For the cost function, find the marginal cost at the given production level *x*. State the units of measurement. (All costs are in dollars.)

*C*(*x*) = 20,000 + 40*x* + 

10,000

*x*

;    *x* = 100

  ----------Select Units---------- dollars per item dollars items dollars per dollar per item





To find the marginal cost, we need to compute the derivative of the cost function \( C(x) \) with respect to \( x \). The marginal cost represents the rate of change of the total cost with respect to the production level \( x \), which is simply the first derivative of the cost function.

Given the cost function:

\[
C(x) = 20,000 + 40x + \frac{10,000}{x}
\]

### Step 1: Find the derivative \( C'(x) \)

1. The derivative of \( 20,000 \) (a constant) is 0.
2. The derivative of \( 40x \) is 40.
3. The derivative of \( \frac{10,000}{x} \) is \( -\frac{10,000}{x^2} \).

So, the marginal cost \( C'(x) \) is:

\[
C'(x) = 40 - \frac{10,000}{x^2}
\]

### Step 2: Evaluate the marginal cost at \( x = 100 \)

\[
C'(100) = 40 - \frac{10,000}{100^2} = 40 - \frac{10,000}{10,000} = 40 - 1 = 39
\]

### Step 3: Determine the units of measurement

Since the cost function is in dollars and the production level \( x \) is in items, the units of marginal cost are **dollars per item**.

Thus, the marginal cost at \( x = 100 \) is **39 dollars per item**.

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Would you like further clarification or details on this process?

Here are 5 related questions to expand your understanding:

1. What is the significance of the marginal cost in production analysis?
2. How does the marginal cost function behave as production increases?
3. What happens to the marginal cost if the term \( \frac{10,000}{x} \) were absent?
4. Can you find the average cost at \( x = 100 \)?
5. How would the cost function change if there were economies of scale?

**Tip:** Marginal cost is essential for making decisions on how much to produce, as it helps determine if producing an additional unit is profitable.

For the cost function, find the marginal cost at the given production level x. State the units of measurement. (All costs are in dollars.) C(x) = 20,000 + 40x + 10,000/x; x = 100

Learn how to find the marginal cost for the cost function C(x) = 20,000 + 40x + 10,000/x at x = 100. This problem involves calculus concepts such as differentiation and interpreting the derivative as the marginal cost in economics. Suitable for students in Grades 11-12.

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