
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 workrelated absences will not be excused.
You are encouraged to work with other students in the class, but your
final writeup must be your own and must be based on your own
understanding. Moreover, writeups must be legible, must be written
in correct, understandable English, and must adhere to the usual
standards of rigor for an advanced mathematics course.
Writeups
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 wellapproximated 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_{n1}), 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 BolzanoWeierstrass 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.
Back to my home page.
Jason Starr
4108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 117943651
Phone: 6316328270
Fax: 6316327631
Jason Starr