Principles of Analysis

Homework Assignment #8

Due Wednesday, March 27, 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 3.4
Problems:   1 (parts b, c, g, l), 11b, 11d

Hint: When computing the necessary limits in problem #1, try rearranging the expression to make use of limits we already know. For example, in 1a (not assigned) you could multiply top and bottom of the fraction by 2, and write a(x) = 2 times (sin 2x)/(2x). We then can apply the Big Limit Theorem to conclude lim x -> 0 of a(x) = 2.

Section 3.5
Problems:   6, 12, 13, 17

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. Remember that for the True/False questions, if its True, give a proof, if its False, give a counterexample.

Section 4.1
Problems:   1b, 5a, 5d, 9, 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.