Principles of Analysis

Homework Assignment #8

Due Wednesday, November 6, START of Class

Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to Bilodeau and Thie's book, the required text for the course. You should write up solutions neatly to all problems, paying particular attention to your arguments and proofs. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be your own work.

Section 3.5
Problems:   9, 12, 13

Hint: Problem #13 is a standard problem in Dynamical Systems. For part (a), apply the IVT to the function g(x) = f(x) - x. You may assume that f(0) > 0 and f(1) < 1, otherwise the proof is trivial. (Why?) For part (b), draw a picture.

Section 4.1
Problems:   1a, 5c, 5d, 9, 11a, 18b, 18c

Hint: Problem #9 is a rigorous proof that the derivative of sin x is cos x and the derivative of cos x is -sin x. For part (a), try multiplying top and bottom of the fraction by 1 + cos h. For problem #18, use the definition of the derivative carefully.

Section 4.2
Problems:   1, 4, 7

Hint: Problems #4 and #7 are focusing on the chain rule.