What is the Antiderivative of a Marginal Function?

To undo a marginal function, we need to find the antiderivative of the marginal function. In other words, the antiderivative of marginal cost is cost.

Problem 1 Find the cost function for the marginal cost function

\displaystyle {C}'(x)={{x}^{{\scriptstyle{}^{1}\diagup{}_{2};}}}

where 16 units cost $45.

Problem 2 Find the cost function for the marginal cost function

\displaystyle {C}'(x)={{x}^{{\scriptstyle{}^{2}\diagup{}_{3};}}}+2x

where 8 units cost $58.

Similarly, the antiderivative of the marginal revenue is revenue. To find the corresponding demand function, we need to divide by x,

\displaystyle p(x)=\frac{R(x)}{x}

Problem 3 Find the demand function for the marginal revenue function

\displaystyle {R}'(x)=175-0.02x-0.03{{x}^{2}}

Problem 4 Find the demand function for the marginal revenue function

\displaystyle {R}'(x)=50-5{{x}^{{\scriptstyle{}^{2}\diagup{}_{3};}}}