# 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

# Curriculum

## Textbooks/literature

We use the following teaching materials:

• C.H. Edwards & D.E. Penney (E&P), Calculus, 7th Edition, Prentice Hall 2008.
• E.B. Saff et al. Complex numbers and differential equations, Custom print (2nd edition), Pearson, 2010. (Bought in a bundle with E&P at Factum Books)

## Curriculum

### From Edwards and Penney:

• Appendix C, A-13 through A-17
• Section 6.8 until the middle of page 493
• Section 9.2
• Section 10.4 until Taylor series, page 749
• Section 11.5
• Section 11.6 until the middle of page 869
• Section 12.1
• Section 12.2
• Section 12.3
• Section 12.4
• Section 12.5
• Section 12.7 up to and including page 956
• Section 12.8
• Section 13.1
• Section 13.2
• Section 13.3
• Section 13.4
• Section 13.5, except Pappus' theorem
• Section 13.6

### From Saff et al. (MN)

Complex numbers:

• Chapter 1, sections 1.1, 1.2, 1.3, 1.4, 1.5

Differential equations:

• Chapter 1, sections 1.1 and 1.2 (motivation and the concept of solutions)
• Chapter 2, sections 2.2 and 2.3
• Chapter 4, sections 4.1, 4.2, 4.3, 4.4, 4.5, 4.9

## 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. All you have to do is to sign up by sending an email to the assistant at least 24 hours before the planned session. If the assistant hasn't received any email by that time Math Cafe is cancelled without further notice. So you can only expect help if you have sent an email in due time and recieved a response!. Please indicate in the email what you need help with (typically jst a specific exercise) without writing a long email about the details of you problem.

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 (email: mfindi13@student.aau.dk)

Here the math cafe in general runs Tuesday or Wednesday afternoon.
Currently the allocated dates if you have signed up by email are (will be updated throughout the semester):
• Wednesday 20/9-17 16:15-17:45 in Auditorium 1.
• Tuesday 26/9-17 16:15-17:45 in Auditorium 1.
• Wednesday 27/9-17 16:15-17:45 in Auditorium 1.
• Tuesday 3/10-17 16:15-17:45 in Auditorium 1.
• Wednesday 4/10-17 16:15-17:45 in Auditorium 1.
• Tuesday 10/10-17 16:15-17:45 in Auditorium 1.
• Tuesday 17/10-17 16:15-17:45 in Auditorium 1.
• Wednesday 18/10-17 16:15-17:45 in Auditorium 1.
• Tuesday 24/10-17 16:15-17:45 in Auditorium 1.
• Wednesday 25/10-17 16:15-17:45 in Auditorium 3.
• Tuesday 7/11-17 16:15-17:45 in Auditorium 1.
• Wednesday 8/11-17 16:15-17:45 in Auditorium 1.
• Tuesday 14/11-17 16:15-17:45 in Auditorium 1.
• Wednesday 15/11-17 16:15-17:45 in Auditorium 1.
• Tuesday 21/11-17 16:15-17:45 in Auditorium 1.
• Wednesday 22/11-17 16:15-17:45 in Auditorium 1.
• Tuesday 28/11-17 16:15-17:45 in Auditorium 1.
• Wednesday 29/11-17 16:15-17:45 in Auditorium 1.
• Wednesday 6/12-17 16:15-17:45 in Auditorium 1.
• Tuesday 12/12-17 16:15-17:45 in Auditorium 1.

### Esbjerg (email: baq@civil.aau.dk)

Here the math cafe in general runs Tuesday afternoon approximately every other week.
Currently the allocated dates if you have signed up by email are (will be updated throughout the semester):
• Wednesday 20/9-17 16:15-17:45 in room B202.
• Wednesday 27/9-17 16:15-17:45 in room B202.
• Wednesday 4/10-17 16:15-17:45 in room B202.
• Wednesday 11/10-17 16:15-17:45 in room B202.
• Wednesday 18/10-17 16:15-17:45 in room B202.
• Wednesday 25/10-17 16:15-17:45 in room B202.
• Wednesday 1/11-17 16:15-17:45 in room B202.
• Wednesday 8/11-17 16:15-17:45 in room B202.
• Wednesday 15/11-17 16:15-17:45 in room B202.
• Wednesday 22/11-17 16:15-17:45 in room B202.
• Wednesday 29/11-17 16:15-17:45 in room B202.
• Wednesday 13/12-17 16:15-17:45 in room B202.

Here the math cafe in general runs Tuesday afternoon.
Currently the allocated dates if you have signed up by email are (will be updated throughout the semester):
• Tuesday 19/9-17 16:15-17:45 in room 0.108 Fkj. 10A.
• Tuesday 26/9-17 16:15-17:45 in room 0.108 Fkj. 10A.
• Tuesday 17/10-17 16:15-17:45 in room 0.108 Fkj. 10A.
• Tuesday 24/10-17 16:15-17:45 in room 0.108 Fkj. 10A.
• Tuesday 31/10-17 16:15-17:45 in room 0.108 Fkj. 10A.
• Tuesday 7/11-17 16:15-17:45 in room 0.108 Fkj. 10A.
• Tuesday 14/11-17 16:15-17:45 in room 0.108 Fkj. 10A.
• Tuesday 21/11-17 16:15-17:45 in room 0.108 Fkj. 10A.
• Tuesday 28/11-17 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 18th of November 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 Tuesday the 14th of November 2017.

## Preparation for the exam

The curriculum for the exam can be found under the tab "Curriculum", and the exercises at the exam will be within these topics. It is a good idea to cover the entire curriculum by using the overview of each lecture.

Example: The exercises about curvature are divided into:

• Calculation of arc length.
• Calculation of curvature.
• Determining the time of maximal curvature.
• Determining the tangent and unit normal vectors.
• Determining the circle of curvature.
• Make sure that you can solve exercises of each of these types. Read the examples in the corresponding chapter as well.

Reflect on the following general principles.
Which topics are connected/build upon others? Make an overview to yourself, and/or discuss it in your group.

Remember True/False.
Use these exercises to figure out the details of the curriculum.

Then solve previous exam questions - purpose: To see how the exercises are phrased. To practice the different types of multiple choice questions. Note that exam questions from previous exams which were not multiple choice can easily be relevant; the only difference is the way, the answer is given.

We offer assistance with the exam preparation. You will get the largest benefit if you have solved many exercises in advance and know where you need help.

## Dates for Q&A session

The Q&A sessions before the reexam in Calculus are given below.

### Aalborg

Two teaching assistants will be available to help you while you prepare for the exam. They are present in AUD 1 on Monday the 19th of February and in AUD 2 on Tuesday the 20th of February, both days at 14:15-17:15

### Esbjerg

There is a Q&A session thursday the 8th of February at 12:30-14:15. This takes place in C1.119