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Question

# The weekly demand for the Pulsar 25 color LED television is$p=600-0.05 x \quad(0 \leq x \leq 12,000)$where $p$ denotes the wholesale unit price in dollars and $x$ denotes the quantity demanded. The weekly total cost function associated with manufacturing the Pulsar 25 is given by$C(x)=0.000002 x^3-0.03 x^2+400 x+80,000$where $C(x)$ denotes the total cost incurred in producing $x$ sets.Compute $C^{\prime}(2000), R^{\prime}(2000)$, and $P^{\prime}(2000)$ and interpret your results.

Solution

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Let's solve given problem.

\begin{aligned} p &= 600 - 0.05x\quad (0\leq x\leq 12,000) \\ C(x) &= 0.000002x^3 - 0.03x^2 + 400x + 80,000 \end{aligned}

In a previous part of this exercise, we derived: $C'(x) = 0.000006x^2 - 0.06x + 400$ ; $R'(x) = 600 - 0.1x$

$C'(2000)=0.000006\cdot 2000^2-0.06\cdot 2000+400={304}$

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