Studdy AI Logo
Math  /  Calculus

QuestionA company determined that the marginal cost, C′(x)C^{\prime}(x) of producing the xx th unit of a product is given by C′(x)=x3−4xC^{\prime}(x)=x^{3}-4 x. Find the total cost function CC, assuming that C(x)C(x) is in dollars and that fixed costs are $8000\$ 8000.

Studdy Solution

STEP 1

1. The marginal cost function is given by C′(x)=x3−4x C^{\prime}(x) = x^3 - 4x .
2. The fixed cost is \$8000.
3. We need to find the total cost function \( C(x) \).

STEP 2

1. Integrate the marginal cost function to find the total cost function C(x) C(x) .
2. Determine the constant of integration using the fixed cost information.
3. Write the complete expression for the total cost function C(x) C(x) .

STEP 3

Integrate the marginal cost function C′(x)=x3−4x C^{\prime}(x) = x^3 - 4x to find C(x) C(x) :
C(x)=∫(x3−4x) dx C(x) = \int (x^3 - 4x) \, dx

STEP 4

Perform the integration:
C(x)=∫x3 dx−∫4x dx C(x) = \int x^3 \, dx - \int 4x \, dx
C(x)=x44−2x2+C0 C(x) = \frac{x^4}{4} - 2x^2 + C_0
where C0 C_0 is the constant of integration.

STEP 5

Use the fixed cost information to determine the constant of integration. Given that the fixed cost is \$8000, we have:
C(0)=8000 C(0) = 8000
Substitute x=0 x = 0 into the total cost function:
C(0)=044−2(0)2+C0=8000 C(0) = \frac{0^4}{4} - 2(0)^2 + C_0 = 8000
C0=8000 C_0 = 8000

STEP 6

Substitute C0=8000 C_0 = 8000 back into the expression for C(x) C(x) :
C(x)=x44−2x2+8000 C(x) = \frac{x^4}{4} - 2x^2 + 8000
The total cost function C(x) C(x) is:
C(x)=x44−2x2+8000 C(x) = \frac{x^4}{4} - 2x^2 + 8000

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord