Use Taylor’s theorem, retaining powers of x − x0 sufficiently large to approximate the val...

Use Taylor’s theorem, retaining powers of x − x0 sufficiently large to approximate the values of y accurately to two decimal places on the given interval using the prescribed increments in x. In Exercises 1 through 6, compare the estimated values with the correct values obtained by solving the problem exactly using elementary methods.

Use Taylor’s series to determine to three places the value of the solution of the problem

y' = − xy2; when x = 0, y = 1,

for x = 0.1, 0.2, and 0.3. Compare your results with the values obtained by solving the problem by elementary means.