I have a problem here relating to derivatives:
An accessories company finds that the cost, in dollars, of producing x belts is given by C(x) = 740 + 30x - 0.068x^2
Find the rate at which average cost is changing when 171 bells have been produced. First find the rate at which the average cost is changing when x belts have been produced. barC'(x) = ?
The question it wants me to fill in is what is confusing me, because in the C'(x) = ?, there is a small bar right above the C. What does that little bar mean?
The bar above the letter "C" indicates that it is the average cost, denoted by "C-bar" or "barC". When you take the derivative of the average cost function, you get the rate at which the average cost is changing with respect to the number of belts produced. Therefore, the expression you're looking for is "barC'(x)".
C(x) is the total cost, so the average cost is
C̅(x) = C(x)/x = 740/x + 30 - 0.068x
C̅'(x) = -740/x^2 - 0.068
so now we have
C̅'(171) = -0.0933
The small bar, known as the "overbar," represents the notation for the average value or average rate of change. In this context, the notation "C'(x)" represents the derivative of the cost function with respect to x, while "barC'(x)" represents the derivative of the average cost function with respect to x.
To find the rate at which the average cost is changing, we need to differentiate the average cost function with respect to x.
The average cost function is given by:
average cost = C(x)/x
To find the derivative of the average cost function, we need to differentiate C(x)/x with respect to x using the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then its derivative is given by:
f'(x) = (g'(x)h(x) - g(x)h'(x))/[h(x)]^2
Applying the quotient rule to the average cost function, we get:
barC'(x) = [C'(x)x - C(x)]/x^2
Now, we can differentiate the cost function C(x) to find the derivative C'(x). Taking the derivative of each term in C(x) with respect to x, we get:
C'(x) = 30 - 0.136x
Substituting C'(x) into the equation for barC'(x), we have:
barC'(x) = [(30 - 0.136x)x - (740 + 30x - 0.068x^2)]/x^2
Simplifying this expression gives:
barC'(x) = (30x - 0.136x^2 - 740 - 30x + 0.068x^2)/x^2
Combine like terms:
barC'(x) = (-0.068x^2 + 0.068x^2 + 30x - 30x - 740)/x^2
barC'(x) = -740/x^2
Therefore, the rate at which the average cost is changing when x belts have been produced is given by -740/x^2.
It simply means to take the derivative of C(x), also stated as
d C(x)/dx or simply C'(x)
C'(x) = 30 - .136x
so C'(171) = 30 - .136(171) = $6.744