 ### MAT 513 Problem Sets Analysis for Teachers I

Spring 2016

There will be 10 weekly problem sets. Each problem set is due at the beginning of lecture on Thursday. The lowest problem set score from the semester will be dropped when computing final course grades. Problems due one week will often practice material covered in previous weeks. It is important to do the problems in a timely manner; do not postpone problems until the last minute.

### UNDER NO CIRCUMSTANCES WILL LATE HOMEWORK BE ACCEPTED.

In exceptional circumstances, such as a documented medical absence, etc., the instructor will excuse the missing problem set and compute the problem set total using the remaining problem sets. Problem sets missed because of work-related absences will not be excused.

You are encouraged to work with other students in the class, but your final write-up must be your own and must be based on your own understanding. Moreover, write-ups must be legible, must be written in correct, understandable English, and must adhere to the usual standards of rigor for an advanced mathematics course. Write-ups which prove too difficult to read or understand may be marked incorrect and / or may be returned to the student to be rewritten. All questions regarding grading of a problem set should be addressed to the instructor in a timely fashion.

Some of the problems below are just for practice. It is a good idea to know how to solve these problems; similar problems may appear in the weekly quizzes.

### PLEASE ONLY WRITE UP AND TURN IN THE RED PROBLEMS.

• Problem Set 1 is due Thursday, February 4
Section 3.1, pp. 108—113. 3, 4, 7, 14, 15.
Section 3.2, pp. 120—122. 3, 4, 8, 9.
Also write a few paragraphs responding to the following hypothetical student question: "By the Archimedean Property / Density Property, to within the precision of our measuring devices, every real number measurement is well-approximated by a fraction. Thus, in science and engineering, it suffices to work only with real numbers that are fractions."
• Problem Set 2 is due Thursday, February 18
Section 3.2, pp. 120—122. 2, 10, 11.
Section 3.3, pp. 131—134. 3(d),(f),(h),(k),(n), 3(a),(b),(c),(e),(g),(i),(j),(l),(m), 8, 11, 13, 16.
Section 3.4, pp. 140—143. 3, 4, 5.
Also write a paragraph or so responding to the following hypothetical student question: "Since the decimal representation 0.999... has the property that each of its finite truncations is strictly less than 1, also the real number with this decimal representation is strictly less than 1." Please make some connection between your answer and the Archimedean Property.
• Problem Set 3 is due Thursday, February 25
Section 3.4, pp. 140—143. 6, 7, 10, 13, 19, 22.
Section 3.5, pp. 148—151. 3, 4, 11, 12.
Also write a paragraph or so explaining how the Nested Intervals Theorem implies that every decimal expansion determines a unique real number (even though some real numbers have more than one decimal expansion). For the Nested Interval Theorem, consider "decimal intervals" of the form A_n = [a_n/10^n, (a_n+1)/10^n], where each a_n is an integer.
• Problem Set 4 is due Thursday, March 3
Section 3.4, pp. 140 — 143. 14, 20, 21.
Section 3.5, pp. 148 — 151. 5, 6, 7, 8.
Section 4.1, pp. 169 — 170. 6 only (a),(c),(e), 7, 13.
Also write a paragraph or so explaining how the notion of a convergent sequence explains the (apparent) paradox of Zeno. For Achilles, A, who moves twice as fast as the tortoise, T, what can you say about the sequence of times (x_n) defined by A(x_n) = T(x_{n-1}), assuming x_0 = 0, A(0) = 0, and T(0) = 1. Is this sequence convergent?
• Problem Set 5 is due Thursday, March 24
Section 4.1, pp. 169—170. 3, 4, 5, 9, 12.
Section 4.2, pp. 177—179. 3, 4, 5 only (d) and (g), 15 only (a) and (c), 17, 18.
Section 4.3, pp. 184—186. 3 only (a) and (d), 9, 10, 13, 15.
Also write a paragraph or so explaining why the Bolzano-Weierstrass theorem implies (and, in fact, is equivalent to) the following result: for every function f defined on a bounded, closed interval [a,b], if the values of f are not bounded above, then there exists a sequence (x_n) in [a,b] that converges to a limit x such that (f(x_n)) is an unbounded, increasing sequence.
• Problem Set 6 is due Thursday, March 31
Section 5.1, pp. 203—305. 7, 8, 9 only (a) and (c), 20.
Section 5.2, pp. 212—214. 6 13.
Section 5.3, pp. 220—221. 5, 6, 7, 15.
Please read carefully the statement of Problem 7 on p. 220, as well as example 5.3.8. Then write a paragraph or so explaining the following (somewhat surprising) corollary of the Intermediate Value Theorem: assuming that the temperature f(x) of a point x on the equator of the Earth is a continuous function, at every moment there exists some point x on the equator such that for its opposite point -x (through the center of the Earth), f(-x) equals f(x), i.e., the opposite points have the same temperature.
• Problem Set 7 is due Thursday, April 7
Section 5.3, pp. 220—221. 8, 9, 13, 16.
Section 5.4, pp. 227—229. 3 only (f) and (h), 5, 10, 11.
Section 6.1, pp. 245—248. 4 only (b) and (d), 6, 8, 10, 12, 18.
Please write a paragraph or so explaining the following geometric meaning of the tangent line to a curve C at a point p: if the tangent line L exists, it is the unique line such that the angle formed by L with the secant line segment connecting p to a distinct point q of C limits to zero as the point q limits to p.
• Problem Set 8 is due Thursday, April 14
Section 6.2, pp. 255—258. 3, 4, 5 only (a) and (d), 6, 9, 11, 14, 20.
Section 6.3, pp. 263—266. 4 only (b) and (g), 6.
Please write a paragraph explaining the intuitive meaning of the Mean Value Theorem to an educated person who does not have a strong background in mathematics.
• Problem Set 9 is due Thursday, April 28
Section 6.4, pp. 272—274. 3, 5, 9.
Section 7.1, pp. 281—283. 5, 6.
Section 7.2, pp. 290—292. 5 6, 11.
Section 7.3, pp. 298—300. 5 only (a) and (d), 14, 19.
• Problem Set 10 is due Thursday, May 5
Section 7.3, pp. 298—300. 6 only (a) and (c), 15.
Section 8.1, pp. 307—309. 3, 4 only (a) and (c), 5 only (b), (e), and (k) (please confer Example 8.1.1).
Section 8.2, pp. 316—319. 3 only (b) and (p), 5.
Please write a paragraph explaining the intuitive heuristic underlying the Fundamental Theorem of Calculus: for the function F of x that measures the area under the graph of f from a to x, the derivative of this function at x equals the height y = f(x) of the graph at x, since the approximate change in F from x to x+dx equals the area of the rectangle with base dx and height y.

Jason Starr
4-108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651
Phone: 631-632-8270
Fax: 631-632-7631
Jason Starr