Docencia \ Grado en Ingeniería Informática

Calculus 2M-I. Spring 2017

Class Information

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

Office Hours: by appointment. Office: 1317.

Class Hours: Tuesday 09:00-11:00, Thursday 13:00-14:00 and Friday 11:00-13:00. Room: Artá.


Announcements

June 12, 2017: Final Grades as they will show on records.

 


 

 


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 the mean value theorem and its consequences to analyze functions of one variable.
  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 two hour lecture will be devoted to one of the topics listed on the syllabus but there could be some overlapping between lectures. The one hour lecture will be used to compensate for exams and festivities.

Lecture Topics
 

Lecture Notes
Exercise/Lab Sets and Assignments

Extension of Differential Calculus

  1. Derivatives, tangent and normal lines
  2. Differentials and linear approximations
  3. Fundamental Theorems and applications
  4. Optimization

Derivatives
Exercise Set 1: 1.1, 1.3, 1.4; 2.1, 2.2, 2.5; 3.2, 3.4, 3.5; 4.2, 4.3, 4.4
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, 3.4, 3.5; 4.1, 4.2, 4.3
Lab Practice 5: 4, 5, 6, 8

Mid Term 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, 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., 1.2., 1.5, 2.1, 2.3, 2.4, 3.5, 3.6
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: Iteration. (Exercise Set 6 due)
  2. Double Integrals in polar coordinates.
  3. Applications and triple integrals.

Exercise Set 7: 1.1

End Term 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

Grading

  • The final grade will be based on the following percentages:
    • Exercise set assignments: 20%.
    • Lab practice: 20%.
    • Two 60 minute exams: 30% each.
  • Class attendance is required.
  • There are no make-up exams.
  • Hand in only assignments in bold the rest are in class assignments.
  • Do not hand in Lab practice assignments, they will be assessed in class before the mid-term and end-term exams.
  • Students are expected to devote 4-5 additional hours a week to personal study and to complete home assignments.
  • Students may work by teams but must submit their work individually including their own explanations.
  • Mathematical software can be used to solve exercise and lab practice sets.

Exams

End-Term Exam 2012
July Exam 2012

End-Term Exam 2013
July Exam 2013

Mid-Term Exam 2014
End-Term Exam 2014

Mid-Term Exam 2015
End-Term Exam 2015

Mid-Term Exam 2016
End-Term Exam 2016

Mid-Term Exam 2017
End-Term Exam 2017