## Calculus 2M. Spring 2020

### Class Information

Instructor: Prof. June Amillo. E-mail: amillo@fi.upm.es

Office Hours: by appointment. Office: 1317.

Class Hours: Tuesday 12:00-14:00 and Friday 10:00-12:00. Room: 6106.

### Announcements

July 31, 2020:  This course has been discontinued and this website will not be updated.

### Course Description

This is an accelerated course of one and several variable calculus. The first part of the course is devoted to those topics of one variable calculus not covered in pre-university math courses: applications of differential and integral calculus and the study of series. The second part deals with differential calculus of more than one variable and multiple integrals leaving out integration in vector fields.

The course is applications oriented and the presentation is not rigorous: some results are proven formally but others are only justified intuitively. The applications are drawn from the world of engineering and economics. Mathematical software will be used extensively as a tool to understand concepts and solve problems.

### Learning Outcomes

By completion of the course the student will have achieved competency in the following skills:

1. Use derivatives to compute differentials and approximate linearly functions of one variable.
2. Apply derivatives to analyze the graph of a function.
3. Apply differential calculus to solve optimization problems.
4. Find anti derivatives and evaluate integrals.
5. Apply integration to find the area of plane regions and volumes of revolution.
6. Analyze convergence of improper integrals and evaluate them.
7. Find limits of sequences and establish the order of growth.
8. Analyze the convergence of series and sum geometric series.
9. Find the power series expansion of functions.
10. Represent two and three dimensional curves and find tangent lines.
11. Calculate arc length and surface of revolution.
12. Graph functions of several variables and understand contour lines.
13. Compute partial derivatives of functions of several variables.
14. Understand gradients and use them to find tangent planes and normal lines.
15. Apply differential calculus of several variables to solve optimization problems.
16. Apply the Lagrange method to solve max/min problems with constraints.
17. Compute double and triple integrals and apply them to find volumes.

### Syllabus

Each topic below corresponds to each two hour lecture but there could be some overlapping between lectures.

### Lecture Notes Exercise/Lab Sets and Assignments

Extension of Differential Calculus

1. Derivatives, tangent and normal lines
2. Differentials and linear approximations
3. Graphing, max. and min.
4. Optimization

Derivatives
Exercise Set 1: 1.1, 1.3, 1.4; 2.1, 2.2, 2.5; 3.2, 3.4, 3.5, 3.7; 4.2, 4.3, 4.6
Lab Practice 1: Lab Practice 2: (for review only)
Lab Practice 3: 5, 6, 7, 8

Extension of Integral Calculus

1. Indefinite integrals. (Exercise Set 1 due)
2. Initial value problems
3. Definite Integrals
4. Areas and volumes
5. Improper Integrals

Integrals
Exercise Set 2: 3.1, 3.2, 3.3, 3.4, 3.6; 4.1, 4.4, 4.5; 5.1, 5.3, 5.4, 5.5, 5.6; 6.1, 6.2, 6.5
Lab Practice 4: 1, 2, 3, 4, 7

Series

1. Sequences. (Exercise Set 2 due)
2. Series: The geometric series
3. Convergence criteria and harmonic series
4. Power series and Taylor expansions

Series
Exercise Set 3: 1.2, 1.4, 1.5; 2.2, 2.3, 2.4, 2.5; 3.3, 3.4d, 3.5d; 4.1, 4.2, 4.3
Lab Practice 5: 4, 5, 6, 8

Midterm Exam

1. Covers lectures 1-13. (Exercise Set 3 due)

Curves and Vector Functions

1. Parametric equations and vector functions.
2. Arc length and area of a surface of revolution.
3. Polar coordinates.

Curves
Exercise Set 4: 1.3,1.4,1.5, 2.1 d&f, 2.2, 2.3, 2.4, 3.2, 3.3.
Lab Practice 6: 1, 2, 7, 9, 10

Partial Derivatives

1. Domains, level curves and limits of two variable functions. (Exercise Set 4 due)
2. Partial Derivatives and differentials.
3. Chain rule and implicit differentiation.
4. Directional derivatives and gradients. (Exercise Set 5 due)
5. Maxima and minima.
6. Lagrange multipliers.
7. Review

Partial Derivatives
Exercise Set 5: 1.1, 1.2d, 1.3d, 2.1, 2.3b, 2.4c
Exercise Set 6: 1.1, 1.2, 1.3, 1.5, 1.6, 2.1, 2.2.a, 2.5, 3.1, 3.2, 3.4, 3.5, 3.6
Lab Practice 7: Section 2.1: 1, 2, 3, 5, 6; Section 2.2: 1.a, 2.a, 3.b, 5

Multiple Integration

1. Double Integrals. (Exercise Set 6 due)
2. Double Integrals in polar coordinates.
3. Applications and triple integrals.

Multiple Integrals

Final Exam

1. Covers lectures 15-27. (Exercise Set 7 due)

### Textbook

There is no specific textbook. For complimentary reading you can use one of the following popular books:

• Salas S. L. & E. Hille, Calculus: One and Several Variables, 9th Edition, John Wiley, New York, 2002
• Stewart J., Calculus,  6th  Edition, Brooks Cole, Toronto, 2007
• Strang G., Calculus, Online Text, http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/
• Thomas G. B. & R. L. Finney, Calculus and Analytic Geometry, 9th Edition, Addison-Wesley Reading, Massachusetts,1999

• Default scheme:
• Midterm Exam: 40%.
• Final Exam: 60%.
• Minimum grade required in each part: 3 out of 10.
• Passing grade: weighted average score of 5 out of 10.
• The midterm exam can be retaken on the date scheduled by the Dean's office.
• Alternative scheme: (requires an attendance ratio greater than 0.85)
• Home assignments: 20%.
• Take Home Projects: 20%.
• Midterm Exam: 24%.
• Final Exam: 36%.
• Passing grade: weighted average score of 5 out of 10.
• Hand in only assignments in bold type.
• Students may work in teams but must write up their work independently using their own words.
• Mathematical software can be used to solve exercises and lab practices.
• Students are expected to devote five or six hours a week to the course outside the class.