The marginal cost of a certain commodity is C'(x)= 3x^2 -36x + 120
If it costs 5 dollars to produce 1 unit, then find the total cost of producing x units is...
Please how do I find the total cost?
if C'(x) = 3x^2 - 36x + 120
then
C(x) = x^3 - 18x^2 + 120x + c , (I integrated)
if x=1 then C(1) = 5
5 = 1 - 18 + 120 + c
c = -98
C(x) = x^3 - 18x^2 + 120x - 98
To find the total cost of producing x units, you need to integrate the marginal cost function, C'(x), and then add the initial cost of producing the first unit. In this case, the initial cost of producing the first unit is $5.
First, integrate the marginal cost function, C'(x), with respect to x:
C(x) = ∫(3x^2 - 36x + 120) dx
To integrate, use the power rule for integration:
∫x^n dx = (1/n+1) * x^(n+1) + C
Applying the power rule, integrate each term separately:
C(x) = ∫(3x^2) dx - ∫(36x) dx + ∫(120) dx
C(x) = (3/3) * x^3 - (36/2) * x^2 + (120 * x) + C
Simplifying further:
C(x) = x^3 - 18x^2 + 120x + C
Now, add the initial cost of producing the first unit (which is $5):
C(x) = x^3 - 18x^2 + 120x + 5
Therefore, the total cost of producing x units is given by the function C(x) = x^3 - 18x^2 + 120x + 5.