Completely anwer all the questions.

1. Consider the line l that passes through the point (0, ff). l is perpendicular to the curve y = 23:2 at a point in the rst quadrant. What is the equation of l? 2. Legendre's equation has the following form: (1 1:2)y\" 2my' + l(l + 1)y = 0. Here, l is an integer. Rodrigues' formula states that the following is a solution for y: 1 d1 _ __ 2 _ z 131(32) ' 211! dxl (a: 1) ' We will verify the formula in the following fashion: let v(:r) = (1:2 1)l. First show that: d (:1:2 1) v($) = 21130013). d1: Now differentiate the above l + 1 times using Leibniz rule, and then simplify to get: div div div (1x2)(d;L'l) _2x(d$l) +l(l+1)dml =0. (Hint: Use the Leibniz formula directly. I will not accept your solution if you reach the conclusion by looking for a pattern.) . You are the manager of an apartment complex with 50 apartments. When you set rent at $800 per month all apartments are rented. As you increase rent by $25 per month one fewer apartment is rented. Maintenance costs run $50 per month for each occupied apartment. What is the rent that maximizes the total amount of prot