Let the total cost function C(x) be defined as follows.
C(x) = 0.0008x3 - 0.04x2 + 99x + 4400
Find the average cost function C.
C(x) = ??
Find the marginal average cost function C '.
C '(x) = ??
To find the average cost function, we need to divide the total cost function C(x) by the number of units x:
C(x) = (0.0008x^3 - 0.04x^2 + 99x + 4400) / x
Now, we can simplify the function:
C(x) = 0.0008x^2 - 0.04x + 99 + 4400/x
This is the average cost function.
Now, we'll find the marginal average cost function, which is the derivative of the average cost function:
C'(x) = d(C(x))/dx = d(0.0008x^2 - 0.04x + 99 + 4400/x)/dx
To find the derivative, we'll differentiate each term separately:
d(0.0008x^2)/dx = 2 * 0.0008x = 0.0016x
d(-0.04x)/dx = -0.04
d(99)/dx = 0
d(4400/x)/dx = -4400/x^2
Now, we can add these derivatives to find the marginal average cost function:
C'(x) = 0.0016x - 0.04 - 4400/x^2
To find the average cost function C(x), you need to divide the total cost function C(x) by the quantity x. The formula for the average cost function is:
C(x) = (1/x) * C(x)
Let's substitute the given total cost function C(x) into the formula:
C(x) = (1/x) * (0.0008x^3 - 0.04x^2 + 99x + 4400)
Simplifying this expression will give us the average cost function C(x).
Now, let's find the marginal average cost function, denoted as C '(x), which represents the derivative of the average cost function with respect to x.
To find C '(x), we need to take the derivative of the average cost function C(x) with respect to x. Differentiating each term of the average cost function will yield us the marginal average cost function.
I will now find C(x) and C '(x) for you.
To find the average cost function, C(x), you divide the total cost function, C(x), by the quantity, x.
So, C(x) = (0.0008x^3 - 0.04x^2 + 99x + 4400) / x
To simplify this expression, you need to divide each term by x:
C(x) = 0.0008x^3 / x - 0.04x^2 / x + 99x / x + 4400 / x
Simplifying further:
C(x) = 0.0008x^2 - 0.04x + 99 + (4400 / x)
Therefore, the average cost function, C(x), is given by:
C(x) = 0.0008x^2 - 0.04x + 99 + (4400 / x)
To find the marginal average cost function, C'(x), you need to find the derivative of the average cost function with respect to x.
Differentiating each term individually, we get:
C'(x) = d/dx (0.0008x^2) - d/dx(0.04x) + d/dx(99) + d/dx(4400 / x)
Using the power rule, we get:
C'(x) = 0.0016x - 0.04 + 0 - (4400 / x^2)
Therefore, the marginal average cost function, C'(x), is given by:
C'(x) = 0.0016x - 0.04 - (4400 / x^2)