Reference no: EM131077345
Math 121A: Homework 2-
1. Calculate the Taylor series at x = 0 of the following functions up to the term in x3.
(a) exsin x
(b) 1/1+x+x2
(c) sin(log(1 + x))
2. By considering Taylor's theorem with remainder, show that the Taylor series for sin x converges for all x.
3. (a) The Taylor series for arctangent is
tan-1x = x - x3/3 + x5/5 - x7/7 + . . . .
By evaluating the first eight terms in the series for x = 1, calculate an approximation for π.
(b) Optional for the enthusiasts. Use a computer to approximate π using the first million terms of the series in (a).
(c) Calculate (3 + i)2(7 + i) and use your result to deduce that
π/4 = 2 tan-1 1/3 + tan-1 1/7.
(d) By considering the first four terms the Taylor series for each arctangent, determine an approximation for π, and compare its accuracy with part (a).
4. The complex numbers z and w are related by
w = 1 + iz/i + z.
Write z = x + iy and w = u + iv where u, v, x, and y are real.
(a) Find expressions for u and v in terms of x and y.
(b) By writing x = tan(θ/2), or otherwise, show that if the locus of z is the real axis y = 0, -∞ < x < ∞, then the locus of w is the circle u2 + v2 = 1, with one point omitted.