Unformatted text preview: Mid Term October 2010 Math 222 FALL 1. Determine (with justiﬁcation) in each case whether the series diverges or converges — conditionally or absolutely: (a) (b) (c) (d)
1+2+4 1+2+4+8 1+2 1+3 + 1+3+9 + 1+3+9+27 + · · · ; ∞ √1 n=200 n+ n+log n ; ∞ ∞ 1n = n=100 (−1)n n=100 −1 + n ∞ n (2n)! n=10 (−1) n!(2n)n 1− 1n n ; . 2. With the aid of known standard power series expansion, give the power series (powers of x) for the following functions, and in each case give the interval of convergence: (a) x2 cos x; (b)
1 x sin(x3 ); x , giving 2+x (c) Obtain the Taylor series in powers of x + 1 for f (x) = the general term and the interval of convergence. 3. Let L1 and L2 denote the lines (a) Find the shortest distance between the lines L1 : OR1 = 2i−3j+k+t(2i−3j+k) and L2 : OR2 = −i+3j+2k+u(i+k) (b) Find the equation of the plane parallel to the above two lines and containing the point (3, −1, 4) 4. Consider the trajectory of a particle whose position at time t ≥ 0 is given by r(t) = −2t, −t2 , 1 t3 . 3 (a) Find the velocity v(t), acceleration a(t) and speed (b) Find the curvature κ(t). (c) Find the distance traveled between t = 0 and t = 3. (d) Find the tangential and normal components of the acceleration vector.
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