How Do You Undo A Marginal Function?

Marginal function like marginal cost, marginal revenue, and marginal profit are all derivatives. This means that we can undo these derivatives to obtain the cost, revenue, and profit functions by taking their antiderivatives.

For example, suppose the marginal cost for a product is given by

\displaystyle {C}'(x)=0.2{{x}^{2}}+5x

where x is the number of units produced. Also suppose the fixed cost are $1000. The antiderivative of the marginal cost is

\displaystyle C(x)=\frac{0.2}{3}{{x}^{3}}+\frac{5}{2}{{x}^{2}}+K

where K is an arbitrary constant. By requiring that the fixed cost is $1000, we know that the cost of producing no items is $1000 or C(0) = 1000. This means that

\displaystyle C(0)=\frac{0.2}{3}\cdot {{0}^{3}}+\frac{5}{2}\cdot {{0}^{2}}+K=1000

or K = 1000. The cost function matching the marginal cost and fixed cost is

\displaystyle C(x)=\frac{0.2}{3}{{x}^{3}}+\frac{5}{2}{{x}^{2}}+1000

Here are several more examples worked out by students.

Example 1 Find the demand function corresponding to the marginal revenue

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

Remember. no revenue is incurred when no items are sold.

anti_marg_revenue_1

Example 2 Find the cost function corresponding to the marginal cost function

\displaystyle {C}'(x)={{x}^{\tfrac{1}{2}}}

Assume that 16 units costs $45.

anti_marg_cost_1

Example 3 The marginal profit in dollars per pound on Brie cheese is

\displaystyle {P}'(x)=x\left( 50{{x}^{2}}+30x \right)

where x is the amount of cheese sold in hundreds of pounds. Assume the profit is -40 when no cheese is sold.

a. Find the profit function.

b. Find the profit from selling 200 pounds of cheese.

anti_marg_profit_1Since the variable x is in hundreds of pounds, the profit is found by substituting 2 (not 200) into the function.

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};}}}