The cost of producing x units of stuffed alligator toys is C(x)=0.001x^(2)+8x+5000. Find the marginal cost at the production level of 1000 units
AAAaannndd the bot gets it wrong yet again!
C'(x)=0.002x+8
C'(1000)=0.002*1000+8 = 10
It is not easy to multiply by 1000
To find the marginal cost at the production level of 1000 units, we can use the concept of the derivative. The marginal cost represents the rate of change of the cost function with respect to the number of units produced.
First, let's find the derivative of the cost function C(x) with respect to x:
C'(x) = d/dx (0.001x^2 + 8x + 5000)
To do this, we can apply the power rule for derivatives. The power rule states that if f(x) = ax^n, then f'(x) = nax^(n-1).
Using the power rule, we find:
C'(x) = 0.001 * 2x^(2-1) + 8 * 1x^(1-1) + 0
Simplifying further:
C'(x) = 0.002x + 8
Now that we have the derivative of the cost function, we can substitute the production level of 1000 units (x = 1000) into the derivative equation to find the marginal cost.
C'(1000) = 0.002 * 1000 + 8
Calculating the expression:
C'(1000) = 2 + 8
C'(1000) = 10
Therefore, the marginal cost at the production level of 1000 units is $10.