# Calculus

## Calculus

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.

### Literature

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:

## Problems

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.

### Lecture 1: Trigonometric functions.

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
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.

### Lecture 2: Polar coordinates.

Material: Section 9.2 in E&P.

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

Exercises:

Section 9.2 true/false study guide, s. 670.

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.

### Lecture 3: Curves in space.

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.

### Lecture 4: Arc length and curvature.

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: 1,  3, 5.
Calculate curvature: 7, 9, 10, 11.
Find time of maximal curvature: 15, 16.
Calculate unit normal and unit tangential vectors: 17, 19.
Determine the circle of curvature: 29.

### Lecture 5: Mini project 1 (self-study).

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

### Lecture 6: Introduction to functions of several variables.

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.

### Lecture 7: Partial derivatives.

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. True/false study guide.

Calculate partial derivatives: 1, 3, 5, 15.
Mixed partial derivatives and the equation zxy = zyx: 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).

### Lecture 8: Optimization.

Material: Section 12.5 in E&P.
Topic: Review of Section 12.5 in E&P. The topic is optimization.
Exercises:
Section 12.5, True/false study guide, page 939.
Section 12.5:
Find horizontal tangent plance: 3, 5, 9.
Find the "highest" and "lowest" point on a surface: 13.
Find min and max of a given function: 23, 25, 27.
Minimize cost: 41 & 47.

### Lecture 9: The chain rule.

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.

### Lecture 10: The gradient and the directional derivative.

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 true/false study guide, page 970.

Section 12.8, page 971.
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.

### Lecture 11: Mini project 2 (self-study).

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

### Lecture 12: Integration of functions of two variables.

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).

### Lecture 13: More on integration of functions of two variables.

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.

### Lecture 14: The double integral in polar coordinates.

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 true/false study guide, page 1025.

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.

### Lecture 15: Triple integrals.

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.

### Lecture 16: Mini project 3 (self-studie).

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

### Lecture 17: Complex numbers.

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:

From MN (notice: answers to odd-numbered problems on page 373)
MN §1.1 - Express a complex number in the form a+ib: 1, 5, 7, 9, 11, 13.
MN §1.1 - Laws of exponents: 15, 17.
MN §1.2 - Geometric intrepretation of complex numbers: 3, 7.
MN §1.3 - Polar form of complex numbers: 5, 7.

### Lecture 18: The Complex Exponential Function

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+ib: 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.

### Lecture 19: First order differential equations.

Material: Sections 2.2 and 2.3 in MN
Topic: Separable and linear first order differential equations.
Exercises:

MN §2.2 -- separable diff. equations: 1, 3, 5.
MN §2.2 -- solve separable diff. eqs: 7, 9, 10.
MN §2.2 -- Initial value problems: 17, 19

MN §2.3 -- linear diff. eqs: 1, 3, 5.
MN §2.3 -- Solve linear diff. eqs: 7, 10, 13.
MN §2.3 -- Initial value problems: 17.

### Lecture 20: Second order differential equations.

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.

### Lecture 21: Inhomogeneous second order differential equations and the superposition principle.

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.

### Lecture 22: Mini project 4 (self-study).

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

## Miniprojects

Work is done in your group room.

### Miniproject 1: Taylor polynomials

#### Description

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)

#### Exercises

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.

#### Exercise

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.

### Miniprojekt 4: Applications second order differential equations: harmonic oscillator

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.

## Old exams

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

### Previous exams

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

## 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.

## Mathematics during the weekend

#### Do you both want to improve your math skills before the exam and also see how the math at the first study year can be applied?

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.

## Preparation for exam

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.

### Aalborg - Calculus (and CALI for GBE)

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.)

### Aalborg - Linear algebra

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.)

### Esbjerg - Linear algebra

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

### Copenhagen - Calculus

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