Suppose the cost C(q) (in dollars) of producing a quantity q of a product
equals
C(q) = 500 + 2q +1/5q^2
The marginal cost M(q) equals the instantaneous rate of change of the
total cost. Find the marginal cost when a quantity of 10 items are being
produced.
To find the marginal cost, we need to take the derivative of the cost function C(q) with respect to q.
C(q) = 500 + 2q + 1/5q^2
Taking the derivative with respect to q:
dC(q)/dq = d(500 + 2q + 1/5q^2)/dq
Using the power rule, we have:
dC(q)/dq = 0 + 2 + (2/5)q
Simplifying, we get:
dC(q)/dq = 2 + (2/5)q
Now we need to evaluate the derivative at q = 10. Plugging in q = 10 into the derivative:
dC(q)/dq = 2 + (2/5)(10)
= 2 + 4
= 6
Therefore, the marginal cost when a quantity of 10 items are being produced is $6.
To find the marginal cost when a quantity of 10 items are being produced, we need to determine the derivative of the cost function C(q) with respect to q.
The cost function is given as C(q) = 500 + 2q + 1/5q^2. Taking the derivative of this function will give us the marginal cost function.
Step 1: Differentiate the constant term
The derivative of a constant term is always zero. In this case, the constant term is 500.
d/dq (500) = 0
Step 2: Differentiate the linear term
The derivative of a linear term is equal to its coefficient. In this case, the linear term is 2q.
d/dq (2q) = 2
Step 3: Differentiate the quadratic term
To differentiate the quadratic term, we use the power rule. The power rule states that the derivative of x^n is nx^(n-1). In this case, the quadratic term is 1/5q^2.
d/dq (1/5q^2) = (1/5) * 2q^(2-1) = (2/5)q
Step 4: Add up the derivatives
Now, we add up the derivatives we obtained in steps 1, 2, and 3 to find the derivative of the cost function.
dC/dq = d/dq (500 + 2q + 1/5q^2) = 0 + 2 + (2/5)q
Simplifying the expression gives us:
dC/dq = 2 + (2/5)q
Step 5: Substitute the quantity q = 10 into the marginal cost function
To find the marginal cost when a quantity of 10 items are being produced, we substitute q = 10 into the derivative function.
M(q) = 2 + (2/5)q
M(10) = 2 + (2/5)(10)
M(10) = 2 + 4
M(10) = 6
Therefore, the marginal cost when a quantity of 10 items are being produced is $6.