Given the function \( C'(x) = 3 - 0.04x + 0.00013x^2 \), this represents the derivative of a cost function \( C(x) \). The derivative \( C'(x) \) typically gives the marginal cost, meaning the rate at which the cost changes with respect to the quantity \( x \).

### Steps to find the cost function \( C(x) \):
To find the cost function \( C(x) \), we need to integrate \( C'(x) \) with respect to \( x \):

\[
C(x) = \int \left( 3 - 0.04x + 0.00013x^2 \right) dx
\]

### Integration:
1. **Integrate term by term:**

\[
\int 3 \, dx = 3x
\]

\[
\int -0.04x \, dx = -0.02x^2
\]

\[
\int 0.00013x^2 \, dx = 0.0000433x^3
\]

2. **Combine all terms and add the constant of integration \( C_0 \):**

\[
C(x) = 3x - 0.02x^2 + 0.0000433x^3 + C_0
\]

### Final Cost Function:
\[
C(x) = 3x - 0.02x^2 + 0.0000433x^3 + C_0
\]

Here, \( C_0 \) is the constant of integration and represents the fixed costs or initial costs when \( x = 0 \).

Would you like more details on the solution or have any questions?

---

Here are 5 related questions to explore further:

1. How do you determine the constant \( C_0 \) in the cost function?
2. What does the marginal cost tell you in economic terms?
3. How would you calculate the total cost for producing a specific number of units?
4. What is the significance of the cubic term \( 0.0000433x^3 \) in the cost function?
5. How does the behavior of the cost function change as \( x \) increases?

**Tip:** When integrating, remember to always add the constant of integration, as it represents initial or fixed values that aren’t accounted for by the derivative alone.

Explore how to integrate the derivative function C'(x) = 3 - 0.04x + 0.00013x^2 to find the cost function C(x). This problem delves into calculus concepts, including derivatives and integration. Suitable for advanced high school students and college-level learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation