Principles of Analysis

Homework Assignment #9

Due Monday, April 8, 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 4.1
Problems:   11, 13

Hint: For these problems, start by determining what must be true to make the function continuous, then what must be true to make them differentiable, etc. Try defining the derivative as a piecewise function. A picture may be helpful as well.

Section 4.2
Problems:   1, 4, 7

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

Section 4.3
Problems:   1, 11a, 11c, 11e, 12, 17, 18, 19a

Hint: Problems #17, 18 and 19a are all nice applications of the Mean Value Theorem (or Rolle's Theorem depending on the problem). Try using proof by contradiction for numbers 17 and 18. For example, on #17, if there is another root r in [a,b], what does Rolle's Theorem applied to the interval [a,r] tell us?

Note: Problem #18 suggests a nice extra credit problem, proved in most dynamical systems courses. See the Extra Credit problems for details.