Introduction: Production, Price, Demand, Revenue, \& Profit
A technology startup's market research department is tasked with determining the market viability of a new smartphone device. After suitable testing on the interest in a new smartphone, the research department determines the following price-demand equation:
$$x=3.2\times {10}^6-500p$$
where $x$ is the amount of units (smartphones) in demand at price $p$ (in dollars).
For example, if the price of the new smartphone is set at $p=\mathrm{\$}100$, then the amount of new smartphones in demand should be:
$$x=3.2\times {10}^6-500(100)=3150000\text{ units }$$
In artdition, the financial department provides the cost function measured in dollars:
$$C(x)=85x+50000$$
where $x$ is the number of smartphones produced. Note that $\mathrm{\$}50000$ is the fixed costs of production (maintenance, overhead, etc.) and $\mathrm{\$}85$ is the cost (labor, materials, marketing, transportation, storage, etc.) per smartphone.
1. (5pts) Use your work in Problem 2 to find the marginal revenue $R(x)$. Then compute and interpret the following: $R^{\prime }(500000),R^{\prime }(1600000)$, and $R^{\prime }(2300000)$. Expression for $\mathrm{R}(\mathrm{x})={\mathrm{x}}^{\ast }(6400-0.002\mathrm{x})$