Johnny's Tumblers sells plastic cups in bulk for start-up restaurants. Their cost for producing these is modeled by: c(x) = 0.02x^2 + 0.5x + 40
a) Since the marginal cost is found by taking the derivative, the formula for c'(x)= ? (My answer: 0.04x + 0.5)
b) The marginal average cost would be the derivative of the average cost function. Find this marginal average cost function. The symbolic notation for this is: c'(x) (with a line over the c) = ?????
This is the part I need help with.
a. correct
b. average cost function= costtotal/x= .02x+.05+40/x
marginal average cost=d/dx C(x)
= .02-40/x^2
To find the marginal average cost function, we need to differentiate the average cost function with respect to x.
The average cost function, denoted as C(x), is given by:
C(x) = c(x)/x
To find the derivative of C(x), we can use the quotient rule. The quotient rule states:
d/dx (f(x)/g(x)) = (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
Applying the quotient rule to C(x), we have:
C'(x) = (c'(x) * x - c(x) * 1) / (x^2)
Now, let's substitute c(x) into the equation:
C'(x) = (c'(x) * x - (0.02x^2 + 0.5x + 40)) / (x^2)
Simplifying further, we have:
C'(x) = (c'(x) * x - 0.02x^2 - 0.5x - 40) / (x^2)
Thus, the symbolic notation for the marginal average cost function is:
c'(x) [with a line over the c] = (c'(x) * x - 0.02x^2 - 0.5x - 40) / (x^2)
To find the marginal average cost function, we need to first determine the average cost function, and then take its derivative.
The average cost (AC) can be found by dividing the total cost C(x) by the quantity x:
AC(x) = C(x) / x
Given that the cost function is C(x) = 0.02x^2 + 0.5x + 40, plugging this into the average cost formula, we have:
AC(x) = (0.02x^2 + 0.5x + 40) / x
To find the marginal average cost, we need to take the derivative of the average cost function AC(x) with respect to x:
c'(x̄) = d/dx (AC(x))
To differentiate the AC(x) expression, we will use the quotient rule. The quotient rule states that if we have the function f(x) = g(x) / h(x), then the derivative f'(x) can be found using the following formula:
f'(x) = [g'(x)h(x) - g(x)h'(x)] / (h(x))^2
Applying the quotient rule to AC(x), we get:
c'(x̄) = [ (d/dx)(0.02x^2 + 0.5x + 40)(x) - (0.02x^2 + 0.5x + 40)(d/dx)(x) ] / x^2
Simplifying this expression, we have:
c'(x̄) = [ (0.04x + 0.5)(x) - (0.02x^2 + 0.5x + 40) ] / x^2
Expanding and rearranging terms:
c'(x̄) = (0.04x^2 + 0.5x - 0.02x^2 - 0.5x - 40) / x^2
Simplifying further:
c'(x̄) = (0.02x^2 - 40) / x^2
Therefore, the marginal average cost function is c'(x̄) = (0.02x^2 - 40) / x^2.
a) ok
b) The average cost is the cost per unit, or c̅(x) = c(x)/x
So, c̅(x) = 0.02x + 0.5 + 40/x
The marginal average cost is thus
c̅'(x) = 0.02 - 40/x^2