Determine the largest interval (a,b) for which Theorem 1 guarantees the existence of a unique solution on (a,b) to the initial value problem below. ху" - 5y' + eХу%3х -3, у(8)-1, у'(8) -D0, у" (8)-2

Suppose that for a company manufacturing calculators, the cost, revenue, and profit equations are given by x² 20' C = 90,000+ 40x, R=200x - where the production output in 1 week is x calculators. If production is increasing at a rate of 400 calculators per week when production output is 6,000 calculators. Find the rate of increase (decrease) in cost, revenue, and profit. A) Costs are increasing at the rate of $ (Simplify your answer.) B) Revenue is (Simplify your answer.) C) Profits are (Simplify your answer.) at the rate of $ P=R-C at the rate of $ per week at this production level. per week at this production level. per week at this production level.

4) The estimated cost of producing x bikes is given by ((x) = 0.05x² + 6x + 9000. Ya find and interpret (`(450) Ub) find the actual cost of manufacturing the 451st bike and compare with C(450)

Solve the initial value problem y = []» ym = [*.] -7 4 У, -6 7 у, у(0) — || -4