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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) = ee^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.
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