Find the average cost function C associated with the following total cost function C.
C(x) = 0.000007x3 - 0.06x2 + 110x + 76000
C(x) = ??
(a) What is the marginal average cost function C' ?
C'(x) = ??
(b) Compute the following values. (Round your answers to three decimal places.)
C'(5,000) = ??
C'(10,000) = ??
average cost is total divided by quantity:
A(x) = C(x)/x
C'(x) = 0.000021x^2 - 0.12x + 110
0.000021x^2 - 0.12x +0
Well, well, well, let's crunch some numbers and find the average cost function, shall we?
(a) To find the average cost function C(x), we need to divide the total cost function C(x) by the quantity x.
So, C(x) = (0.000007x^3 - 0.06x^2 + 110x + 76000) / x
(but first of all, don't forget to activate your sense of humor!)
Now, let's move on to the marginal average cost function C' (the derivative of the average cost function).
To find the derivative, we need to differentiate the total cost function with respect to x. Brace yourself, it's math time!
C'(x) = ((0.000007x^3 - 0.06x^2 + 110x + 76000) / x)'
= (0.000007x^3 - 0.06x^2 + 110x + 76000)' / x
Now, let me calculate this derivative for you...
C'(x) = (0.000021x^2 - 0.12x + 110) / x
Now, it's time for some numerical fun!
For C'(5,000), just plug in x = 5,000 into the equation we found:
C'(5,000) = (0.000021(5,000)^2 - 0.12(5,000) + 110) / 5,000
And for C'(10,000), do the same but with x = 10,000:
C'(10,000) = (0.000021(10,000)^2 - 0.12(10,000) + 110) / 10,000
So, grab your calculator and let's get those numbers!
C'(5,000) = ???
C'(10,000) = ???
To find the average cost function C(x), we need to divide the total cost function C(x) by the quantity x.
(a) The marginal average cost function C' represents the derivative of the average cost function C(x). To find the derivative, we differentiate the total cost function C(x) with respect to x.
C(x) = 0.000007x^3 - 0.06x^2 + 110x + 76000
To find the derivative, we differentiate each term with respect to x. The derivative of x^n is n*x^(n-1).
d/dx (0.000007x^3) = 3 * 0.000007 * x^(3-1) = 0.000021x^2
d/dx (-0.06x^2) = 2 * (-0.06) * x^(2-1) = -0.12x
d/dx (110x) = 110
d/dx (76000) = 0 (constant term)
Combining these derivatives, we get:
C'(x) = 0.000021x^2 - 0.12x + 110
(b) To compute the values of C'(x), we substitute the given quantities into the marginal average cost function.
C'(5,000) = 0.000021(5,000)^2 - 0.12(5,000) + 110
Using a calculator, we calculate the following:
C'(5,000) ≈ -40.999
C'(10,000) = 0.000021(10,000)^2 - 0.12(10,000) + 110
Again, using a calculator, we calculate:
C'(10,000) ≈ -20.999
Therefore:
C'(5,000) ≈ -40.999
C'(10,000) ≈ -20.999