Please recall that we have a math-cafe where you can get help with your unsolved exercises. Next session is:

- Aalborg: Not scheduled yet.
- Copenhagen: Not scheduled yet.

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

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.

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.

- 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

- 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)

**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.

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

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

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

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.

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

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.

The rest of the problems from session no. 1.

- Session 2, screencast 1. The derivative of a function.
- Session 2, screencast 2. Example of a derivative.
- Session 2, screencast 3. Derivatives of special functions.
- Session 2, screencast 4. Differentiation rules.
- Session 2, screencast 5. Examples of the use of differentiation rules.
- Session 2, screencast 6. Rates of change.
- Session 2, screencast 7. Example with rates of change.

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

Section 3.1, page 71, problems no. 23, 27.

Section 3.2, page 82, problems no. 3, 9, 11, 60.

Section 3.1, page 71, problems no. 21, 30-35, 38, 39.

Section 3.2, page 82, problems no. 13, 19.

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.

- Session 3, screencast 1. The derivatives of sine and cosine.
- Session 3, screencast 2. The derivatives of the other trigonometric functions.
- Session 3, screencast 3. The chain rule.
- Session 3, screencast 4. Examples of the use of the chain rule.
- Session 3, screencast 5. The chain rule and sine and cosine functions.

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

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

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

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

Miniproject 1

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

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

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

- Session 6, screencast 1. Areas of curvilinear figures.
- Session 6, screencast 2. Areas under graphs.
- Session 6, screencast 3. Summation notation.
- Session 6, screencast 4. Rules and formulas for summation.
- Session 6, screencast 5. Areas.

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

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

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.

- Compute h
_{1}and h_{2}. - Compute the area of ΔOAC and the area of ΔABC.
- Use your results from 2. to find an underestimate for π.
- 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?

- Session 7, screencast 1. Riemann sums.
- Session 7, screencast 2. Integrals.
- Session 7, screencast 3. Evaluation of integrals.
- Session 7, screencast 4. Integrals and anti-derivatives.
- Session 7, screencast 5. Examples of integrals.
- Session 7, screencast 6. Properties of integrals

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

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

Section 5.5, page 227, problems no. 35.

Section 5.7, page 245, problems no. 51, 53, 61.

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

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

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

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

Miniproject 2

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

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

Section 11.3, page 304, problems no. 1, 14.

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).

- Session 11, screencast 1. Parametric equation for a line.
- Session 11, screencast 2. Examples of lines given by parametric equations.
- Session 11, screencast 3. Intersections of lines.
- Session 11, screencast 4. Symmetric equations for a line.
- Session 11, screencast 5. Planes in 3D-space.
- Session 11, screencast 6. Angles between planes.

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

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

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

- Session 12, screencast 1. Parametric curves.
- Session 12, screencast 2. Vector functions.
- Session 12, screencast 3. Calculating derivatives of vector functions.
- Session 12, screencast 4. Rules for derivatives of vector functions.

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

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

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

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.

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.

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

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

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

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

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

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

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

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

Solve the following system of linear equations:

$$ \begin{align} x_1 - 3x_3 &= 8 \\ 2x_1 + 2x_2 + 9x_3 &= 7 \\ x_2 + 5x_3 &= -2 \end{align} $$

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

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

Miniproject 3

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

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

Section 1.4, page 444. problem no. 15.

From the AAU first year mathematics webpage

Regular exam 2013

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

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

Section 2.1 page 494, problems no. 29, 63.

From the AAU first year mathematics webpage,

Regular exam 2013 (as in session no. 19).

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.

Section 2.3 page 520, problems no. 1, 2.

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

Section 2.4 page 532, problem no. 5, 9, 11.

From the AAU first year mathematics webpage,

Regular exam 2012.

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

Miniproject 4

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

This miniproject is supported by screencast 3 (MATLAB).

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).

- 2017
- 2016
- Test set

- 2011
- 2012
- 2013
- 2014
- 2015

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.

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.

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.

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.