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The growth rate function represents the rate at which a population is either increasing or decreasing. The population function describes the population size as a function of time. In order to find the population function from the growth rate function, we need to integrate the growth rate function over time. This will give us an expression for the population function.

We are asked to make a sketch of the population function that results from the following growth rate functions: \(r_1(t) = k\), \(r_2(t) = kt\), and \(r_3(t) = k \cos{t}\). Where \(k\) is some constant.

We will integrate each growth rate function to obtain the population function. For the first growth rate function: \(P_1(t) = \int r_1(t) dt = \int k dt = kt + C_1\) For the second growth rate function: \(P_2(t) = \int r_2(t) dt = \int kt dt = \frac{1}{2}kt^2 + C_2\) For the third growth rate function: \(P_3(t) = \int r_3(t) dt = \int k\cos{t} dt = k\sin{t} + C_3\) Here, \(C_1\), \(C_2\), and \(C_3\) are the constants of integration.

From the exercise, we know that the population at time \(t=0\) starts at some positive value. Thus, we can determine \(C_1\), \(C_2\), and \(C_3\). Let's assume the initial population size is \(P_0\). \(P_1(0) = P_0 \Rightarrow k\cdot 0 + C_1 = P_0 \Rightarrow C_1=P_0\) \(P_2(0) = P_0 \Rightarrow \frac{1}{2}k\cdot 0^2 + C_2 = P_0 \Rightarrow C_2 = P_0\) \(P_3(0) = P_0 \Rightarrow k\sin{0} + C_3 = P_0 \Rightarrow C_3 = P_0\)

Now, we have obtained the population functions that include the initial population size: \(P_1(t) = kt + P_0\) \(P_2(t) = \frac{1}{2}kt^2 + P_0\) \(P_3(t) = k\sin{t} + P_0\)

Now, we will sketch the population functions. For \(P_1(t)\): This function represents a linear population growth, with a constant slope of \(k\), and the population will increase or decrease depending on the sign of \(k\). For \(P_2(t)\): This function is a quadratic population growth. If \(k > 0\), it represents a convex upwards parabola, meaning an accelerating increase in population. If \(k < 0\), it represents a convex downwards parabola, indicating an accelerating decrease in population. For \(P_3(t)\): The third function is a sinusoidal population growth. As \(t\) increases, the population oscillates between growth and decline in a periodic manner, centered around the initial value \(P_0\). In conclusion, integrating the growth rate functions give us the corresponding population functions, and understanding the properties of the functions allows us to sketch them.