The marginal revenue function on sales of q units of a product is R^'(q)=200-12 √(q) dollars per

step 1 Our first task is to graph the marginal revenue function. Here it is. Now, our second task, Part

The marginal revenue function on sales of q units of a...

The marginal revenue function on sales of $q$ units of a product is $R^{\prime}(q)=200-12 \sqrt{q}$ dollars per unit. (a) Graph $R^{\prime}(q)$ (b) Estimate the total revenue if sales are 100 units. (c) What is the marginal revenue at 100 units? Use this value and your answer to part (b) to estimate the total revenue if sales are 101 units.

The marginal revenue function on sales of $q$ units of a product is $R^{\prime}(q)=200-12 \sqrt{q}$ dollars per unit.
(a) Graph $R^{\prime}(q)$
(b) Estimate the total revenue if sales are 100 units.
(c) What is the marginal revenue at 100 units? Use this value and your answer to part (b) to estimate the total revenue if sales are 101 units.

Key Concepts

Marginal Revenue Function

The marginal revenue function represents the instantaneous rate of change of total revenue with respect to the quantity sold. It is essentially the derivative of the total revenue function, indicating how much additional revenue is generated by selling one more unit. This concept is fundamental in understanding how revenue changes as sales volume increases.

Integration to Calculate Total Revenue

Total revenue can be estimated by integrating the marginal revenue function over a specified interval of units sold. This integration process accumulates all the incremental revenues provided by the marginal function, thereby reconstructing the total revenue over that range. It is a core technique in calculus for recovering a function from its derivative.

Graphing Functions

Graphing a function provides a visual representation of the relationship between variables, such as quantity and marginal revenue. Plotting the marginal revenue function helps in understanding its behavior, such as slopes, intercepts, and trends over different values of quantity, making it easier to interpret and analyze the underlying economic relationships.

Differential Approximations

Differential approximations, or linear approximations, use the derivative (marginal revenue) to estimate changes in the total revenue for small changes in quantity. By applying the marginal revenue value at a specific point, one can quickly estimate the effect of selling one additional unit, which is useful in decision-making and forecasting scenarios.

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