Problem
The cost function for a certain commodity is \[ C(q)=88+0.19 q-0.007 q^{2}+0.0008 q^{3} . \] (a) Find $C^{\prime}(100)$. Interpret $C^{\prime}(100)$. This is the rate at which costs are increasing with respect to the production level when $x=100$. This is the amount of time, in minutes, it takes to produce 100 items. This is the number of items that must be produced before the costs reach 100 . This is the cost of making 100 items. This is the rate at which the production level is decreasing with respect to the cost when $x=100$. (b) Find the actual cost of manufacturing the 101st item. (Round your answer to the nearest cent.) $\$$ Need Help? Read It
Solution
Step 1 ：Given the cost function \(C(q)=88+0.19 q-0.007 q^{2}+0.0008 q^{3}\)
Step 2 ：We need to find the derivative of the cost function, \(C'(q)\), which gives the rate of change of the cost with respect to the quantity produced.
Step 3 ：Using the power rule for differentiation, the derivative of \(C(q)\) is \(C'(q) = 0.19 - 0.014q + 0.0024q^{2}\)
Step 4 ：Substitute \(q=100\) into the derivative function to find the rate of change of the cost at \(q=100\), \(C'(100) = 22.79\)
Step 5 ：Final Answer: The rate of change of the cost with respect to the quantity produced when the quantity is 100 is \(\boxed{22.79}\)
