# 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.

### Teaching assistant service

There will be opened a Teaching Assistant service concerning answers to questions regarding Calculus exercises during the period May 26 to June 5 (including both dates).

Questions may be sent to the following email address: calculushjaelp@math.aau.dk. The number of the exercise that you need help for should be stated as subject, e.g. "Summer examination 2014, exercise 7" or "E&P chapter xxx exercise YYY".

Three teaching assistants are responsible for answering your questions. You may not expect an immediate answer. It depends on the number of questions asked as well as the availability of the teaching assistants.

On Friday June 5 it will be possible to meet one of the teaching assistants in Aalborg. You are to book time via above stated mail address no later than Thursday June 4, at 12 noon. "Subject" should be stated as follows: "Meeting Friday?" and the mail must include a specific question that you need help for and no more than three questions.

## Problems

Plan for Calculus F2014.

Literature:

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

Saff et al. - E.B. Saff et al. Complex numbers and differential equations, Custom print (2nd edition), Pearson, 2010.

### 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.
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 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 Saff et al.
Topic: Introduktion to a new topic; complex numbers. Review of Sections 1.1, 1.2 and 1.3 in Saff et al.

Exercises:

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

Section 13.6:
Evaluate triple-integrals: 1, 3, 7, 9.
Find volume using triple-integrals: 17, 23, 31.

### Lecture 18: The Complex Exponential Function

Material: Section 1.4 in Saff et al.
Topic: Review of Section 1.4 regarding the complex exponential function.
Exercises: From Saff et al. (notice: answers to odd-numbered problems on page 373)
Saff §1.1 - Express a complex number in the form a+ib: 1, 5, 7, 9, 11, 13.
Saff §1.1 - Laws of exponents: 15, 17.
Saff §1.2 - Geometric intrepretation of complex numbers: 3, 7.
Saff §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:

Saff §1.4 - Express a complex number in the form a+ib: 1.
Saff §1.4 - Express a complex number in polar form: 3.
Saff §1.4 - Polar form of complex numbers: 5.
Saff §1.4 - The complex exponentioal: 7, 8, 9, 10, 11.
Saff §1.4 - Trigonometric identities: 12, 13.

Then any remaning "old" problem on complex numbers.

### Lecture 20: Second order differential equations.

Material: Saff §4.1- §4.3.
Topic: The topic is second order differential equations: Saff §4.1- §4.3.
Exercises: Problem sheet on complex polynomials.

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

Material: Saff §4.4 and §4.5.
Topic: Review of Sections 4.4 og 4.5 in Saff regarding inhomogeneous second order differential equations and the superposition principle.
Exercises: Saff §4.2 - Find the complete solution to differential eq.: 1, 3, 5, 7.
Saff §4.2 - Solve initial value problem: 13, 15, 17.
Saff §4.2 - Linear independence: 27, 29.
Saff §4.3 - The characteristic equation: 1, 3, 5, 9.
Saff §4.3 - Find the complete solution to differential eq.: 11.
Saff §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

Ved alle miniprojekter skal I arbejde i jeres grupperum (der er ikke nogen forelæsning).

## 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