Optimizing with derivatives Question 1. The problem statement, all variables and given/known data You're on the moon and you need a bit from the base 900 feet away so you can continue your experiment. The base has a roover/launcher used for retrieving the bit. The roover can move a max dist of 600 feet. The launcher is controled by angle. At this time the angle is inclined at 70 degrees. It cost $375 per foot to move the roover, and $20,000 per degree to change the launch angle. What are the optimal settings (what angle should I launch from where) to minimize cost? acceleration due to grav on moon = 5.31 ft/s^2 initial velocity of bit leaving launcher = 75 ft/s 2. Relevant equations R = V^2(sin(2x))/g ; where R = total distance, V= Velocity, x= theta (some angle) and g= gravitational acceleration. 3. The attempt at a solution I made an equation for constraint and pne for the optimaization. Constraint: 900 = [[(75^2)*sin(2x)]/5.31]+ y ; where y = distance roover moves and x = theta optimization minCost = [20000*|70-x|]+2y(375) ; |...| = absolute value, 2y because the roover has to return to base. I remember doing problems like this in HS but I need help with this one. I solved for y in the constraint equation, then substiituted it in for y in the optimization equation. Here's what I got: minCost= [-7.945*(10^5)*sin(2x)+675000+20000*|x-70|] Now if im correct I get the 1st derivative of this and I will get the extrema or something like that which will be the optimal angle. Then I plug that into the first equation and get y.... But how do I find the first derivative. This might not even be the right method so please help thatnks.