Profit The marginal profit of a small chicken farm, in...

Profit The marginal profit of a small chicken farm, in dollars,
is given by
$P^{\prime}(x)=3 \sqrt{x}+2$
where $x$ is the sales volume in hundreds of chicken. The profit is $-\$ 2000$ when no chicken are sold. Find the profit function.

Key Concepts

Integration

Integration is the mathematical process used to recover an original function from its derivative. In the context of economics, integrating a marginal function allows us to obtain the cumulative measure of profit or cost, taking into account the constant of integration which represents the initial state.

Marginal Analysis

Marginal analysis involves examining the instantaneous rate of change in a function, such as profit, as a result of a small change in some variable. It helps in understanding how incremental changes affect total outcomes, and is essential in decision-making processes.

Initial Value Problems

An initial value problem is a type of differential equation that includes additional information, such as a specific value of the function at a given point, to determine the constant of integration. This additional condition ensures that the resulting function is uniquely determined.

Transcript

00:01 So we want to find the profit function given this marginal function here.

00:10 The first thing we're going to have to do is figure out how the marginal function relates to the profit.

00:16 So remember that marginal is really just talking about a very small change in something.

00:24 And the way that we describe very small changes in calculus is with the derivative.

00:30 So what this is really saying is look at we're given the derivative of the profit function.

00:40 So if i want to find the profit function from the marginal, we'll want to take the antiderivative or integrate p prime of x here.

00:54 Let's go ahead and do that.

01:02 So it's not really willing to cooperate.

01:04 So i'll just rewrite again.

01:06 P prime of x is equal to 3 square root of x plus 2.

01:15 So we want to integrate this so we can find our profit function.

01:25 So i'll just go ahead and integrate each side of this function with respect to x.

01:36 So integrating p prime of x will just give us p of x.

01:42 So our profit, and then if we want to take the integral of 3 square root of 2 x plus 2, well first we can go ahead and rewrite this so we can put it into terms of things we do know the integral of.

02:04 So remember the square root of x is really x to the one half power, and any time we have a constant being multiplied, or just a constant, it's really being multiplied by that variable to the zero of power.

02:21 So this will be two instances of power rule.

02:24 So the first thing we want to do is go ahead and use the sum and scalar property of integration and rewrite this as.

02:33 So three, so i'll just go ahead and rewrite the equation first, plus 2 x to 0, and then distribute the integration with our variables.

02:48 And now we can apply the power rule.

02:52 So it will be 3, and then power rule for antiderivatives.

02:59 So it's 1 half, and then i add a power of 1, and then divide by the new power, so 1 half plus 1 is 3 halves.

03:09 Then i add some constant c1...