1. Consider the differential equation dy/dx = 3(x^2) / e^2y
(a) Find a solution for y = f(x) to the differential equation satisfying f(0) = ½.
(b) Find the domain and range of the function f found in part (a). Justify your answer.
2. Suppose that the function f has a continuous second derivative for all x, and that f(0) = 2, f ‘ (0) = -3, and f “ (0) = 0. Let g be a function whose derivative is given by g'(x) = e^-2x(3f(x) + 2f'(x)) for all x.
(a) Write an equation of the line tangent to the graph of f at the point where x = 0.
(c) Given that g(0) = 4, write an equation of the line tangent to the graph of g at the point where x = 0.
(d) Show that g''(x) = e^-2x(-6f(x) 2f''(x)) At x = 0, is g ‘ (x) = 0 and g “ (x) < 0? Justify your answer.
I apologize for not thanking whoever it was that helped me with the last questions i had. I ended up checking my answers with my friend, and kind of forgot to check here.