# 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

Above-mentioned 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:

## Exam

Information about the exam can be found under the tab Exam information.

### Hand-ins

During the course, written exercises will be given. For the degree programs listed below, the enrolled students can only attend the exam if at least 10 out of 18 of these hand-ins are approved. The extent of each exercise is expected to be around one handwritten sheet of A4-paper.

If the degree programme is not listed, it is still possible to hand-in exercises and receive feedback.

The hand-in exercises for each lecture will be listed at the Problems page.

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## 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 and 3 regarding function values.
Differentiate inverse trig. functions: 5, 6, 17.
Integrate inverse trig. functions: 31,  35.

Hand-in assignments:

• A: 20, 43 in Appendix C; 5, 31 in Section 6.8
• B: 15, 43 in Appendix C; 5, 31 in Section 6.8

### 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 from 9.2: 29, 31, 39, 41, 53 & 63.

Hand-in assignments:

• A: 7, 11, 13, 21 in Section 9.2
• B: 2.f, 7, 21 in Section 9.2

### 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 acceleration vectors: 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.

Hand-in assignments:

• A: 1, 11, 13, 21 in Section 11.5
• B: 13,23 in Section 11.5

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.

Hand-in assignments:

• A: 1, 7, 11, 15 in Section 11.6
• B: 1, 9 in Section 11.6

### Lecture 5: Introduction to functions of several variables.

Material: Section 12.1, 12.2 and possibly parts of 12.3 in 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.

Hand-in assignments:

• A: 7, 29, 41 in Section 12.2; 4, 43 in Section 12.3
• B: 3, 25, 41 in Section 12.2

### Lecture 6: 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).

Hand-in assignments:

• A: 5, 21, 31 in Section 12.4
• B: 3, 21, 31 in Section 12.4

### Lecture 7: 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 plane: 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.

Hand-in assignments:

• A: 9, 25 in Section 12.5
• B: 21, 23 in Section 12.5

### Lecture 8: 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.

Hand-in assignments:

• A: 1, 21 in Section 12.7
• B: 1, 19 in Section 12.7

### Lecture 9: 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.

Hand-in assignments:

• A: 11, 21, 33 in Section 12.8
• B: 5, 21, 29 in Section 12.8

### Lecture 10: 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).

Hand-in assignments:

• A: 15, 19, 37 in Section 13.1
• B: 11, 15, 19 in Section 13.1

### Lecture 11: 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.

Hand-in assignments:

• A: 19, 31 in Section 13.2; 11 in Section 13.3
• B: 31 in Section 13.2; 11 in Section 13.3

### Lecture 12: 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.

Hand-in assignments:

• A: 1, 9, 13 in Section 13.4
• B: 9, 13 in Section 13.4

### Lecture 13: 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.

Hand-in assignments:

• A: 3, 17 in Section 13.6
• B: 3, 17 in Section 13.6

### Lecture 14: Complex numbers.

Material: Section 1.1, 1.2, and 1.3 in MN
Topic: Introduction 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 interpretation of complex numbers: 3, 7.
MN §1.3 - Polar form of complex numbers: 5, 7.

Hand-in assignments:

• A: 7, 11 in Section 1.1; 5(a,d), 7(a-c, g) in Section 1.3
• B: 11, 15 in Section 1.1; 5 in Section 1.2

### Lecture 15: 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 exponential: 7, 8, 9, 10, 11.
MN §1.4 - Trigonometric identities: 12, 13.

Hand-in assignments:

• A: 3, 13 in Section 1.4
• B: 1b, 3b, 9 in Section 1.4

### Lecture 16: 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.

Hand-in assignments:

• A: 7, 17 in Section 2.2; 7, 17 in Section 2.3
• B: 3, 17 in Section 2.2; 7 in Section 2.3

### Lecture 17: 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.

Hand-in assignments:

• A: 1, 13 in Section 4.2; 3, 21 in Section 4.3
• B: 13 in Section 4.2; 21 in Section 4.3

### Lecture 18: Inhomogeneous second order differential equations and the superposition principle.

Material: MN §4.4 and §4.5.
Topic: Review of Sections 4.4 and 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 and 17.
MN §4.5 -- Use the superposition principle: 1, 2
MN §4.5 -- Find the general solution: 3 and 5.

Hand-in assignments:

• A: 11 in Section 4.4; 1, 5 in Section 4.5
• B: 11 in Section 4.4; 3 in Section 4.5

### Self-study session 1.

This session concerns Taylor polynomials. Details can be found here.

### Self-study 2.

This session reviews Chapter 12 in E&P. Details can be found here.

### Self-study 3 (self-study).

This session concerns an application of double integrals in calculating mass and centre of gravity (centroid). Details can be found here.

### Self-study 4.

This session concerns applications of second order differential equations. Details can be found here.

## Self-study sessions

Work is done in your group room.

### Self-study session 1: Taylor polynomials

#### Description

Program of the day:

• Read Section 10.4, pages 743-749, in E&P regarding Taylor polynomials and Taylor's 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.

### Self-study session 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 two days 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 received a response!. Please indicate in the email what you need help with (typically just 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.

Information on when and where the math cafe will take place is coming soon.

### Aalborg (email: dhaug16@student.aau.dk)

Currently the allocated dates if you have signed up by email are (will be updated throughout the semester):

• Tuesday 22/10-19 16:15-17:45 in Auditorium 1.
• Tuesday 29/10-19 16:15-17:45 in Auditorium 1.
• Wednesday 6/11-19 16:15-17:45 in Auditorium 1.
• Tuesday 12/11-19 16:15-17:45 in Auditorium 1.
• Tuesday 19/11-19 16:15-17:45 in A309, Strandvejen 12–14.
• Tuesday 26/11-19 16:15-17:45 in A309, Strandvejen 12–14.
• Tuesday 3/12-19 16:15-17:45 in A309, Strandvejen 12–14.
• Tuesday 10/12-19 16:15-17:45 in A309, Strandvejen 12–14.
• Tuesday 17/12-19 16:15-17:45 in A309, Strandvejen 12–14.

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## Exam

The exam will be a digital exam with invigilation. That means that you have to show up like an ordinary written exam, but that the exam questions are answered online through Moodle.

All students must bring their own computer with internet access, but only the use of DigitalEksamen and Moodle is allowed – digital notes are not allowed. To prevent cheating the program ITX-Flex must be running during the exam. This must be installed in advance; how this is done can be found in the official guidelines.

We recommend using one of the following browsers to answer the questions in Moodle: Chrome, Firefox, Opera, or Safari. It is, in principle, possible to answer the questions using Internet Explorer or Edge, but the question layout may be inconvenient.

### During the exam

At the start of the exam you are required to log in to both DigitalEksamen and ITX-Flex. Here, you will find a link to Moodle, where the exam questions themselves will be answered.

In Moodle, you are asked to choose between Danish and English exam questions – this can only be chosen once. It is a good idea to select a language before the exam in order to avoid delay on the exam day. Once the language has been selected, the corresponding exam questions will be unlocked at the start of the exam. Answer the questions like you would in any multiple-choice exam.

### Submission

After having finished your attempt, you must first submit it in Moodle. Afterwards, you must download one of the forms found on the ‘Set of exam questions’ in DigitalEksamen/ITX-Flex. This is to be filled in with name and student-number and then uploaded and submitted in DigitalEksamen. This is important, as your hand-in cannot be graded otherwise. Once the submission in Moodle closes, you have an additional 10 minutes to finish your submission in DigitalEksamen.

### What is allowed?

You are allowed to use handwritten, printed, and copied notes, as well as textbooks.

You are not allowed to use electronic devices, except for accessing DigitalEksamen and the exam page in Moodle. Visiting other webpages is not allowed either.

For additional information about the exam and the current rules, we refer to the guidelines that may be found on the Moodle page for exams on the first year of study.

## 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.

## Dates for Q&A-sessions

We offer assistance with the exam preparation in both calculus and linear algebra at all three campi. This consists of a Q&A-session, where it is possible to ask questions within the syllabus and receive help in solving concrete exercises. During this session, it is also possible solve exercises on your own, and then ask for hints if you get stuck. The session takes as its starting point the old exam questions, which may be found here at first.math.aau.dk. We recommend that you solve as many as you can beforehand, such that you know where you come short. Note that the teaching assistants will not visit you in your group rooms. Instead, everyone will be solving exercises individually or in small groups in the rooms specified below.

### Aalborg

Teaching assistants will be available to help you while you prepare for the exam. They are present in AUD 7 on Friday the 10th of January and Monday the 13th of January, both days at 9:00–12:00 and 12:30–15:30.

Before the re-exam there will be a Q&A-session on Monday the 24th of February at 8:00–12:00. This takes place in Auditorium 3.

### Esbjerg

There will be a Q&A-session Monday the 24th of February. This happens at 13:00–15:00 and will take place in B108.