The marginal cost function is the derivative of the cost function with respect to the quantity of the product being produced. To find the marginal cost function, we must first find the derivative of the cost function.

In this case, the cost function is:

c = 500 + 10q²

To find the derivative, we use the power rule, which states that the derivative of a function of the form f(x) = x^n is n * x^(n-1). In this case, we have:

dc/dq = 10 * 2 * q^(2-1) = 20q

Therefore, the marginal cost function is:

mc = 20q

To find the marginal cost at a given value of q, we simply plug that value into the marginal cost function. For example, if q = 100, the marginal cost is:

mc = 20 * 100 = 2000.

In summary, the marginal cost function is mc = 20q, and the marginal cost at q = 100 is 2000.