Calculus II, Spring 2024 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework 02

1. Section 8.8: Problems: 35, 39, 41, 45, 55, 65, 66.

2. Section 8.8:

Let f(x) = \frac{1}{x^{0.5} ln x}. Is the improper integral ∫_{e}^{∞} f(x) dx convergent? Hint: Try not to find the anti-derivative of f(x), but instead find a suitable g(x) for comparison using Theorem 2 or Theorem 3. See also Example 5-7 of Lecture 02 on how to find g(x).

3. Section 8.8:

Do the same for ∫0^{\frac{1}{e}} \frac{1}{x^{2}| ln x|} dx.

4. Section 10.1: Problems 46, 53, 59, 63, 67, 69, 73, 81, 87, 89. Hint for problem 73: Read Appendix A.5, ”Limit 6”. Hint for problem 87: Multiply by \frac{n + \sqrt{n^{2} − n}}{n + \sqrt{n^{2} − n}}. Hint for problem 89: Use Theorem 4 and Fundamental Theorem of Calculus + L’Hôpital’s Rule.

5. Section 10.2: Problems 31, 33, 43, 61, 63, 65, 71, 78.