Calculus 2 Science

Final Exam Answers

Fall 1998

1) arcsin(2x)

2a)

2b)

2c)

2d)

2e) p

2f) ln(x+3) - (x-1)^-1 + C

3a) Area = 71/3

3bi)

3bii)

4a) 0

4b) e³

4c) ¥

5a) Integral is -1/ln(x) so it diverges

5b) Integral is 2/3 arctan(x/3) so it converges to p /3

6a) Converges by Ratio Test

6b) Diverges using n^th term test

6c) Converges by Integral Test

6d) Diverges by Limit Comparison Test (use b_n = n^-1/2 )

7a) Not absolutely convergent, but does Converge by Alternating Series Test

7b) Absolutely Convergent and hence Convergent using Comparison with b_n = 1/n²

8a) 1/10 , 1/8 , 3/22 , 1/7

8b)

8c) 1/6

9) Converges from -5 to 1. Converges at -5, but does not converge at 1.

10a) Converges for all real numbers by Ratio Test

10b) x² - 1/6 x⁶ + 1/5! x¹⁰ - 1/7! x¹⁴ + 1/9! x¹⁸ + O(x²⁰)