For the cost function​ below, where c is the cost of producing q units of a​ product, find the​ marginal-cost function. What is the marginal cost at the given values of​ q?
c left parenthesis q right parenthesis equals 18 q squared minus 14 q plus 6​; qequals3​, qequals6​, and qequals13
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Part 1
c prime left parenthesis q right parenthesisequals

enter your response here
Part 2
c prime left parenthesis 3 right parenthesisequals

enter your response here ​(Simplify your​ answer.)
Part 3
c prime left parenthesis 6 right parenthesisequals

enter your response here ​(Simplify your​ answer.)
Part 4
c prime left parenthesis 13 right parenthesisequals

enter your response here ​(Simplify your​ answer.)

Let's solve the problem step by step.

### Part 1: Find the Marginal-Cost Function

The marginal-cost function is the derivative of the cost function \( c(q) \) with respect to \( q \). Given the cost function:

\[
c(q) = 18q^2 - 14q + 6
\]

We find the derivative of \( c(q) \):

\[
c'(q) = \frac{d}{dq}\left(18q^2 - 14q + 6\right)
\]

Let's compute this derivative:

\[
c'(q) = 36q - 14
\]

So, the marginal-cost function is:

\[
c'(q) = 36q - 14
\]

### Part 2: Calculate \( c'(3) \)

Now, substitute \( q = 3 \) into the marginal-cost function:

\[
c'(3) = 36(3) - 14
\]

\[
c'(3) = 108 - 14 = 94
\]

So, \( c'(3) = 94 \).

### Part 3: Calculate \( c'(6) \)

Next, substitute \( q = 6 \) into the marginal-cost function:

\[
c'(6) = 36(6) - 14
\]

\[
c'(6) = 216 - 14 = 202
\]

So, \( c'(6) = 202 \).

### Part 4: Calculate \( c'(13) \)

Finally, substitute \( q = 13 \) into the marginal-cost function:

\[
c'(13) = 36(13) - 14
\]

\[
c'(13) = 468 - 14 = 454
\]

So, \( c'(13) = 454 \).

### Summary:

1. The marginal-cost function is \( c'(q) = 36q - 14 \).
2. \( c'(3) = 94 \)
3. \( c'(6) = 202 \)
4. \( c'(13) = 454 \)

Would you like any further details or have any questions about these calculations?

Here are some related questions to expand your understanding:

1. What does the marginal-cost function tell us about the production process?
2. How would the marginal-cost function change if the cost function had an additional cubic term?
3. How do marginal costs relate to average costs in production?
4. What is the economic interpretation of a negative value in the marginal-cost function?
5. How does the marginal cost change with respect to \( q \) in this specific problem?

**Tip:** Always double-check your derivatives and substitutions to ensure accurate results in calculus problems.

Learn how to find the marginal-cost function from a given cost function using calculus. Detailed step-by-step solutions provided for specific values of q: 3, 6, and 13. Understand the concepts of derivatives and marginal cost in economics and production analysis.

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