Mathematics for Multimedia Applications

Literature

Edwards, Penney, Spence, Insel, Friedberg: Mathematics for Multimedia Applications, Compiled by Iver Ottosen Aalborg University, Pearson (2014), ISBN 978-1-78399-714-5

Exam and hand-ins

Individual written 4 hours examination. Books and notes are allowed, but no calculators, computers, cell phones or the like.

To be eligible to take the exam, you must have at least 3 accepted self-study assignments (miniprojects) and 10 accepted ordinary hand-in exercises.

Practical remarks regarding hand-ins and miniprojects

Hand-in exercises and miniprojects must be uploaded in Moodle as pdf-files or jpg-files (do not use archive files). Scan in your individual (hand-written) solution or take a picture of it. The file name must include the session number for the hand-in. MATLAB figures can be saved as pdf-files via File -> Save As -> choose pdf format.

There is a deadline for uploading each assignment. After the deadline, the teacher will check a sample of the hand-ins. If an assignment cannot be accepted, the teacher will notify you by email.

Late hand-ins can be uploaded at a special slot, however the teacher can refuse to accept these.

Syllabus

Part 1: Calculus

  • Appendix C, page 1-5
  • Section 3.1, 3.2, 3.3, page 60-91
  • Section 3.7, 3.8, page 123-146
  • Section 5.3, page 195-203
  • Section 5.4, page 207-212 (middle)
  • Section 5.5, page 218-227
  • Section 5.9, page 259-267 (top)
  • Section 11.1, 11.2, 11.3, page 280-301 (middle)
  • Section 11.4, 11.5, page 305-323

Part 2: Linear Algebra

  • Section 1.1, 1.2, 1.3, 1.4, page 395-444
  • Section 2.1, page 485-494
  • Section 2.3, page 512-515
  • Section 2.4, page 525-527 (bottom)
  • Section 2.7, page 557-565
  • Section 2.8, page 575 (middle)-577
  • Section 6.9, page 587-589 (bottom)

Miniprojects

Remark that the notes are designed for the lectures, where they are accompanied by an oral presentation. They do not constitute a textbook.

Click here to download a collection of formulas for the first part of course.

1. Session: Review of Trigonometry

Lecture

Appendix C, Review of Trigonometry, page 1-5. Notes.

Hand-in Exercises

Appendix C, page 5, problems no. 11, 15, 21.

Exercises for 1. and 2. session

Appendix C, page 5, problems no. 1, 3, 7, 9, 17, 26, 29, 37 (don't use the addition formula in part (c)), 39, 41, 43.

2. Session: The Derivative and Rates of Change

Lecture

Section 3.1, The Derivative and Rates of Change, page 60-70, and Section 3.2, Basic Differentiation Rules, page 73-82. Notes.

Hand-in Exercises

Prove the following formula:
sin(α)sin(β) = ½ (cos(α–β)–cos(α+β))
Hint: Start with the right hand side of the equation. Use the addition formula for cosine.

Exercises

The rest of the problems from session no. 1.

3. Session: Derivatives of Trigonometric Functions

Lecture

Section 3.7, Derivatives of Trigonometric Functions, page 123-131, and Section 3.3, The Chain Rule, page 84-91. Notes.

Hand-in Exercises

Section 3.1, page 71, problems no. 23, 27.
Section 3.2, page 82, problems no. 3, 9, 11, 60.

Exercises

Section 3.1, page 71, problems no. 21, 30-35, 38, 39.
Section 3.2, page 82, problems no. 13, 19.

MATLAB installation

Please download and install MATLAB on your computer before session no. 5. Screencast 1 on the AAU first year mathematics webpage describes how to do it. Use your AAU email account. Otherwise the installation will not work.

4. Session: Exponential and Logarithmic Functions

Lecture

Section 3.8, Exponential and Logarithmic Functions, page 134-146. Notes.

Hand-in Exercises

Section 3.7, page 131, problems no. 1, 3, 5, 9, 15, 65.

Exercises

Section 3.7, page 131, problems no. 7, 11, 17, 19, 21, 67, 73, 77.

5. Session: Trigonometric Functions and Sound

Group work on Miniproject 1 which can be found on the AAU first year mathematics webpage.

Hand-in

Miniproject 1

6. Session: Areas, Sums and Integrals I

Lecture

Section 5.3, Elementary Area Computations, page 195-203 (top). Notes.

Hand-in Exercises

Section 3.8, page 146, problems no. 1, 3, 5, 21, 27, 33, 39, 63.

Exercises

Section 3.8, page 146, problems no. 23, 41, 43, 47, 48, 59, 64.

7. Session: Areas, Sums and Integrals II

Lecture

Section 5.4, Riemann Sums and the Integral, page 207-212 (middle), and Section 5.5, Evaluation of Integrals, page 218-227. Notes.

Hand-in Exercises

Section 5.3, page 205, problems no. 1, 3, 6, 9, 11, 19, 43.

Exercises

Section 5.3, page 205, problems no. 15, 23, 39.

Exercise: The number pi

In this exercise we estimate π via regular polygons inscribed in the unit circle.

  1. Compute h1 and h2.
  2. Compute the area of ΔOAC and the area of ΔABC.
  3. Use your results from 2. to find an underestimate for π.
  4. Let M be the midpoint of the (shortest) arc from B to C. Compute the area of ΔBMC
    and find a better estimate for π. How could one improve this estimate?

8. Session: Vectors

Lecture

Section 11.1, Vectors in the Plane, page 280-285, and Section 11.2, Three-Dimensional Vectors, page 286-290 (middle). Notes.

Hand-in Exercises

Section 5.5, page 227, problems no. 5, 7, 9, 11, 13, 21, 25, 31.

Exercises

Section 5.5, page 227, problems no. 35.
Section 5.7, page 245, problems no. 51, 53, 61.

9. Session: Dot Product and Cross Product

Lecture

Section 11.2, Three-Dimensional Vectors, page 290 (middle)-295, and Section 11.3, The Cross Product of Vectors, page 297-301 (middle). Notes.

Hand-in Exercises

Section 11.1, page 285, problems no. 1, 3, 5, 7, 9, 13, 19.

Exercises

Section 11.1, page 285, problems no. 31, 37, 43.

10. Session: Numerical Integration

Group work on Miniproject 2 which can be found on the AAU first year mathematics webpage.

Hand-in

Miniproject 2

11. Session: Lines and Planes in Space

Lecture

Section 11.4, Lines and Planes in Space, page 305-311. Notes.

Hand-in Exercises

Section 11.2, page 295, problems no. 1 (c), 5 (c), 62.
Section 11.3, page 304, problems no. 1, 14.

Exercises

Section 11.2, page 295, problems no. 39, 43, 47.
Exercise: Find the perpendicular projection of the vector a = (4, 4, -7) onto the vector b = (2, -1, -2).

12. Session: Curves and Motion in Space I

Lecture

Section 11.5, Curves and Motion in Space, page 313-316. Notes.

Hand-in Exercises

Section 11.4, page 311, problems no. 1, 3, 7, 23, 39.

Exercises

Section 11.4, page 311, problems no. 9, 17, 25, 31, 37.

13. Session: Curves and Motion in Space II

Lecture

Section 11.5, Curves and Motion in Space, page 316-323. Notes.

Hand-in Exercises

Section 11.5, page 324, problems no. 3 (t ≥ 0), 7, 8, 54.

Exercises

Section 11.5, page 324, problems no. 1, 2, 4, 21, 23.

14. Session: Matrices and Vectors

Lecture

Section 1.1, Matrices and Vectors, page 395-403, and Section 1.2, Linear Combinations, Matrix-Vector Products, and Special Matrices, page 405-411. Notes.

Hand-in Exercises

Section 11.5, page 324, problems no. 11, 61.
Hint for 61 (a): Calculate the velocity vector v(t) and show that v(t)•r(t) = 0.

Exercises

Section 11.5, page 324, problems no. 13, 45 (g = 32 ft/s²).

15. Session: Matrix-Vector Products

Lecture

Section 1.2, Linear Combinations, Matrix-Vector Products, and Special Matrices, page 411-416. Notes.

Hand-in Exercises

Section 1.1, page 403, problems no. 1, 3, 5, 7, 9, 11, 19, 21.

Exercises

Section 1.1, page 403, problems no. 79, 80, 81.

16. Session: Systems of Linear Equations

Lecture

Section 1.3, Systems of Linear Equations, page 419-430. Notes.

Hand-in Exercises

Section 1.2, page 417, problems no. 1, 3, 5, 7, 9, 17, 19, 29, 33.

Exercises

Section 1.2, page 417, problems no. 13, 15, 31.

17. Session: Gaussian Elimination

Lecture

Section 1.4, Gaussian Elimination and The Row Reduction Algorithm, page 433-444. Notes.

Hand-in Exercises

Solve the following system of linear equations:
$$ \begin{align} x_1 - 3x_3 &= 8 \\ 2x_1 + 2x_2 + 9x_3 &= 7 \\ x_2 + 5x_3 &= -2 \end{align} $$

Exercises

Section 1.3, page 430, problems no. 1, 3, 5, 7, 9, 11, 23, 25, 81.

18. Session: Matrices, Vectors and Systems of Linear Equations

Group work on Miniproject 3 which can be found on the AAU first year mathematics webpage.

Hand-in

Miniproject 3

19. Session: Matrix Multiplication

Lecture

Section 2.1, Matrix Multiplication, page 485-494. Notes.

Hand-in Exercises

Section 1.4 page 444, problems no. 17, 19, 21, 27.

Exercises

Section 1.4, page 444. problem no. 15.
From the AAU first year mathematics webpage
Regular exam 2013

20. Session: The Inverse of a Matrix

Lecture

Section 2.3, Invertibility and Elementary Matrices, page 512-515, and Section 2.4, The Inverse of a Matrix, page 525-527. Notes.

Hand-in Exercises

Section 2.1 page 494, problems no. 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 25, 27.

Exercises

Section 2.1 page 494, problems no. 29, 63.
From the AAU first year mathematics webpage,
Regular exam 2013 (as in session no. 19).

21. Session: Linear Transformations and Matrices

Lecture

Section 2.7, Linear Transformations and Matrices, page 557-565, and Section 2.8, Composition and Invertibility of Linear Transformations, page 575 (middle)-577. Notes.

Hand-in Exercises

Section 2.3 page 520, problems no. 1, 2.
Section 2.4 page 532, problems no. 1, 3, 7, 13.

Exercises

Section 2.4 page 532, problem no. 5, 9, 11.
From the AAU first year mathematics webpage,
Regular exam 2012.

22. Session: Applications to Computer Graphics

Group work on Miniproject 4 which can be found on the AAU first year mathematics webpage.

Hand-in

Miniproject 4
 

Miniprojects

Miniproject 1: Trigonometric Functions and Sound

This miniproject is supported by screencast 1 and 2 with installation and basic usage of MATLAB.

Miniprojekt 2: Numerical Integration

Miniprojekt 3: Matrices, Vectors and Systems of Linear Equations

This miniproject is supported by screencast 3 (MATLAB).

Miniprojekt 4: Applications to Computer Graphics

Click here to download tower.m (remember to rename to tower.m after download) and here to download linesegmentplot.m (remember to rename to linesegmentplot.m after download).