Interpretation of the Derivative — Question 4

Question 4

The cost (in dollars) to produce $x$ items is given by the function: $$C(x) = 50x + 300 \sqrt{x}$$

• Find the marginal cost function.

• Interpret the meaning of $C'(100)$.

• Approximate the cost of producing the 101st item using the derivative.

• Compute the actual cost of producing the 101st item and compare with your approximation.

Question 4 - Solution

Marginal cost. For $x>0$,

$$\boxed{C'(x)=50+150/\sqrt{x}}.$$

Thus $C'(100)=65$ dollars per item. This is the instantaneous cost rate and predicts a cost of approximately $\boxed{\$65}$ for one additional item.

Actual additional cost.

$$C(101)-C(100)=50+300(\sqrt{101}-10)\approx\boxed{\$64.96}.$$

The derivative estimate exceeds the actual cost by about $\$0.04$. This agrees with $C''(x)=-75x^{-3/2}<0$: the marginal cost decreases over the next item.