# 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.

### Screencasts

Note that screencasts come with subtitles (both when played on YouTube and when downloaded).

## 1st 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 1st and 2nd 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.

Notes regarding the exercises.

## 2nd 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.

## 3rd 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.

## 4th 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.

## 5th Session: Trigonometric Functions and Sound

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

Miniproject 1

## 6th 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.

## 7th 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?

## 8th 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.

## 9th 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.

## 10th Session: Numerical Integration

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

Miniproject 2

## 11th 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).

## 12th 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.

## 13th 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.

## 14th 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).

## 15th 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.

## 16th 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.

## 17th 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.

## 18th Session: Matrices, Vectors and Systems of Linear Equations

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

Miniproject 3

## 19th 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

## 20th 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).

## 21st 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.

## 22nd Session: Applications to Computer Graphics

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

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 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).

## Old exam problems

Note: new structure in the organisation of the exam. Relevant from spring 2016 and onwards.
• 2017
• 2016
• Test set

• 2011
• 2012
• 2013
• 2014
• 2015

## Math cafe

#### Do you have a hard time understanding linear algebra and/or calculus at the first study year, and are you determined to do something about it?

Then Math cafe is just the right thing for you. It is held throughout the semester at all three campuses (specific times and places are listed below). It is an extra possibility for getting help with maths. A teaching assistant is available to help you with exercises from the last few lectures. The teaching assistants are preparing to help with the material from the last few lectures, and they might not be able to help with all your math questions, but feel free to ask. This is a new initiative and its success is partly measured by the amount of students coming to the math cafe. If there is a great interest in this initiative we will schedule more than the ones planed now. On the other hand if attendance is very low cancellation may occur.

Note: This is an extra curricular activity, so it is NOT a valid excuse for not participating in other course activities or project work.

### Aalborg

Here the math cafe in general runs either Tuesday or Thursday afternoon every week.
Current scheduled dates (will be updated throughout the semester):
• Tuesday 14/3-17 16:15-17:45 in auditorium 1.
• Thursday 23/3-17 16:15-17:45 in auditorium 1.
• Tuesday 28/3-17 16:15-17:45 in auditorium 1.
• Thursday 6/4-17 16:15-17:45 in auditorium 1.
• Thursday 20/4-17 16:15-17:45 in auditorium 1.
• Tuesday 25/4-17 16:15-17:45 in auditorium 1.
• Thursday 4/5-17 16:15-17:45 in auditorium 1.
• Tuesday 9/5-17 16:15-17:45 in auditorium 1.
• Thursday 18/5-17 16:15-17:45 in auditorium 1.
• Tuesday 30/5-17 16:15-17:45 in auditorium 1.

### Esbjerg

Here the math cafe in general runs Tuesday afternoon approximately every other week.
Scheduled dates so far (will be updated throughout the semester):
• Tuesday 21/3-17 16:15-17:45 in room C1.119.
• Tuesday 4/4-17 16:15-17:45 in room C1.119.
• Tuesday 25/4-17 16:15-17:45 in room C1.119.
• Tuesday 9/5-17 16:15-17:45 in room C1.119.
• Tuesday 23/5-17 16:15-17:45 in room C1.119.

### Copenhagen

Here the math cafe in general runs Friday afternoon approximately every other week.
Scheduled dates so far (will be updated throughout the semester):
• Friday 17/3-17 16:15-17:45 in room 0.108, FKJ 10A.
• Friday 7/4-17 16:15-17:45 in room 0.108, FKJ 10A.
• Friday 28/4-17 16:15-17:45 in room 0.108, FKJ 10A.
• Friday 19/5-17 16:15-17:45 in room 0.108, FKJ 10A.
• Friday 2/6-17 16:15-17:45 in room 0.108, FKJ 10A.