Assignment 8 find the taylor polynomials of the indicated

Question

Assignment 8-

1. (a) Prove that a sphere with radius r has volume 4/3 πr³.

(b) Determine the volume of the solid obtained by rotating, about the y-axis, the region in the first quadrant bounded by the curves y = x and y = x².

2. Find the Taylor polynomials (of the indicated degree and at the indicated point) for the following functions.

(a) f(x) = e^{e^x}; degree 3, at 0.

(b) f(x) = 1/1+x²; degree 2n + 1, at 0.

(c) f(x) = cos(x), degree 2n, at π/2.

3. (a) Find the Taylor polynomial of degree n for the function f(x) = log(1 - x) at x = 0 and use this to evaluate log(5/4) with an error of at most 10^-4.

(b) Show that for x > 0,

|3√(1 + x) - 1 - x/3 + x²/9| ≤ 5x³/81.