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

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

This course builds upon the curriculum from high school. Hence, it is important to be very familiar with this material. If you have the need for repetition, Rob Ghrist's Coursera course 'Calculus 1 variable' may be a good help in addition to revisiting problems from high school.

In this Calculus course we use a BUNDLE OF TWO BOOKS, namely:

- MN: An Introduction to Complex Numbers and Differential Equations", Second Edition, Compiled by Morten Nielsen, Pearson
- E&P: "Calculus: Early Transcendentals", Seventh Edition, Edwards and Penny, Pearson

Abovementioned BUNDLE has ISBN number 9781784499075 and is available in the book store (previously it had ISBN number 1-783-99028-7). The compilation by Morten Nielsen is not sold elsewhere.

In addition, the following material is used:

Please, look at Moodle.

Below you will find suggested exercises for the various Calculus sessions. In case your teacher prepares a separate problem list, please follow that
list. Generally, each student is personally responsible for acquiring sufficient
problem solving skills. A large number of exercises is listed for each session,
so it is recommended that you begin with the 'prioritized' exercises marked with **bold**.
Skills obtained from a given session is often needed for solving the exercises of
subsequent sessions, so therefore it is advisable not to postpone exercises for a given session.

**Plan for Calculus.**

**Literature:**

**E&P ** - Edwards and Penney: Calculus - Early
Transcendentals. 7th edition. Prentice Hall.

**MN** - An Introduction to Complex Numbers and
Differential Equations", Second Edition, Compiled by Morten
Nielsen, Pearson.

Exercises: Prioritized problems are marked with bold.

**Material:** Appendix C, A13-A17 and Section 6.8 in
E&P.

**Topic:** Introduction to Calculus. Then a review
of Appendix C, A13-A17 and Section 6.8 in E&P.

**Exercises:**

Angle conversion: **1**, 3, 5, 7, 9.

Periodic properties of trig. functions: **15,** 20.

Trigonometric identities: 26

Evaluate trig. functions: **29**, 33

Addition formulas: 37

Trig. equations: **43**, 47.

Section 6.8, pages 496-498 in E&P.

True/false study guide, 1-8.

All questions i Exercises 1, 2 og 3 regarding function values.

Differentiate inverse trig. functions: **5**, 6, 17.

Integrate inverse trig. functions: **31**, 35.

**Material:** Section 9.2 in E&P.

**Topic:** Review of Section 9.2 E&P. The topic
is polar coordinates.

**Exercises:**

Section 9.2:

Conversion between polar and rectangular coords.: **1(a)(b)(c)(f),
2(a)(e)(f).**

Conversion of equations from rectangular to polar coords.: **3**,
6, **7**.

Conversion of equations from polar to rectangular coords.: **11**,
**13**, 17.

Find the equation of a curve in polar coords.: **21** &
27.

Supplemental Exercises fra 9.2: 29, 31, 39, 41, 53 & 63.

**Material:** Section 11.5 in E&P.

**Topic:** Review of Section 11.5 E&P. Parametric
description of curves in space. One can also review parts of
Section 9.4 on plane curves.

**Exercises:** Section 11.5, True/false study guide,
s. 861

Section 11.5:

Derivatives of parametric curves: **1**,9.

Determine velocity- and accelerationvectors: **13**, 15.

Integration of parametric curves: **17**.

Formulas for differentiation of parametric curves: **21, 23.**

Velocity, speed, and acceleration of particle: 35.

Trajectory of a projectile: 43, 45. Use that 1 mile equals
1609.344 meter, and that *g* = 9.80665
m/s2.

Finally: Previous Exercises.

**Material:** Section 11.6 (until page 869 middle) in
E&P.

**Topic:** Review of Section 11.6 (until page 869 in
the middle) in E&P. The topics are arc length and curvature of
plane curves.

**Exercises:
**Section 11.6:

Calculate arc length:

Calculate curvature:

Find time of maximal curvature:

Calculate unit normal and unit tangential vectors:

Determine the circle of curvature: 29.

The mini project concern Taylor polynomials. Details can be found here.

**Material:** Section 12.1, 12.2 og evt. noget af
12.3 i E&P.

**Topic:** Introduction to the theory of functions of
several variables. Review of sections 12.1, 12.2 and possible
parts of 12.3 in E&P.

**Exercises:**

Section 12.2, **True/false study guide**, s. 907.

Section 12.2:

Determine the domain of a function: **1, 3**, 5, 7, 8.

Describe the graph of a given function: 22, **25**, 29.

Describe level curves of a given function: **41**, 44.

Match graph and level curves: **53**, 54, **55**, 56, **57**,
58.

Section 12.3:

Limits and continuity: 1,5.

Existence of limits: 21, 43.

**Material:** Section 12.4 in E&P

**Topic:** Review of Section 12.4 i E&P. The
topic is partial derivatives.

**Exercises:
**Section 12.4, page 927-931 in E&P.

Calculate partial derivatives: **1, 3, 5**, 15.

Mixed partial derivatives and the equation *z*_{xy}
= *z*_{yx}: **21**,
25.

Determine tangent plance: **31**, 38.

Existence of functions with given partial derivatives: **41**,
43.

Verify solutions to partial differential equations: 55, 58 (feel
free to use Maple).

**Material:** Section 12.5 in E&P.

**Topic:** Review of Section 12.5 in E&P. The
topic is optimization.

**Exercises:
**Section 12.5,

Section 12.5:

Find horizontal tangent plance:

Find the "highest" and "lowest" point on a surface:

Find min and max of a given function:

Minimize cost:

**Material:** Section 12.7 in E&P.

**Topic:** Review of Section 12.7 in E&P. The
topic is the chain rule.

**Exercises:**

Section 12.7 **true/false study guide**, page 959.

Section 12.7:

Use the chain rule to find partial derivatives: **1, 3.**

Write down the chain rule in a given setup: 13.

Implicit differentiation: **19, 21**, 23.

The chain rule and partial differential equations: 40, 43.

**Material:** Section 12.8 in E&P.

**Topic:** Review of Section 12.8 i E&P. Topics
are the directional derivative and the gradient.

**Exercises:
**Section 12.8

Section 12.8, page 971.

Calculate the gradient: **3, 5**.

Calculate the directional derivative: **11**, 15.

Find the maximal directional derivative: **21**, 23.

Find tangent line/plane to a curve/surface: **29, 31**, 33.

Find the tangent line for a conical section: 41.

The mini project reviews Chapter 12 in E&P. Details can be found here.

**Material:** Section 13.1 and part of Section 13.2
in E&P.

**Topic:** Review of Section 13.1 and part of 13.2 in
E&P. The topic is integration of functions of two variables.

**Exercises:**

Section 13.1** true/false study guide**, page 1003.

Section 13.1:

Evaluate iterated integrals: **11, 13, 15**, 17, **19**,
21, 25, 27, 29, 31. Riemann sum: 37.

Previous Exercises from 12.8 ( 21, 23, 29 & 31).

**Material:** Last part Section 13.2 and Section 13.3
in E&P.

**Topic:** Review of the remaining part of Section
13.2 and Section 13.3 i E&P. The topic is integration of
functions of two variables and applications to area and volume.

**Exercises:**

Section 13.3 true/false study guide, page 1017.

Section 13.2:

Evaluate iterated integrals: **3**, 7 13.

Evaluate a double integral over a given region:** 19.**

Switch the order of integration: **31** & 33.

Section 13.3:

Calculate the area: **3**, 5, 9.

Find the volume of a solid: **11**, 15.

Find the volume of a solid (more advanced): 31 & 42.

**Material:** Section 13.4 in E&P.

**Topic:** Review of Section 13.4 in E&P. The
topic is the double integral in polar coordinates.

**Exercises:
**Section 13.4

Section 13.4:

Find area of region given by polar curves: **1, 3,** 4.

Calculate the area of a solid: **9**, 11 (start here) and 29,
33 (more challenging).

Change to polar coords. in double integral: **13**, 15.

Section 13.3:

Calculate the volume of solids: **29** & 35.

**Material:** Section 13.6 in E&P.

**Topic:** Review of Section 13.6 i E&P. The
topic is triple integrals.

**Exercises:**

Section 13.6, **true/false study guide**, s. 1045.

Section 13.6:

Evaluate triple-integrals: **1, 3**, 7, 9.

Find volume/centroid using triple-integrals: **17,** 23, 31.

The mini project concerns an application of double integrals in
calculating mass and center of gravity (centroid). Details can be found
here.

**Material:** Section 1.1, 1.2, and 1.3 in MN

**Topic:** Introduktion to a new topic; complex
numbers. Review of Sections 1.1, 1.2 and 1.3 in MN

**Exercises:**

MN §1.1 - Express a complex number in the form a+

MN §1.1 - Laws of exponents:

MN §1.2 - Geometric intrepretation of complex numbers:

MN §1.3 - Polar form of complex numbers:

**Material:** Section 1.4 in MN

**Topic:** Review of Section 1.4 regarding the
complex exponential function.

**Exercises:**

MN §1.4 - Express a complex number in the form a+**i**b:
**1.**

MN §1.4 - Express a complex number in polar form: **3**.

MN §1.4 - Polar form of complex numbers: 5.

MN §1.4 - The complex exponentioal: **7**, 8, **9**, 10,
11.

MN §1.4 - Trigonometric identities: 12, **13**.

**Material:** Sections 2.2 and 2.3 in MN

**Topic:** Separable and linear first order
differential equations.

**Exercises:**

MN §2.2 -- solve separable diff. eqs:

MN §2.2 -- Initial value problems:

MN §2.3 -- linear diff. eqs:

MN §2.3 -- Solve linear diff. eqs:

MN §2.3 -- Initial value problems: 17.

**Material:** MN §4.1- §4.3.

**Topic:** The topic is second order differential
equations: MN §4.1- §4.3.

**Exercises:**

MN §4.2 - Find the complete solution to differential eq.: **1**,
3, 5, 7.

MN §4.2 - Solve initial value problem: **13,** 15, 17.

MN §4.2 - Linear independence: 27, 29.

MN §4.3 - The characteristic equation: **1, 3**, 5, 9.

MN §4.3 - Find the complete solution to differential eq.: **11.**

MN §4.3 - Solve initial value problem: **21**, 23, 25.

**Material:** MN §4.4 and §4.5.

**Topic:** Review of Sections 4.4 og 4.5 in MN
regarding inhomogeneous second order differential equations and
the superposition principle.

**Exercises:**

MN §4.4 -- Solve inhomogeneous diff. eqs: **9,** **11**,
13, 15 og 17.

MN §4.5 -- Use the superposition principle: **1**, 2

MN §4.5 -- Find the general solution: **3** og 5.

The mini project concerns applications of second order
differential equations. Details can be found
here.

Work is done in your group room.

Program of the day:

- Read Section 10.4, pages 743-749, in E&P regarding Taylor polynomials and Taylors formulas with remainder. You may skip the remarks about infinite sequences at pages 743-744. Start with section "Polynomial Approximations", p. 744.
- Calculate the exercises given below. Although electronic equipments are not allowed to exam, it is still important that engineering students have feelings about numerical calculations. Therefore some of the exercises require calculators (or Matlab or Maple)

Solve the exercises in the given order. Regarding exercise 6 below: The numeric calculations can be done by using calculators. It is also the same for the last exercise.

- Section 10.4, page 755 in E&P: Exercises 1, 3, 4, 13, 16.
- Section 10.4, pages 755 in E&P: Exercises 5, 6.

Write an expression for the general Taylor polynomial of degree `n`

for the function cos(x) expanded around a=0. Write also an expression for the general remainder. Use this to decide an n, such that the four first decimals in the approximation with the value of Taylor polynomial to cos(0.1) is correct.

Note! One shall discuss for the value of `n`

with help of estimates on remainders. It is not enough to make a numerical experiment to decide `n`

. But it is reasonable to make a numerical computation to confirm that one has found a useful value `n`

.

Read Section 4.9 (from page 281) in Saff et al.

Exercises Section 4.4 (p. 247): 9, 11, 13, Section 4.9 (p. 290): 1, 2, 3, 5, 7.

- 2017 spring
- Regular exam (answers)
- Reexam (answers)
- L edition with Laplace transformations:
- 2016 autumn
- 2016 spring
- Regular exam (answers where some parts may be in Danish)
- Reexam (answers where some parts may be in Danish)
- Test set

Notice that the topics integration in cylinder- and spherical coordinates and the binomial equation are no longer covered in the Calculus course. You may therefore disregard any problem related to these topics below. Specifically: Trial exam 1; Ex. 9, Exam 2011; Ex. 9, Exam 2012; Ex. 9 and Exam 2013, Ex. 4.

- 2011
- 2012
- 2013
- 2014
- 2015 spring
- 2015 autumn

Please, find the Danish solutions for some of the exams at this page.

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.

Then **Math Saturday** the **8th of April 2017 at 9:30-15:00** is just what you need. The main part of this event is held as a workshop in **Aud. 1, Badehusvej, Aalborg**.

The day will consist of two mini-projects where the teacher will give a short presentation of each subject (one before lunch and one after), and afterwards the teacher will assist you as needed during the project. Throug both "pen and paper" exercises and MATLAB exercises the projects will stregthen your math skills. Hence, this is a great occacion to practice Calculus and prepare for the exam.

It is possible to participate as non-Danish speaker since the course material and exercises will be in English, but the short intro by the teacher will be held in Danish.
The two subjects are **Map projections** and **Numerical methods**, and a more detailed description of each is available on the Danish version of this page (you may try your luck with Google Translate or ask a fellow student that understands Danish).

A **free sandwich** is served for lunch and therefore you need to **sign up** by filling out the form below **no later than Wednesday the 5th of April 2017**.

It is no longer possible to sign up for the event.

You are offered help to prepare for the coming exam in both calculus and linear algebra at all three campuses.
The idea is much like the exercise session during a normal lecture, where you **on your own** solve exercises and can get help from a teaching assistant present to help you.
The point of departure for this exam preparation is the old exam sets available at this website, and you are encouraged to solve as much as possible on your own before showing up, so you know which parts you find difficult.
Please note that **the teaching assistants are not coming to the group rooms**; rather everybody sits in the same room where the teaching assistants are present as indicated below.

**Note:** In Aalborg the sessions Thursday and Friday are split up since there are fewer teaching assistants these days, so please take note of when you are supposed to come.

Thursday 8 June 16:15 - 18:45 in Auditorium 6 Badehusvej 5-13.

(Only for the classes taught by Horia Cornean, Nikolaj Hess-Nielsen and Athanasios Georgiadis.)

Friday 9 June 16:15 - 18:45 in Auditorium 6 Badehusvej 5-13.

(Only for the classes taught by Diego Ruano and Jon Johnsen.)

Saturday 10 June 10:00 - 15:00 in Auditorium 6 Badehusvej 5-13.

(For everyone.)

Sunday 11 June 10:00 - 15:00 in Auditorium 6 Badehusvej 5-13.

(For everyone.)

Thursday 8 June 16:15 - 18:45 in Auditorium 7 Badehusvej 5-13.

(Only for the class taught by Jacob Broe.)

Friday 9 June 16:15 - 18:45 in Auditorium 7 Badehusvej 5-13.

(Only for the class taught by Nikolaj Hess-Nielsen.)

Saturday 10 June 10:00 - 15:00 in Auditorium 7 Badehusvej 5-13.

(For everyone.)

Sunday 11 June 10:00 - 15:00 in Auditorium 7 Badehusvej 5-13.

(For everyone.)

Wednesday 7 June 8:15 - 10:15 in own group rooms.

Wednesday 31. maj 9:00 - 13:00 in room 3.161 FKJ 10A.