rate function, \(G(t)\).

Step-by-step solution

Understanding the growth rate function

The growth rate function determines how fast the population is changing at any given time. In this exercise, we are asked to find the population function, given the growth rate function. To do this, we need to find the integral of the growth rate function with respect to time. Assume the growth rate function is \(G(t)\) and the population function is \(P(t)\). We can write the relationship between the two functions as: \[P(t) = P(0) + \int_0^t G(u) du\] where \(P(0)\) is the population at time \(t=0\), and \(u\) is a dummy variable used for integration.

Integrating the growth rate function

We don't have an explicit form for the growth rate function, \(G(t)\), but we can describe the procedure in general terms. To find the population function, \(P(t)\), we should integrate the growth rate function \(G(t)\) with respect to time, using the following integral: \[P(t) = P(0) + \int_0^t G(u) du\]

Sketching the population function

Once we have found the population function, \(P(t)\), we can sketch its graph as a function of time. Here are some general guidelines for sketching the population function given its relationship with the growth rate function: 1. If the growth rate function is positive, the population function will be increasing. 2. If the growth rate function is negative, the population function will be decreasing. 3. If the growth rate function is increasing, the population function will be concave up (i.e., bending upwards). 4. If the growth rate function is decreasing, the population function will be concave down (i.e., bending downwards). Using these guidelines, we can sketch the population function as a function of time. In conclusion, without having a specific growth rate function, \(G(t)\), we cannot provide a detailed step-by-step solution for the problem. However, we have outlined the general process one should follow to obtain the population function, \(P(t)\), and sketch it as a function of time. The most important aspect when solving such problems is to remember that the population function is the integral of the growth rate function with respect to time.