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)) )