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Let's go through the problem step by step using the provided cost function and answer the questions.

The total cost function is given as:
\[
C(x) = 2000 + 2x - 0.0001x^2 \quad \text{where} \quad 0 \leq x \leq 6000
\]

### (a) Actual cost of producing the 971st and 1821st discs
The actual cost of producing the 971st disc and 1821st disc can be found by plugging $x = 971$ and $x = 1821$ into the cost function $C(x)$.

#### Step 1: Compute $C(971)$
\[
C(971) = 2000 + 2(971) - 0.0001(971)^2
\]
Simplifying this will give the cost for producing the 971st disc.

#### Step 2: Compute $C(1821)$
\[
C(1821) = 2000 + 2(1821) - 0.0001(1821)^2
\]
This will give the cost for producing the 1821st disc.

### (b) Marginal cost when $x = 970$ and $x = 1820$
Marginal cost refers to the derivative of the cost function with respect to $x$, denoted as $C'(x)$.

We first differentiate the given cost function:
\[
C'(x) = \frac{d}{dx}(2000 + 2x - 0.0001x^2)
\]
This simplifies to:
\[
C'(x) = 2 - 0.0002x
\]

Now, we can compute the marginal cost at $x = 970$ and $x = 1820$.

#### Step 3: Compute $C'(970)$
\[
C'(970) = 2 - 0.0002(970)
\]

#### Step 4: Compute $C'(1820)$
\[
C'(1820) = 2 - 0.0002(1820)
\]

I'll now calculate these values for you.Here are the results:

### (a) Actual cost:
- The cost of producing the **971st disc** is **\$3847.72**.
- The cost of producing the **1821st disc** is **\$5310.40**.

### (b) Marginal cost:
- The marginal cost when $ x = 970 $ is **\$1.81**.
- The marginal cost when $ x = 1820 $ is **\$1.64**.

Would you like further explanation on any of these calculations or details on how I derived the results?

Here are 5 related questions to deepen your understanding:
1. How is marginal cost useful in production analysis?
2. What is the relationship between total cost and marginal cost?
3. How would the cost function change if the quadratic term were different?
4. What does a decreasing marginal cost imply about production efficiency?
5. How can the marginal cost function be used to find the optimal production level?

**Tip:** Marginal cost helps companies decide how many additional units to produce, as it shows the cost impact of increasing production by one more unit.

The total weekly cost incurred by Lincoln Records in pressing x compact discs is given by C(x) = 2000 + 2x - 0.0001x^2. What is the actual cost incurred in producing the 971st and 1821st disc? What is the marginal cost when x = 970 and x = 1820?

Learn how to calculate the actual cost of producing the 971st and 1821st discs using the cost function C(x) = 2000 + 2x - 0.0001x^2. Also, explore marginal cost calculations at x = 970 and x = 1820 through differentiation.

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