Population Growth A population grows at a rate P^'(t)=500 t e^-r t / 5, where P(t) is the popula

Population Growth A population grows at a rate P^'(t)=500 t...

Population Growth A population grows at a rate $$P^{\prime}(t)=500 t e^{-r t / 5},$$ where $P(t)$ is the population after $t$ months. $\begin{array}{l}{\text { (a) Find a formula for the population size after } t \text { months, given }} \\ {\text { that the population is } 2000 \text { at } t=0 \text { . }} \\ {\text { (b) Use your answer from part (a) to find the population size }} \\ {\text { after } 3 \text { months. }}\end{array}$

Population Growth A population grows at a rate
$$P^{\prime}(t)=500 t e^{-r t / 5},$$
where $P(t)$ is the population after $t$ months.
$\begin{array}{l}{\text { (a) Find a formula for the population size after } t \text { months, given }} \\ {\text { that the population is } 2000 \text { at } t=0 \text { . }} \\ {\text { (b) Use your answer from part (a) to find the population size }} \\ {\text { after } 3 \text { months. }}\end{array}$

Key Concepts

Exponential Functions

Exponential functions involve expressions where a constant base is raised to a power that contains a variable. These functions are important in modeling situations with continuous growth or decay, capturing the essence of processes that change multiplicatively over time.

Integration

Integration is the reverse process of differentiation and is used to find the original function when its rate of change is known. In the context of solving differential equations, integration allows us to recover the expression for a quantity (like population) based on its rate of change.

Evaluation of Solutions

After obtaining a general solution from the integration process, the next step is to substitute particular values (such as a specific time) into the solution to evaluate the model's behavior at that point. This evaluation is essential for understanding the dynamics of the model over discrete intervals.

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives, describing how a process evolves over time. They are fundamental in modeling dynamic systems such as population growth, where the rate of change of the population is given by a function of time and the current population size.

Initial Value Problems

An initial value problem involves solving a differential equation together with a specific condition provided at the start of the time period (such as an initial population size). This additional information is crucial for determining the particular solution that accurately reflects the situation being modeled.

Transcript

00:01 145, we're asked to find a formula for population size after t months, given that the population is 2000 at t equals 0.

00:09 So to do that, we're going to have to find the integral because p prime of t is the derivative.

00:18 So to find p of t, you take the integral of p prime of t.

00:26 And to do that, we're going to use u substitution, and we're going to let you equal.

00:32 Let's let you equal this.

00:40 So, u equals e to the negative t squared over 5.

00:54 And then du would be negative 2t, e to the negative t squared over 5.

01:10 And over here i'm just doing the power rule.

01:15 So this over 5 d t so now i'm going to go ahead and substitute you back into the equation so it would be the integral of i'm going to move this negative sign to the front which makes things easier and then i would say 500 times 5 over 2 since i moved the 5 over 2 to the other side let me write that down.

01:59 It would be 5 over 2 du equals t to the negative t over 2.

02:17 So i moved it, since i moved it to the other side, i can do 5 on the numerator over 2.

02:25 And then the du would be this part.

02:32 So it includes the t in the original equation.

02:37 So now we can go ahead and simplify this.

02:40 And we can also move it to the front since it is a constant...

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