Let $R(x), C(x)$, and $P(x)$ be, respectively, the revenue, cost, and profit, in dollars, from the production and sale of $x$ items. If $R(x)=4 x$ and $C(x)=0.004 x^{2}+1.2 x+60$, find each of the following. a) $P(x)$ b) $R(200), C(200)$, and $P(200)$ c) $R^{\prime}(x), C^{\prime}(x)$, and $P^{\prime}(x)$ d) $R^{\prime}(200), C^{\prime}(200)$, and $P^{\prime}(200)$ a) $P(x)=-0.004 x^{2}+2.8 x-60$ (Use integers or decimals for any numbers in the expression.) b) $\mathrm{R}(200)=\$$ (Type an integer or a decimal.)

Step 1 ：First, we know that profit is calculated by subtracting cost from revenue. So, we have $P(x) = R(x) - C(x)$.

Step 2 ：Substitute $R(x) = 4x$ and $C(x) = 0.004x^2 + 1.2x + 60$ into the equation, we get $P(x) = 4x - (0.004x^2 + 1.2x + 60)$.

Step 3 ：Simplify the equation, we get $P(x) = -0.004x^2 + 2.8x - 60$.

Step 4 ：For part b, we need to substitute $x = 200$ into $R(x)$, $C(x)$, and $P(x)$.

Step 5 ：Substitute $x = 200$ into $R(x) = 4x$, we get $R(200) = 4*200 = 800$.

Step 6 ：Substitute $x = 200$ into $C(x) = 0.004x^2 + 1.2x + 60$, we get $C(200) = 0.004*200^2 + 1.2*200 + 60 = 260$.

Step 7 ：Substitute $x = 200$ into $P(x) = -0.004x^2 + 2.8x - 60$, we get $P(200) = -0.004*200^2 + 2.8*200 - 60 = 540$.

Step 8 ：For part c, we need to find the derivative of $R(x)$, $C(x)$, and $P(x)$.

Step 9 ：The derivative of $R(x) = 4x$ is $R'(x) = 4$.

Step 10 ：The derivative of $C(x) = 0.004x^2 + 1.2x + 60$ is $C'(x) = 0.008x + 1.2$.

Step 11 ：The derivative of $P(x) = -0.004x^2 + 2.8x - 60$ is $P'(x) = -0.008x + 2.8$.

Step 12 ：For part d, we need to substitute $x = 200$ into $R'(x)$, $C'(x)$, and $P'(x)$.

Step 13 ：Substitute $x = 200$ into $R'(x) = 4$, we get $R'(200) = 4$.

Step 14 ：Substitute $x = 200$ into $C'(x) = 0.008x + 1.2$, we get $C'(200) = 0.008*200 + 1.2 = 2.8$.

Step 15 ：Substitute $x = 200$ into $P'(x) = -0.008x + 2.8$, we get $P'(200) = -0.008*200 + 2.8 = 1.2$.