MTH 4/585 Index

MTH 4/585 Course Web Page

Fall 2009

Week 1: The Real Numbers

1. No assignments prior to class.
2. Class Notes
3. Homework Problems from Chapter 2 in the Usiskin book: You should be able to complete and reason through all the problems that have solutions in the book (the solutions are presented right after the chapter problems). Largely, these are procedural or practice. You should be able to complete the following:
    p. 25, #5, 6, 7, 8, 9, 12
    p. 33, #2, 6
    p. 39, # 4, 5, 9
    p. 46, #3, 4, 5, 7, 8, 9, 10
(We will discuss some of these in class next week.)

Write up to hand in p. 33 #6 and one of: p. 25 #5 or p. 46 #3 or p. 46 #4. You will hand in this assignment at the beginning of class on Thursday, October 8.

4. Good Projects: (p. 63): 1, 2, 3

5. Associated Readings/Critical Writing (all of these are available electronically in journals through the PSU library):
· Sirotic, N and Zazkis, A (2007) Irrational numbers: The gap between formal and intuitive knowledge. Educational Studies in Mathematics, 65 (1) 49-76.

· Tall, D.O. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48(2–3), 199–238.

If you choose to read and write a 3-4 page paper on one of these readings, the paper is due (by email) by Tuesday, October 6. See the syllabus for further instructions and details on the critical writing assignments.

Week 2: Functions

1. Pre-class assignment: Read Chapter 3 in the Usiskin textbook. Send me a one-page reflection on your reading (via email) by Tuesday, October 6.
2. Class Notes
3. Homework Problems from Chapter 3 in the Usiskin book: You should be able to complete all problems that have solutions and the following problems:
    p. 75, #1, 2, 3, 4, 5, 9
    p. 79, #3, 4, 5, 7
    p. 85, #1, 3
    p. 94, # 2, 3, 4, 5, 7, 8, 9, 10, 11, 13
    p. 100, #4, 6, 9, 10, 13
    p. 105, #2, 3, 4, 5, 6, 7, 11

Also: Take a look in a calculus or pre-calculus book (or books, if you have access to more than one). Record how function is defined. Record how inverse is defined.
(We will discuss some of these in class.)

Write up to hand in p. 79 #7, p. 94 # 7, p. 100 #13, and one of the following: p. 75 #5, p. 85 #1, p. 100 #4. You will hand in this assignment at the beginning of class on Thursday, October 22.

4. Good Projects: (p. 130): 1, 3, 4, 5, 6

5. Associated Readings/Critical Writing (all of these are available electronically in journals through the PSU library):
· Breidenbach, D., Dubinsky, E., Hawks, J., and Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23, 247-285.
Notes: A more in-depth description of the action and process view of function described in class.

· Kleiner, I. (1989). Evolution of the function concept: A brief survey. The College Mathematics Journal, 20(4), 282-300.
Notes: An historical analysis of how function developed throughout mathematical problem solving and thought from 4000 years ago.

· Schoenfeld, A. and Arcavi, A. (1988). On the meaning of variable. Mathematics Teacher, 81(6), 420-427.
Notes: Describes 10 different definitions of variable between 1710 and 1984 and comments on implications on student learning.

· Schwartz, B. and Hershkowitz, R. (1999). Prototypes: Brakes or levers in learning the function concept? The role of computer tools. Journal for Research in Mathematics Education, 30(4), 362-389.
Notes: More theoretical writing than some of the other papers. Investigates the role of technology in learning function.

If you choose to read and write a 3-4 page paper on one of these readings, the paper is due (by email) by Tuesday, October 22..

Week 3: Limits and Continuity

1. Pre-class assignment: Review Chapter 3, Section 3.2.4. No reflection due this week.
2. In addition to the problems assigned in Week 2, write up one of the following: p. 112, #8, 10, 11. Hand this in with your assignment due Thursday, October 22.

Week 4: Equation Solving, the Intermediate Value Theorem, and Continuity

1. Pre-class assignment: Read Section 4.3 in the Usiskin textbook (you may find Sections 4.1 and 4.2 a helpful review of some algebraic topics, but we won't discuss these two sections). Send me a one-page reflection on your reading (via email) by Tuesday, October 20.

2. Class Notes

3. Homework Problems from Chapter 4 in the Usiskin book: You should be able to complete all problems that have solutions and the following problems:
    p. 166, #1, 3
    p. 173, #3, 4, 5

Write up to hand in p. 166 #1(f) and from the list below: One of (a) or (b); (c); (d). You will hand in this assignment at the beginning of class on Thursday, October 29.

a) Prove that the function defined by f(x) = sin(1/x), x not equal to 0; f(0) = 0 is not continuous at x = 0 (we began a discussion of this in class in small groups).
b) Prove that the function defined by f(x) = 1 if x is rational and 0 if x is irrational is not continuous at any x.
c) At what values of x is the following function continuous? Justify your answer.
    f(x) = 0 if x is irrational and f(x) = sin (x) if x is rational
d) If f is continuous on [a, b], is |f| also continuous on [a, b]? Prove or show a counterexample. Is the converse true? Again, prove or show a counterexample.

4. Good Projects: (p. 178): 2, 3, 6

5. Associated Readings/Critical Writing (all of these are available electronically in journals through the PSU library):

·Tall, D. and Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169.

·Szydlik, J. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31, 258- 276.

·Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International journal of mathematical education in science and technology, 32, 487-500.

·Juter, K. (2006). Limits of functions as they developed through time and as students learn them today. Mathematical thinking and learning, 8, 407-431.

·Conference paper by Craig Swinyard, available at http://rume.org/crume2009/Swinyard_LONG.pdf

If you choose to read and write a 3-4 page paper on one of these readings, the paper is due (by email) by Tuesday, October 27.

Week 5: The Intermediate Value Theorem, The Mean Value Theorem, and Average Rates

1. Pre-class assignment: Bring to class one or more definitions of function and inverse function from a calculus or pre-calculus textbook. Review Theorem 4.12 on p. 171 of the Usiskin text and Section 4.3.4 (Extended analysis of average speeds). No reflection due this week.

2. Class Notes

3. Homework Problems. You should be able to complete all of the following problems (a) - (d). Choose one of (a) or (c) and one of (b) or (d) to hand in (two problems total: one must be either problem (a) or problem (c) and the other must be either problem (b) or problem (d)). You will hand in this assignment at the beginning of class on Thursday, November 5.

(a) Prove that the equation (1 - x) cos(x) = sin(x) has at least one solution in (0, 1).
(b) Explain why the hypothesis conditions are necessary in the statement of the Modified Converse to the Intermediate Value Theorem (see Class Notes for the Theorem statement): (i) f is piecewise monotonic and (ii) f satisfies the intermediate value property on [a, b].
(c) Let r be a number contained in each of a sequence of nested intervals, [an, bn]. Suppose the width of the intervals goes to 0 as n approaches infinity. Prove that r is unique.
(d) Show that the version of the Intermediate Value Theorem proved in class (see Class Notes) is equivalent to the following statement of the IVT:

Let f be a continuous function on [a, b] and suppose f(a) > 0 and f(b) < 0. Then there exists z in (a, b) such that f(z) = 0.

4. Good Projects: Find at least two of Cauchy's proofs of the Mean Value Theorem and discuss their shortcomings.

5. Associated Readings/Critical Writing (all of these are available electronically in journals through the PSU library):

Schorr, R. (2003). Motion, speed, and other ideas that "should be put into books." The Journal of Mathematical Behavior, 22, 265-477.
Reports on middle school students' learning of linear and quadratic functions with the SimCalc software.

Schnepp, M. & D. Chazan (2004). Incorporating experiences of motion into a calculus classroom. [videopaper, no page numbers]. Educational Studies in Mathematics, 57(3).

Cirillo, M. (2007). Humanizing calculus. The Mathematics Teacher, 101 (1), 23-?
You will need to get a hard copy of this article at the PSU Library or in the Curriculum Resource Lab, Neuberger Hall, room 305. If you are a member of NCTM, you can download this from the NCTM website with your login and password.

Tall, D. and Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169. (The citation for this article was incorrect last week.)

If you choose to read and write a 3-4 page paper on one of these readings, the paper is due (by email) by Tuesday, November 3.

Week 6: Mean Value Theorem and Trigonometric Functions

1. Pre-class assignment: Read Sections 9.1 and 9.2 in the Usiskin text. Send me a one-page reflection on your reading (via email) by Tuesday, November 3.

2. Class Notes

3. Homework Problems. You should be able to complete all of the problems below.

Write up to hand in two problems from the list below: You will hand in this assignment at the beginning of class on Thursday, November 12.

(a) (i) Provide two proofs of the Increasing Function Theorem (see Class Notes for the statement of this Theorem): one that relies on the Mean Value Theorem and one that does not. (ii) Explain why, in the hypothesis of the theorem f must be continuous on a closed interval and differentiable on an open interval. Why does the conclusion of the theorem have f'(x) >0 on an open interval and f increasing on a closed interval?
(b) Prove that the least upper bound of a set is unique if it exists.
(c) Find examples f and [a, b] having exactly one Rolle's Theorem point c, exactly two, and infinitely many. Find an "infinitely many" case so the set of c is not a closed interval.
(d) Suppose that f and g each satisfy the hypotheses of the Mean Value Theorem on [a, b] and that f(a) = g(a) and f(b) = g(b). Prove that there exists a c in (a, b) at which f'(c) = g'(c). Interpret your result graphically.
(e) Prove the Generalized Mean Value Theorem (also known as Cauchy's Theorem or Formula; see Class Notes for the statement of this Theorem). Explain why the hypotheses of the theorem are necessary: (1) f and g are continuous on (a, b); (2) f and g are differentiable on [a, b], and (3) g'(x) is not equal to zero on (a, b).

4. Good Projects: p. 475-476 #2, #5

5. Associated Readings/Critical Writing (all of these are available electronically in journals through the PSU library):

Weber, K. (2005). Students' understanding of trigonometric functions. Mathematics Education Research Journal, 17, 91-112. Download a copy here.

George Sweeny's paper on students interpretation of trigonometric functions in realistic settings, from the 2009 Research in Undergraduate Mathematics Education Conference Proceedings, available at http://rume.org/crume2009/Sweeney_LONG.pdf .

If you choose to read and write a 3-4 page paper on one of these readings, the paper is due (by email) by Tuesday, November 10.

Week 7: Trigonometric Functions

1. No pre-class assignment.

2. Class Notes

3. Homework Problems from Chapter 9 in the Usiskin book: You should be able to complete all problems that have solutions and the following problems:
p. 439 #4
p. 449 # 2, 3, 4, 5

Write up to hand in p. 449 #2 and 3. You will hand in this assignment at the beginning of class on Thursday, November 19.

4. Good Projects: see previous week.

5. For the critical writing assignment due on Tuesday, November 17, you may choose an article from one of the previous week's assignments for which you have not yet submitted a critical response.

Week 8: The Derivative

1. Pre-class assignment: In a calculus book, look up the introduction to derivatives. Send me a one-page description of what the introduction to derivative entails, including what types of functions are used as examples. Send your description to me via email by Tuesday, November 17.

2. Class Notes

3. Homework Problems. You should be able to complete all of the problems below. Please write up all of these problems to turn in on Thursday, December 3.

(a) You are the teacher in a calculus class. You give a quiz in which one of the questions is "Find the derivative of f(x) = x(squared) at x = 3." One of your students writes:
    f(3) = 3(squared) = 9, d/dx(9) = 0.

Write a response to this student.

(b) (i) One definition of derivative is f'(a) = lim (x -->a) [f(x) - f(a)/ (x - a)]. Interpret this definition graphically. That is, create a picture to "explain" this definition, making sure to account for x, a, f, and the limit.

(ii) How does the above definition of derivative relate to this other one: f'(a) = lim (h -->0) [f(a + h) - f(a)]/h? Show that you can derive one from the other.

(iii) How does f'(x) = lim (h -->0) [f(x + h) - f(x)]/h relate to the two limit definitions of derivative above?

(c) Generate an example or explain why it is impossible to have:
    (i) a function that is differentiable and continuous
    (ii) a function that is differentiable and not continuous
    (iii) a function that is not differentiable and continuous
    (iv) a function that is not differentiable and not continuous

Consider differentiability and continuity over some domain (not at a point) as you answer this question.

(d) If f and g are two differentiable functions, is f + g also differentiable? Prove that it is or show a counterexample. If it is differentiable, is the result f' + g', (f + g)', something else? Does your result extend to finite sums of differentiable functions?

Do some digging around in calculus and analysis books and on the web to answer the question: Can we differentiate an infinite series term-by-term? If so, under what conditions can we do this?

4. Associated Readings/Critical Writing:

Lauten, A. D., Graham, K., and Ferrini-Mundy, J. (1994). Student understanding of basic calculus concepts: Interaction with the graphics calculator. Journal of Mathematical Behavior 13 (1994), pp. 225–237.

Marrongelle, K. (2005). Enhancing meaning in mathematics: Drawing on what students know about the physical world. The Mathematics Teacher, 99, 162-272. Available here.

Nemirovsky, R., and Rubin, A. (1992). Students' tendency to assume a relationship between a function and its derivative. Cambridge, MA: TERC. Available here.

Orton, A. (1983). Student's understanding of differentiation. Educational Studies in Mathematics 14 (1983), pp. 235–250.

If you choose to read and write a 3-4 page paper on one of these readings, the paper is due (by email) by Tuesday, December 1.

Week 9: No Class - Thanksgiving

Week 10: Area, Volume, and the Integral

1. Pre-class assignment: Read Sections 10.1 and 10.2 in the Usiskin text, paying particular attention to section 10.1.1 (pp. 477-484), 10.1.4 (pp. 500-504), 10.2.1 (pp. 511-514), and 10.2.3 (pp. 523-527). No reflection on this reading assignment is due.

2. Class Notes

3. Associated Readings/Critical Writing:

Dick, T., Dion, G., and Wright, C. (2003). Case study: Advanced placement calculus in the age of the computer algebra system. The Mathematics Teacher, 96, 588-596.

Engelke, N. and Sealy, V. (2009). The great gorilla jump: A Riemann Sum investigation. Paper presented at the Twelfth Conference on Research in Undergraduate Mathematics Education. The paper is available here.

Haciomeroglu, E., Aspinwall, L. and Presmeg, L. (2009). Visual and analytic thinking in calculus. The Mathematics Teacher, 103, 140-145.

Orton, A. (1983). Students' understanding of integration. Educational Studies in Mathematics, 14, 1-18.

If you choose to read and write a 3-4 page paper on one of these readings, the paper is due (by email) by Tuesday, December 8.

Final Exam

The final exam is scheduled for Thursday, December 10 from 4pm - 6pm (in our usual classroom). I will provide a study guide (will be posted on Friday, 12/4) from which the exam questions will be drawn. To get started preparing early, you should be sure that you know how to do all of the homework problems listed on this web page!

Final Exam Study Guide (pdf)