Fall 2007

Homework Assignments

Problem Set 1: Due Wednesday September 5th

1. Read the Forward and Interview in the “Dialogue” posted on the class website.
2. Identify three contributions Galileo, Isaac Newton, James Clerk Maxwell, Karl Pearson, and Ronald Fisher have each made to your life. How are their contributions related?
3. Show the square root of 7 is irrational.
4. Prove the Archimedes/Heron formula for the area of a triangle.
5. Exercise Set 4.3 (Page 34): 1-5 (Please do not hand in these problems.)
6. Explain why the sentence “I am lying.” is a problem.

Problem Set 2: Due Wednesday September 12th

1. Read Chapter 4 on Geometry (Pages 21-40) in the “Dialogue.”
2. Use a truth table to show the inverse is equivalent to the converse. (The inverse of the statement “If p, then q.” is the statement “If ~p, then ~q.”)
3. Read Section 5.1 on Limits (Pages 41-61) in the “Dialogue.”
4. Exercise Set 5.1 (Page 61): 1-8
5. Read Section 5.2 on the Geometric Series (Pages 61-64) in the “Dialogue.”
6. Exercise Set 5.2 (Page 64): 2-6 (even numbered problems)
7. Read Section 7.1 on the Archimedes/Heron method for computing square roots. (Pages 179-185) in the “Dialogue.”
8. Exercise Set 7.1 (Page 185): 2, 3, 6
9. Use the Archimedes/Heron method to compute the square root of zero. Initialize the method with
   x₀ = 1. How many iterations need to be computed to produce a result with error less than 0.000001?

Problem Set 3: Due Wednesday September 19th

1. Read Section 5.3 on Limit Facts (Pages 64-72) in the “Dialogue.”
2. Read Section 5.4 on the Least Upper Bound Principle (Pages 72-78) in the “Dialogue.”
3. Read Section 5.5 on Cauchy Sequences (Pages 78-85) in the “Dialogue.”
4. Exercise Set 5.3 (Page 72): 3-6
5. Exercise Set 5.4 (Page 78): 1-3
6. Compute the area under the function f(x) = x e^-x on the interval [0, ∞).
7. Compute the mean of the function f(x) = x e^-x on the interval [0, ∞).
8. Compute the variance of the function f(x) = x e^-x on the interval [0, ∞).

Problem Set 4: Due Wednesday October 3rd

1. Explain why the Poisson distribution must be discrete. (No credit will be awarded for stating that the variable x must be an integer for x! to be defined.)
2. Compute the number of cubic miles of melting ice in Greenland will it take to raise the ocean levels one foot. Explain the assumptions in your computation. Your answer is closest to which of the following numbers:
   1. The number of cubic miles of a one mile thick sheet of ice on top of the state of Connecticut.
   2. The number of cubic miles of a one mile thick sheet of ice on top of the state of Colorado.
   3. The number of cubic miles of a one mile thick sheet of ice on top of the state of Texas.
   4. The number of cubic miles of a one mile thick sheet of ice on top of the state of Alaska.

   How many cubic miles of ice cover Antarctica?

3. Exercise Set 5.5 (Page 85): 1-5
4. For what values of x is the series 1 – 3x + 3²x² – 3³x³ + 3⁴x⁴ – 3⁵x⁵ +…+ Cauchy? Explain!

Problem Set 5: Due Wednesday October 10th

1. Comment on the articles:
   Papal stargazers reach for heaven ,
   Pope sacks astronomer over evolution debate, and
   Finding Design in Nature By CHRISTOPH SCHÖNBORN .
   What would Galileo have thought?
   What would Giordano Bruno have thought?
2. Read Section 5.6 on the Limit of a Function (Pages 85-90) in the “Dialogue.”
3. Read Section 5.7 on the Limit Facts for Function (Pages 90-95) in the “Dialogue.”
4. Compute the radius of convergence for the series ∑ xⁿ / n!
5. Compute the radius of convergence for the series ∑ n³ xⁿ
6. Compute the radius of convergence for the series ∑ 2ⁿ xⁿ / n!
7. Compute the radius of convergence for the series ∑ 2ⁿ xⁿ / n ²
8. Compute the radius of convergence for the series ∑ n³ xⁿ / 5 ⁿ
9. Prove: If a_n > 0 for all n and ∑ a_n diverges, then the series ∑ a_n / (1 + a_n ) diverges.

Problem Set 6: Due Wednesday October 17th

1. Read Chapter 9 on Advanced Calculus (Sections 9.1-9.8 or Pages 107-192) in the “Dialogue” dated 10-09-07
2. Exercise Set 9.6 (Page 181): Trigonometric Series 1-3
3. Exercise Set 9.7 (Page 186): 1-2
4. Exercise Set 9.8 (Page 190): 5-7
5. Compute the Trigonometric series for the function
   f(x) = -1 if x in [-&pi,0] and
   f(x) = 1 if x in [0,&pi].
6. Compute the Trigonometric series for the function
   f(x) = 0 if x in [-&pi,0] and
   f(x) = 1 if x in [0,&pi].

Problem Set 6: Due Wednesday November 7th

1. Read the discussion of Connectedness and Compactness in Chapter 10 in the “Dialogue” dated 10-09-07
   (i.e. Pages 201-217)
2. Exercise Set 10.1 (Page 207): 1-9 (odds)
3. Exercise Set 10.2 (Page 212): 1, 3
4. Exercise Set 10.3 (Page 216): 1, 3
5. Use Pythagoras/Parseval to show that &pi⁴ /90 = ∑ 1/n⁴

Problem Set 7: Due Wednesday November 14th

1. Read the discussion of the Mean Value Theorem Section 11.3 (pages 224-227) in Dialogue (dated 10-16-07)
2. Read the discussion of Integration Section 11.5 (pages 232-252) in Dialogue (dated 10-16-07)
3. Exercise Set 11.3 (Page 226): Problems 1-6
4. Exercise Set 11.5 (Page 252): Problems 1-6

Problem Set 8: Due Monday November 26th

1. Read Sections 11.5-11.8 (pages 224-227) in Dialogue (dated 10-16-07)
2. Exercise Set 11.6 (Page 257): Problems 1-2
3. Exercise Set 11.7 (Page 264): Problems 1-7 (odds)
4. Exercise Set 11.8 (Page 269): Problems 1, 3

Bonus Problem: (15 extra points)

Pick a data set from the website (housing is probably a good choice):

Show how to set up a system of equations (with explanation) for a maximum likelihood parameter estimation for the Poisson and/or Gamma distributions. If someone wants to do show logistic regression, that would be acceptable as well. I will use your equations to show how Newton/Raphson can be used to solve the equations. (To earn credit on this problem you need to have the patience to answer all my dumb questions. Thus, you may have to come by my office to explain the details.) This assignment must be completed by December 5th.

Bonus Problem: (10 extra points)

• matlab code

If you are not comfortable with matlab and just want to compute the integral using a spread sheet, I will compute the data for you. This assignment must be completed by December 5th.

 

Student Presentations

Student presentations will be given on Wednesday November 21st.

• Student Presentations (from previous semesters)

Hour Exam Schedule

First Hour Exam:

Date: Wednesday September 26th Topics:

1. Chapters 1-5, (Pages 1-85) in the Dialogue
2. Chapter 7, (Pages 171-186) in the Dialogue

Be able to explain:

1. The difference between an axiom, a postulate, a definition and a theorem
2. modus ponens, modus tollens

Be able to compute:

1. any assigned problem (in particular, those assigned to be handed in for credit)
2. the square root algorithm for a positive number
3. the limit of a sequence and be able to prove your answer (e.g. the quotient of two polynomials)
4. be able to use the definition to prove limits

Be able to define:

1. absolute value
2. limit of sequence (i.e. convergent sequence)
3. increasing
4. decreasing
5. upper bound for set or sequence
6. lower bound for set or sequence
7. least upper bound for set
8. greatest lower bound for set
9. the least upper bound principle
10. Cauchy sequence

Be able to state and prove:

1. the square root or cube root of prime number is irrational.
2. limit of sum = sum of limits (for sequences)
3. limit of product = product of limits
4. the squeezing (or sandwich) theorem
5. the uniqueness theorem for limits
6. the sum formula for the Geometric Series
7. every convergent sequence is bounded
8. every bounded monotone increasing sequence converges
9. the square root algorithm produces a bounded decreasing sequence (& thus converges).
10. Define glb.
11. Find the glb for S = { (-1)ⁿ( 2 – 4/2ⁿ): n = 1, 2, 3, 4, ….}
12. Define a sequence by x₁ and x_n+1 = x_n + (-1/3)ⁿ. Show x_n is Cauchy.
13. Use the definition of convergent sequence to prove lim(2 + 5/n³) = 2.

Practice Exam

• Practice Exam 1 (Spring 2004)
• Practice Exam 1 (Fall 2005)
• Practice Exam 1 (Fall 2006) (For the practice exams, ignore problems that contain the word “subsequence.” You will be responsible for this topic on Exam 2.)

Second Hour Exam:

Date: Wednesday October 24th

Topics:

1. Chapter 9, Sections 9.5 (Cauchy Sequences)-9.8 (Limit Facts for Functions)
   (Pages 146-200) in the Dialogue updated on 10-16-07.
2. Chapter 10, Section 10.1
   (Pages 201-208) in the Dialogue updated on 10-16-07.
3. Cauchy Sequences
4. Series
5. Convergence Tests for Series
6. Power Series
7. Taylor Series
8. Trig/Fourier Series

Be Able to Explain:

1. The connection between Cauchy Sequences and convergent Sequences
2. The idea behind the proof that Cauchy Sequences are convergent Sequences
3. The connection between Cauchy Sequences and convergent power series
4. The connection between the Geometric Series and the Ratio Test
5. The connection between the Geometric Series and the nth Root Test
6. The dangers encountered when differentiating and integrating Power Series
7. The dangers encountered when differentiating and integrating Trig Series

Be able to compute:

1. any assigned problem (in particular, those assigned to be handed in for credit)
2. Ratio Test
3. Root Test
4. Comparison Test
5. Alternating Series Test
6. Radius of Convergence
7. Interval of Convergence
8. Taylor Series for cos(x), sin(x), e
9. Fourier Coefficients (i.e. a_k and b_k) for a given function f(x).
10. How to differentiate and integrate Power Series
11. How to differentiate and integrate Trig Series
12. How to sum series (e.g. 1 + 1/2^2 + 1/3^2 + ,,, + 1/n^2 + = &pi^2/6)

Be able to Define:

1. Cauchy Sequence
2. nth partial sum of a series
3. Convergent Series
4. Euler’s Constant
5. Absolute convergence
6. Limit of Function

Be Able to State and Prove:

1. A series of positive terms that is bounded converges.
2. If a series converges, then its nth term converges to zero.
3. If the nth term of a series converges to zero, then it may or may not converge.
4. the Ratio Test
5. the nth Root Test
6. Alternating Series Test
7. Limit of Sum Equals Sum of Limits
8. Limit of Product Equals Product of Limits
9. The sum of two continuous functions is continuous
10. The product of two continuous functions is continuous
11. The quotient of two continuous functions is continuous

• Practice Exam 2 (Spring 2004)
• Practice Exam 2 (Fall 2005)
• Practice Exam 2 (Fall 2006)

 

(Ignore questions on these practice exams which deal with the Mean Value Theorem, the Intermediate Value Theorem,, and the Fundamental Theorem of Calculus.)

Third Hour Exam:

Date: Friday November 30th

Topics:

1. Chapter 10, (Continuity, Compactness, and Connectedness)
   (Pages 201-216) in the Dialogue updated on 10-16-07.
2. Chapter 11, Sections 11.1-11.3
   (Pages 217-227) in the Dialogue updated on 10-16-07.
3. Chapter 11, Sections 11.5-11.8
   (Pages 232-269) in the Dialogue updated on 10-16-07.

Be Able to State AND Prove:

1. Connectedness/Intermediate Value Theorem
2. Compactness/Extrema
3. Rolle’s Theorem
4. Mean Value Theorem
5. Definition of Integral
6. Intermediate Value Theorem for Integrals
7. Fundamental Theorem of Calculus

Homework Problems:

• Problem Set 6
• Problem Set 7
• Problem Set 8

• Practice Exam 3 (Fall 2004)
• Practice Exam 3 (Fall 2005)
• Practice Exam 3 (Fall 2006)
• Final Exam (Fall 2004)
• Final Exam (Fall 2005)
• Final Exam (Fall 2006)

Final Exam Information

Date: Final Exam Group 13F (Thursday 13 December 2007)
Time: 3:00-5:00PM (Note time change!!!
Location: Little Hall 127

Topics:

1. The final exam is comprehensive and includes all topics from the previous three exams.

Practice Exams:

• Final Exam (Fall 2004)
• Final Exam (Fall 2005)
• Final Exam (Fall 2006)