Math Problem Statement
solve as a business calculus beginner student
Solution
Let's solve the problem step by step.
Given:
The marginal cost function is: Fixed cost is:
Part (a): Find the cost function :
The cost function is obtained by integrating the marginal cost function .
Substitute :
Integrate term by term:
Simplify:
Where is the constant of integration.
From the fixed cost, :
Thus:
So the cost function becomes:
Part (b): Find the cost of producing 20 units:
Substitute into :
Calculate step by step:
- , so
- , so
- Add the fixed cost .
Final Answers:
(a) The cost function is:
(b) The cost of producing 20 units is:
Do you have questions or need more clarification? Let me know!
5 Related Questions:
- What is the interpretation of marginal cost in business terms?
- How do fixed costs impact the total cost function?
- Can marginal costs ever decrease, and how would that affect ?
- How does integrating marginal cost differ from differentiating total cost?
- What happens if fixed costs are reduced to zero in this problem?
Tip:
When solving for a cost function from marginal cost, always remember to add the fixed cost at the end. This ensures the total cost function is accurate.
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Math Problem Analysis
Mathematical Concepts
Calculus
Integration
Marginal Cost Analysis
Formulas
C(x) = ∫C'(x) dx + C₀
C'(x) = 0.5x^2 - 3x
C(x) = x^3 / 6 - 3x^2 / 2 + C₀
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
College Level or Advanced High School (Business Calculus)
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