The marginal cost of producing the xth box of CDs is given by

9 − x/(x^2 + 1)^2.
The total cost to produce 2 boxes is $1,200. Find the total cost function
C(x).

I'm getting 9x-(1/(x^2 - 1))+1181.9 but i guess its wrong

sadofi

To find the total cost function C(x), we need to integrate the marginal cost function over the range of production.

Given that the marginal cost function is 9 - x/(x^2 + 1)^2, we integrate it to find the total cost function C(x):

C(x) = ∫(9 - x/(x^2 + 1)^2) dx

To find the integral, we split the fraction into two parts:

C(x) = ∫(9 - x(x^2 + 1)^-2) dx = ∫(9 - x(x^4 + 2x^2 + 1)^-2) dx

Now, we distribute the x in the numerator:

C(x) = ∫(9 - (x^5 + 2x^3 + x)^-2) dx

Next, we can expand the denominator before integration:

C(x) = ∫(9 - (x^5 + 2x^3 + x)^-2) dx = ∫(9 - x^10 - 4x^8 - 2x^6 - x^2) dx

To find the integral, we can use the power rule of integration:

C(x) = 9x - 1/11x^11 - 4/9x^9 - 1/7x^7 - 1/3x^3 + C

Now we need to find the value of C when x = 2, given that the total cost to produce 2 boxes is $1,200:

C(2) = 9(2) - 1/11(2^11) - 4/9(2^9) - 1/7(2^7) - 1/3(2^3) + C = 1200

Simplifying this equation will give us the value of the constant C:

18 - (2048/11) - (512/9) - (128/7) - (16/3) + C = 1200

Calculating the values:

C = 1200 - 18 + (2048/11) + (512/9) + (128/7) + (16/3)

After performing the calculation, you can find the value of C, which will give you the total cost function C(x).

To find the total cost function C(x), we need to integrate the marginal cost function over the range of boxes produced.

The marginal cost function is given as 9 - x/(x^2 + 1)^2.

Integrating this expression will give us the total cost function C(x).

∫ (9 - x/(x^2 + 1)^2) dx

To integrate this expression, let's use partial fractions.

First, we rewrite the expression as:

∫ (9(x^2 + 1)^2 - x) / (x^2 + 1)^2 dx

Next, we expand (x^2 + 1)^2:

∫ (9x^4 + 18x^2 + 9 - x) / (x^2 + 1)^2 dx

Now, we can split the rational function into partial fractions:

∫ (9x^4 + 18x^2 + 9 - x) / (x^2 + 1)^2 dx
= ∫ (A/x^2 + B/x + C(x^2 + 1) + Dx + E) / (x^2 + 1)^2 dx

Finding the values of A, B, C, D, and E requires solving a system of equations. Once we have the values, we can integrate each term separately.

After performing the integration, the total cost function C(x) will be obtained.

However, it seems that your calculation of the total cost function may be incorrect. I recommend rechecking your integration and the coefficients of the partial fractions.

forgot that pesky factor of 2.

∫-x/(x^2+1)^2 dx
is not 1/(x^2+1)
Check the derivative:
d/dx (x^2+1)^-1 = -2x(x^2+1)^-2

So,

∫9 − x/(x^2 + 1)^2 = 9x + 1/(2(x^2+1)^2)