For each of the following initial value problems, write down a formula for the solution x(t) and determine its maximal interval of existence. (a) i = x² – 4 with x(0) = 0

Consider the following functions. f₁(x) = 0, f₂(x) = x, f(x) = ex g(x) = c₁f₁(x) + C₂f₂(x) + C3f3(x) Solve for C₁, C₂, and c3 so that g(x) = 0 on the interval (-∞, ∞). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution [0, 0, 0}.) ={1 {C₁, C₂, C3} = Determine whether f₁, f₂, f3 are linearly independent on the interval (-∞, ∞). O linearly dependent O linearly independent Need Help? Read It

How would you find the absolute maximum and minimum values of each function over the indicated interval, and indicate the X-values at which they occur? This is for #16

Sand is being dumped from a hopper railcar at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other (both are growing larger as more sand is dumped). How fast is the base diameter increasing when the base is 18 feet long? Round your answer to three decimal places. ft/min