How do you get the antiderivative for C'(x) = 20-(x/200) to be complete when I only know that I'm using Right Riemann sum with n=5 after. I know the antiderivative would be C=20x-(1/400)x^2+K (K substituting for what normally would be C since that is already used) but how do I get K so I can actually use the function?
not sure what you're after here. If you want the area on the interval [0,5] then K does not matter. A definite integral C(5)-C(0) will subtract off the K. Only indefinite integrals have the unknown constant.
If you need to pin down K, then you have to know the value of C for some x. For example, if C(0) = 10, then
C=20x-(1/400)x^2+10
my bad, as I noted on your previous post. The Riemann sum gives the area under the curve for C'(x). That area approximates the function C(x).
So use values of C' for x=1,2,3,4,5.
To find the constant of integration, K, you would need additional information beyond the Right Riemann sum with n=5. The constant of integration, K, is determined by the initial condition or boundary condition of the problem.
In this case, you mentioned that the function represents the rate of change of a quantity, C'(x), and you want to find the antiderivative, C(x). To fully determine the value of K, you would need to know the value of C(x) at a particular point, such as the initial condition or a boundary value.
Once you have the specific value of C(x) for a given x, you can substitute it into the antiderivative equation and solve for K. For example, if you know C(0) = 100, you can substitute x=0 and C=100 into the antiderivative equation:
100 = 20(0) - (1/400)(0)^2 + K
This simplifies to:
100 = K
Therefore, in this specific case, you would find that K = 100. Then the complete function would be C(x) = 20x - (1/400)x^2 + 100.
So, to find the value of K, you would need to know a specific point or condition associated with the quantity being modeled by the function C(x).