Question

why is average cost defined as C(x)/x, where C(x) is your cost function. When you want the average of any function you can use the integral mean value theorem and integrate f(x) / b-a where f(x) is defined over f(x). This is why average velocity, is the integral of velocity / t2 - t1, which gives us x(t2) - x(t1) / t2 - t1

Answer 1

Because, C(x) is the total cost of x units. So you are taking the total cost and dividing it by the number of units. I think what is confusing you is that with velocity and most other functions you have probably worked with, your variable is time. In the cost function your variable is units.

If you had C(t) where t is time, then you could use the equation you are used to because when you are taking the integral of C(t) you are finding the total sum of the parts up until time t.

Hopefully that clears it up a little bit, let me know if you need further clarification.