# Answers for the June 2001 exam

1a: x e^x - e^x + c
1b:  (13 sqrt{13} - 8)/27
1c: the unit vector is (1/sqrt{2}, 1/sqrt{2})

2a: y = -5/3 + c exp(3x)
2b: y = c/sqrt(1+x^2) + 1 - 1/sqrt(1+x^2) log( (1 + sqrt(1+x^2))/|x| )

3a:    P_5 = x + x^2 + x^3 + x^4 + x^5
3b:    [-1, 1)
3c:    0.3313

4b:    8
4c:    3 pi/ 2

5: (0,0) is a saddle point, (1/2, 1/ sqrt{2}) and (1/2, -1/ sqrt{2}) are both local minima

6a: Diverges, by the limit comparison test.
6b: converges by the direct comparison test.  We know that 0 <= sin(x) <= x if x is in [0,pi].  So 0 <= x sin(x) <= x^2 if x is in [0,pi].  Let x = 1/n.  Since n >= 1, we know that x is in (0,pi].  This shows that 0 <= 1/n sin(1/n) <= 1/n^2.  And we're done by the direct comparison test because weknow that the series with c_n = 1/n^2 converges.
6c: converges by ratio test

7. (This is a homework question right out of the book: #12 of section  11.13)
7a:  x^2 - 2 x^4 + 3 x^6 - 4 x^8 + ... = - sum_{n=1}^infinity   (-1)^n  n x^{2n}
7b:    - 9/100

8: 1 - pi/4 + 1/2 ln 2  (or 1 - pi/4 - ln(1/sqrt(2)) )