A particular computing company finds that its weekly profit, in dollars, from the production and sale of $x$ laptop computers is $P(x)=-0.004 x^{3}-0.1 x^{2}+500 x-700$. Currently the company builds and sells 11 laptops weekly.
a) What is the current weekly profit?
b) How much profit would be lost if production and sales dropped to 10 laptops weekly?
c) What is the marginal profit when $x=11$ ?
d) Use the answer from part (a) and (c) to estimate the profit resulting from the production and sale of 12 laptops weekly.

Step 1 ：Let's denote the number of laptops produced and sold weekly as \(x\) and the weekly profit as \(P(x) = -0.004x^{3} - 0.1x^{2} + 500x - 700\).

Step 2 ：For part a), we need to find the current weekly profit when \(x = 11\). Substituting \(x = 11\) into the profit function, we get \(P(11) = -0.004(11)^{3} - 0.1(11)^{2} + 500(11) - 700 = 4782.576\). So, the current weekly profit is \$4782.576.

Step 3 ：For part b), we need to find the profit lost if production and sales dropped to 10 laptops weekly. First, we calculate the profit when \(x = 10\), which is \(P(10) = -0.004(10)^{3} - 0.1(10)^{2} + 500(10) - 700 = 4286.000\). The profit lost is then \(P(11) - P(10) = 4782.576 - 4286.000 = 496.576\). So, the profit lost if production and sales dropped to 10 laptops weekly is \$496.576.

Step 4 ：For part c), we need to find the marginal profit when \(x = 11\). The marginal profit is the derivative of the profit function, which is \(P'(x) = -0.012x^{2} - 0.2x + 500\). Substituting \(x = 11\) into the derivative, we get \(P'(11) = -0.012(11)^{2} - 0.2(11) + 500 = 496.348\). So, the marginal profit when \(x = 11\) is \$496.348.

Step 5 ：For part d), we use the answer from part a) and c) to estimate the profit resulting from the production and sale of 12 laptops weekly. The estimated profit is \(P(11) + P'(11) = 4782.576 + 496.348 = 5278.924\). So, the estimated profit resulting from the production and sale of 12 laptops weekly is \$5278.924.