Math 3321 (section 11136) - Spring 2024

Engineering Mathematics

(face to face)

Notes, reviews

• NOTES (updated February 21, corrections possible)
• Section 3.5 Exercises
• Section 3.6 (Vibrating Mechanical Systems)
• Section 3.7 tools:
  • Long division of polynomials, see e.g. Wikipedia
  • Synthetic division of polynomials, see e.g. Wikipedia
  • Finding rational roots of polynomial equations with integer coefficients: Wikipedia, Rational root theorem. There are also many videos about this.
    • If \(p/q\) in reduced form is a rational root of the polynomial with integer coefficients \(P(x)=a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\), \(a_n\not=0\), then:
      • \(p\) divides the constant term \(a_0\)
      • \(q\) divides the leading coefficient \(a_n\).
    • As one does the computation, additional roots can be ruled out: if \(p/q\) in reduced form is a rational root of \(P(x)\) then for any integer \(k\), \(k q -p \), which is the numerator of \(k-\frac{p}{q}\), divides \(P(k)\).

      Proof : if \(\alpha=p/q\) is a solution of \(P(x)=0\) then \(\beta=p/q-k\) is a solution of \(P(y+k)=0\); compute that \(Q(y):=P(y+k) = a_n y^n + \dots + P(k)\), so the numerator \(p-k q \) of \(\beta=p/q-k\) must divide the constant term of \(Q(y)\), which is \(P(k)\).

      
      Note that for \(k=0\) we obtain the original rule: \(p\) divides the constant term of \(P(x)\). Replacing \(x\) by \(1/x\) and a bit of algebra, this gives that \(q\) divides the leading coefficient of \(P(x)\).
    • See also Example 3 on the Wikipedia page.
    • For example, the rational roots of the polynomial with integer coefficients \(2 x^7+\dots + 3=0\), if any, are among \(\pm1, \pm3, \pm 1/2, \pm 3/2\).
      If we compute \(P(\)1\()\) and get \(5\), then \(3=3/1\) cannot be a root because \(3 - \)1\(*1=2\) does not divide \(5\).
      Similarly, \(1/3\) cannot be a root because \(1-\)1\(*3=-2\) does not divide \(5\).
      But cannot yet rule out \(1/2\) because \(1-\)1\(*2=-1\) divides \(5\).
• Section 5.4: RREF
  • from "row echelon form" to "reduced row echelon form": definition and examples
• Sections 5.5-5.6: Matrices, determinants
  • Ch5.5-5.6 summary (not complete)
• Chapter 6: Systems of Linear Differential Equations
  • Complex & repeated eigenvalues
• plotting solutions of differential equations in dimensions 1 and 2 (some of the plots in the notes)
  • dfield and pplane, graphic tools for numerical solution of ODEs in 1 and 2 dimensions, in Java.
    Developed for MATLAB by John C. Polking, Department of Mathematics, Rice University (see details in the link above).
    
    I recommend the Java version. Last time I tried the MATLAB code, it did not work with the current Matlab.

Exam 1:

• Exam 1 Review questions
• Exam 1 Review questions with answers
  TYPO: The answer to II.7 is \(\quad\dots - 3 y'= -6\)
  TYPO: In the answer to VIII.4(iii), should replace \(t\) by \(t/2\)
• notes from the review on Feb. 13 with some solved problems
• one more solved review problem
• annotations on the course notes

Exam 2:

• Exam 2 Review questions
• Exam 2 Review questions with answers
  [The answer to #5 in the 3.4-3.5 part is to the problem listed there, with RHS \((6x + 2)\); for RHS \(4/x^2\) one answer is \(z=2/3\).]
• old notes with some solved problems
• 03/26/2024 notes

Exam 3:

• Exam 3 Review questions
• Exam 3 Review questions with answers
• 04/23/2024 notes, extend "other older notes" below
• old notes with some solved problems
• other old notes with some solved problems

Final exam:

• list of topics (not all the material is discussed in these notes!)
• Final Exam Review questions (with answers removed)
• Final exam Review questions with answers
  • TYPO at answer of #2: should be   \(\ldots -y'=\ldots\)   instead of   \(\ldots- 2 y'=\ldots\)
  • TYPO at answer of #25: should be \(3x + 6 - \dots\) instead of \(3x + 3 - \dots\)
  • the answer to problem 23 is correct
    error in the notes: when eliminating denominators, should be \(B(s+4)\) instead of \(B(s-4)\)
• Old notes with some problems solved:
  • notes_a
  • notes_b
  • notes_c
  • notes_d

Assignments

General course information

• Classes: MoWeFr 10:00-11:00am, room CBB 108
• Instructor: Andrew Török
• Office: 672 PGH
• Phone: (713) 743-3478 (but not a good way to reach me)
• E-mail: see the home page
• Office hours: by appointment and see the Teaching page
• Course description
  • Prerequisite: MATH 1432 (Calculus II)
  • Course structure (with available help and grading policy)
  • University of Houston Syllabus Language: Fall 2023 (will update with Spring 2024 when/if becomes available)
  • Textbook:
    • The entire text with supporting materials (answers to odd-numbered problems, solutions to selected problems, etc.) is available at Canvas/CourseWare(CCS).
    • An Access Code is required to obtain the course materials/take exams/etc. Students pay for access to CASA as part of their fee bill via CTAP. If you opt out of the CTAP, then the code can be purchased at UH Bookstore. If the code is not entered by the deadline given on CASA, then access to CASA will be denied.
    • Suggested problems
    • Answers to odd-numbered problems
  • Syllabus, with exam topics
  • Calendar with exam dates
  • Exam scheduler, upgraded 2024.01
  • Department course description

• UH CAPS Statement

  Counseling and Psychological Services (CAPS) is free, 24/7 mental health support for all students. We can help with a range of concerns such as, managing stress, adjusting to college, or feeling sad and hopeless. You can learn more about CAPS by visiting our website, uh.edu/caps; calling our office 713-743-5454; or dropping into one of our Let's Talk locations, uh.edu/caps/outreach/lets-talk. After business hours, you can access the on-call clinician by calling, 713-743-5454.

• I will use your e-mail address at my.uh.edu, please have a correct e-mail address listed there.

Connect to CourseWare to access the text, online quizzes and Exam scheduler