Calculus I, Fall 2013 (http://www.math.nthu.edu.tw/˜ wangwc/)

Brief answer to selected problems in Homework 14

1. Section 7.4 Problem 8: Note that ln 2 < 1. Answer (from slow to fast growth): (ln 2)^{x}, x^{2}, 2^{x} ≈ e^{x}. Problem 24: Answer (from slow to fast growth): (log_{2} n)^{2}, \sqrt{n} log_{2} n, n. The slower growth rate, the more efficient.

2. Section 8.1: Problem 65: Define F (x) = ∫_{x}^{b} f(t)dt. It follows that F (b) = 0, F′(x) = −f(x). Then apply intgegration by parts to ∫_{a}^{b} F (x)dx.

3. Section 8.2:

Problem 28: With the hint, = ∫0^{\frac{1}{2}} \frac{ds}{\sqrt{1 − s}} = 2 − \sqrt{2}.

Problem 34: Sol 1: ∫ sec x tan^{2} xdx = ∫ tan xd sec x = tan x sec x − ∫ sec^{3} xdx

= tan x sec x − ∫ sec x tan^{2} xdx − ∫ sec xdx.

Sol 2: ∫ sec x tan^{2} xdx = ∫ \frac{sin^{2} x}{cos^{3} x}dx = ∫ \frac{s^{2}ds}{(1 − s^{2})^{2}} . Then apply technique of partial fractions in Section 8.4.