If x(0)>0 x(t) is increasing if x(t)<50, and decreasing if x(t)>50, and if fact
x(t) -> 50 as t -> infty, though a bit more work is needed to prove this.
approximate dx/dt by [x(t+h)-x(t)]/h, so our finite difference approximationb) Write the finite difference approximation to the equation used in the Euler method?
x(t+h) = x(t) + h*[5*x(t) - 0.1(x^2(t))]
c) For what step size values does the diff. approximation approach a fixed point as n goes to infinity? What is the fixed point?
d) For what step size values does the diff. approximation exhibit oscillations as n goes to infinity?
P2. Derive the following two formulas for approximating the third derivative and find their error terms; which of the two is more accurate?
a) f'''(x) = [1/(h^3)][f(x+3h) - 3f(x+2h) + 3f(x+h) - f(x)]
b) f'''(x) = [1/(2h^3)][f(x+2h) - 2f(x+h) + 2f(x-h) - f(x-2h)]