Interpretation of the Derivative — Question 10

Question 10

Suppose that the function $C(p)$ models the total cost (in dollars) of producing $p$ items, and is given by: $$C(p) = 200 + 8p + 0.05p^2$$

• Find and interpret $C'(p)$.

• Find $C'(50)$ and interpret its meaning in context.

• Determine whether the marginal cost is increasing or decreasing as $p$ increases.

Question 10 - Solution

We are given: $$C(p) = 200 + 8p + 0.05p^2$$

(a) Find and interpret $C'(p)$:

Differentiate: $$C'(p) = \frac{d}{dp}(200 + 8p + 0.05p^2) = 0 + 8 + 0.1p = 8 + 0.1p$$

Interpretation: $C'(p)$ is the marginal cost — the approximate cost of producing one additional item when $p$ items have already been produced.

(b) Compute and interpret $C'(50)$:

$$C'(50) = 8 + 0.1(50) = 8 + 5 = \boxed{13}$$

Interpretation: When 50 items have been produced, the cost of producing one more item is approximately $13.

(c) Is marginal cost increasing or decreasing?

Since: $$C'(p) = 8 + 0.1p$$ and the derivative of this is: $$C''(p) = 0.1 > 0$$

The marginal cost is increasing as $p$ increases.

$$\boxed{\text{Marginal cost increases with production level.}}$$