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

Brief answer to selected problems in HW02

1. Section 2.3: problem 57(b). Take any ϵ < 1 will do.

2. Section 2.5:

Problem 64: If f is discontinuous at g(0), f(g(x)) may still be discontinuous at x = 0. No contradiction Problem 67: The function g(x) = f(x) − x satisfies g(0) · g(1) ≤ 0. If the equality holds, then c is found. Otherwise, apply the Intermediate Value Theorem to g(x) on [0, 1]. Problem 77: The function f(x) = cos(x) − x satisfies f(0) · f(1) < 0.

3. Section 2.6:

Problem 92: If B > 0, take δ = 1/\sqrt{B} will do. If B ≤ 0, take δ to be any positive real number (for example δ = 1) will do. Problem 93: (a): change ’x_{0} < x < x_{0} + δ’ to ’x_{0} − δ < x < x_{0}’. (b): change ’f(x) > B to f(x) < B (or f(x) < −B). (c): apply both changes in (a) and (b).