# Calculus, 2015 autumn

## 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 1-783-99028-7 and is available in the book store. The compilation by Morten Nielsen is not sold elsewhere.

In addition, the following materials are used:

## Exam

Please, look at Moodle.

## Problems

Plan for Calculus F2014.

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.

### 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
Addition formulas: 37
Trig. equations: 43, 47.
Supplemental Exercises: 17, 19, 21, 38, 39.

### 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 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, 8, 17, 18.
Integrate inverse trig. functions: 31, 32, 35, 36, 47, 48.
Supplemental excercise: 56.

### 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 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 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 11.5, 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 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 11.6:
Calculate arc length: 1, 2, 3, 5.
Calculate curvature: 7, 10, 11.
Find time of maximal curvature: 15, 16.
Calculate unit normal and unit tangential vectors: 17, 19.
Determine the circle of curvature: 29.

Supplemental Exercises
Exercises regarding projectiles: 43, 44, 45, 46 from E&P Section 11.5. Use that 1 mile equals 1609.344 meter, and that g = 9.80665 m/s2.

### 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 12.2, 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 8: Optimization.

Material: Section 12.5 in E&P.
Topic: Review of Section 12.5 in E&P. The topic is optimization.
Exercises: Section 12.4, 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 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 12.5, 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 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 12.7 true/false study guide, page 959.

Section 12.7:
Use the chain rule to find partial derivatives: 1, 7.
Write down the chain rule in a given setup: 13.
Implicit differentiation: 19, 21, 23.
The chan rule and partial differential equations: 40, 43.

### 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 12.8 true/false study guide, page 970.

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.

### 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 13.1 true/false study guide, page 1003.

Section 13.1:
Evaluate iterated integrals: 11, 13, 15, 17, 19, 21, 25, 27, 29, 31. Riemann sum: 31.

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

### 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 13.3 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 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 13.4 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 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:

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 18: The Complex Exponential Function

Material: Section 1.4 in MN
Topic: Review of Section 1.4 regarding the complex exponential function.
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 19: Complex polynomials.

Material: Notes on complex polynomials.
Topic: A review of the notes on complex polynomials with special focus on the quadratic equation.
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.

Then any remaning "old" problem on complex numbers.

### 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: Problem sheet on complex polynomials.

### 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 §4.2 - 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 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

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.