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Course

Calculus I - Math UN1101 - Section 002 and 003.
Fall 2020.
Columbia University.

Lectures

Classroom online, on Zoom.
Section 002: Mo, We 10:10am-11:25am.
Section 003: Mo, We 11:40am-12:55pm.

Instructor

Name: Daniele Alessandrini.
Contact: daniele.alessandrini@gmail.com.
Office: 624 Mathematics.

Office hours: Tu 5pm-6pm, Fr 5pm-6pm, online, on Zoom. The Zoom link is posted on Courseworks, in the Syllabus page. Please send me an email in advance or just before connecting to Zoom to make sure I see you.

TAs

The TAs grade the assignments and can answer questions from students. You can ask them questions by e-mail, and they serve in the Virtual Help Room, where you can reach them via Zoom to ask your questions. Their e-mail addresses and Zoom links to the Virtual Help Rooms are available on Courseworks in the Syllabus page.

Name Virtual Help Room
Haodong Yao Fr 9am-12am
Aya Tazi Th 5pm-7pm
Destine Lee Mo, We, 1pm-2pm

Syllabus

The Syllabus for this course.

Content

Required text

Calculus: Early Transcendentals, 8th edition, by James Stewart (CENAGE Learning). The book is available at the Columbia bookstore.

Course Outline

In this course we will describe some basic ideas and techniques that lie at the foundation of all pure and applied mathematics. We will discuss about functions and their limits, dervatives and integrals. We will focus on their meaning, significance, applications and methods of computation. We will use the first six chapters of the course textbook (Calculus, Early Transcendentals, by Stewart).

  1. Functions (Chapter 1).
    • Polynomials and rational functions.
    • Roots.
    • Exponential and logarithm.
    • Trigonometric functions.
  2. Limits (Chapter 2).
    • Computation of limits.
    • Continuous functions.
    • Increasing and decreasing functions.
  3. Derivatives.
    • Introduction to derivatives (Chapter 2).
    • Differentiation rules (Chapter 3).
    • Maxima and minima (Chapter 4).
    • Concavity (Chapter 4).
  4. Integrals.
    • Computation of integrals (Chapter 5).
    • Applications of integrals (Chapter 6).

Lectures

Date No Topic Textbook Reference
20/09/09 01 Prerequisites: Numbers and sets of numbers. n/a
20/09/14 02 Prerequisites: Intervals and equations. n/a
20/09/16 03 Prerequisites: Equations and inequalities. n/a
20/09/21 04 Inequalities. Basics about functions. Section 1.1.
20/09/23 05 Linear and piecewise linear functions. Range. Section 1.1, 1.2.
20/09/28 06 Increasing, decreasing and 1-1 functions. Inverse fuctions. Root functions Section 1.1, 1.2, 1.5.
20/09/30 07 Equations and inverse functions. Section 1.5.
20/10/05 08 Change of variables. Exponential and logarithm. Section 1.4, 1.5.
20/10/07 First Midterm n/a
20/10/12 09 Exponential and trigonometric functions. Section 1.4, Appendix D.
20/10/14 10 Trigonometric equations. Appendix D.
20/10/19 11 Limits at infinity. Section 2.6.
20/10/21 12 Limit laws. Section 2.3.
20/10/26 13 Limits at a finite number, limit laws. Section 2.2, 2.3.
20/10/28 14 Continuous functions, Derivatives. Section 2.5, 2.7.
20/11/02 Holiday. n/a
20/11/04 15 Derivatives, Differentiation Rules. Section 2.5, 3.1, 3.2, 3.3, 3.4.
20/11/09 16 Computing Derivatives, differentiable functions, graphing functions. Section 2.8, 3.6, 4.3, 4.5.
20/11/11 17 Graphing functions, L'Hospital Rule. Section 4.4, 4.5.
20/11/16 18 Absolute and local maxima and minima. Section 4.1.
20/11/18 19 Second derivative, concavity. Integrals. Section 4.3.
20/11/23 Second midterm. n/a
20/11/25 Holiday. n/a
20/11/30 20 The fundamental theorem of calculus. Section 5.3.
20/12/02 21 Substitution Rule. Section 5.5.
20/12/07 22 Integration by parts. Areas between curves. Section 7.1, 6.1
20/12/09 23 Computing volumes with integrals. Section 6.2
20/12/14 24 Review lecture.
Prerequisite

There are no formal pre-requisites; anyway, an understanding of pre-calculus will be assumed. This includes high school mathematics.

Resources

If your pre-calculus is weak, the textbook has some review material in Appendix A and B. Moreover, the section "Principles of problem solving", at the end of Chapter 1 can also be helpful.

You can also consider the book Precalculus, Mathematics for calculus, by Stewart, Redlin, Watson. This book is not required, and not important for most students. It is useful only if you have gaps in your background knowledge.

Also, you can consider the website Khan Academy, it has plenty of useful instruction material freely available.

Homework

Homework sheets

The homework will be published online in form of homework sheets every Wednesday night. There is one week time to submit the written solutions.

Date Number Topic Hand in by Grader Answers
20/09/09 01 Numbers 20/09/16 Haodong Answers 1
20/09/16 02 Equations and inequalities 20/09/23 Aya Answers 2
20/09/23 03 Inequalities and Functions 20/09/30 Haodong Answers 3
20/09/30 04 Inverse function 20/10/07 Destine Answers 4
20/10/07 05 Exponential function 20/10/14 Haodong Answers 5
20/10/14 06 Exponential and trigonometric functions 20/10/21 Aya Answers 6
20/10/21 07 Limits at infinity 20/10/28 Haodong Answers 7
20/10/28 08 Limits at a finite number 20/11/04 Destine Answers 8
20/11/04 09 Derivatives 20/11/11 Haodong Answers 9
20/11/11 10 Graphs of functions 20/11/18 Aya Answers 10
20/11/18 11 Maxima, minima and inflection points 20/12/02 Haodong Answers 11
20/12/02 12 Integrals 20/12/09 Destine Answers 12
20/12/09 13 Applications of integrals not graded Answers 13
20/12/14 Mock Exam not graded

Handing in

Written assignments will be due on Wednesdays at 11:30pm. Hand-in is online, via Courseworks, at the Assignments page. We encourage collaboration on assignments, but all solutions must be written up by you alone in your own words.

Late hand in

We will accept late homework, but we deduct 15% of the points for every day of lateness.

Exams

Midterm exams

There will be midterm exams, on Wednesday October 7th and on Monday November 23, during the usual class times.

Final exam

Projected schedule for the final exam. For Section 002, Wednesday December 23rd, 9am– Noon. For Section 003, Monday December 21st, 9am–Noon. The date will be confirmed in November.

Exam details

The midterms and final exams will be online. Please, check the rules for on-line exams. You must plan to take the midterm and final exams at the scheduled time, so please make your plans accordingly. Besides students with disabilities having prior arrangements with ODS, the only exceptions will be for those with an incapacitating illness, a serious family emergency, or situations of comparable gravity. In the first case you will need a note from a doctor; in the other cases you will need a note from your advising dean. Incompletes can be granted only by your advising dean and only in the circumstances mentioned above. Anyone guilty of academic dishonesty, such as cheating on an exam or helping someone else to cheat, will fail the course and faces further academic discipline.

Grading

Homework 10%, midterms 25%, final 40%.