To find the marginal cost function, we first need to find the derivative of the cost function $C(x)$, which represents the rate at which the cost is changing with respect to $x$.

Given the cost function:

$C(x) = 400,000 + 590x + 0.001x^2$

The marginal cost $MC(x)$ is the derivative of $C(x)$ with respect to $x$:

$MC(x) = \frac{d}{dx}\left(400,000 + 590x + 0.001x^2\right)$

Now, let's differentiate each term:

1. The derivative of $400,000$ is 0 because it's a constant.
2. The derivative of $590x$ is $590$.
3. The derivative of $0.001x^2$ is $2 \times 0.001 \times x = 0.002x$.

So, the marginal cost function is:

$MC(x) = 590 + 0.002x$

Now, to estimate how fast the cost is increasing when $x = 10,000$, substitute $x = 10,000$ into the marginal cost function:

$MC(10,000) = 590 + 0.002(10,000) = 590 + 20 = 610$

Thus, the rate at which the cost is increasing when $x = 10,000$ is $610$.

Summary:

• The marginal cost function is $MC(x) = 590 + 0.002x$.
• When $x = 10,000$, the cost is increasing at a rate of $610$ per unit increase in $x$.

Would you like a more detailed explanation or any additional help with this? Here are some related questions you could explore:

1. How do we interpret the marginal cost function in terms of business operations?
2. How can we find the total cost at a given production level $x$?
3. What is the significance of the constant term (400,000) in the cost function?
4. How would the marginal cost change if we had a different quadratic term in the cost function?
5. How do we compute the average cost function from the given cost function?

Tip: The marginal cost function helps businesses understand the cost increase for producing one more unit. It’s particularly useful in pricing and production decision-making.