Fall 2017 & IAP 2018
Single & Multivariable Calculus
Calculus I and II
Math 18.01A-18.02A, Concourse - MIT

Lectures and Recitations by:
Robert Winters

Office: 16-137
Phone: x3-2050
(but e-mail is better)

Calendar of topics and
homework assignments
(username/password required)

Concourse 18.01A (CC.181a)
Stellar site

Concourse 18.02A (CC.182a)
Stellar site


The text for the course is
Calculus with Analytic Geometry, 2nd Edition by George F. Simmons (ISBN 9780070576421), published by McGraw-Hill
[Click on the image below for prices.]

Simmons, 2nd Edition
You can probably get the best price for this book from MIT students who took this course. The book has not changed in years.

In addition, you will want to purchase or download a copy of the 18.02A Course Notes.

Table of Contents:

• Ch. 1 - Numbers, Functions, and Graphs

• Ch. 2 - The Derivative of a Function

• Ch. 3 - Computation of Derivatives

• Ch. 4 - Applications of Derivatives

• Ch. 5 - Indefinite Integrals and Differential Equations

• Ch. 6 - Definite Integrals

• Ch. 7 - Applications of Integration

• Ch. 8 - Exponential and Logarithm Functions

• Ch. 9 - Trigonometric Functions

18.01A Topics

Ch. 10 - Methods of Integration

Ch. 11 - Further Applications of Integration

Ch. 12 - Indeterminate Forms and Improper Integrals

Ch. 13 - Infinite Series of Constants

Ch. 14 - Power Series

• Ch. 15 - Conic Sections

• Ch. 16 - Polar Coordinates

18.02A Topics

• Ch. 17 - Parametric Equations, Vectors in the Plane

• Ch. 18 - Vectors in Three-Dimensional Space, Surfaces

• Ch. 19 - Partial Derivatives

• Ch. 20 - Multiple Integrals

• Ch. 21 - Line and Surface Integrals, Green's Theorem, Gauss' Theorem, and Stokes' Theorem


IAP Schedule (starting Monday, January 8):
Lecture/Recitation: Mon, Wed 11:00am to 12:00pm; Tues, Thurs, 10:00am to 12:00pm in 16-160.
Additional recitation time informally afternoons in Concourse Lounge.

Problem Set #15 (due Tues, Jan 30)
References: Notes V7 (Laplace's Equation and Harmonic Functions); 15.5 (Surface Integrals); Notes V9 (Surface Integrals); RW-Supplement on surface integrals; Notes V10 (The Divergence Theorem); and Lecture Notes.

Problem Set #16 (due Thurs, Feb 1)
References: Notes V10 (The Divergence Theorem); Notes V13 (Stokes' Theorem); Notes V14 (Some Topological Questions); Notes V15 (Relations to Physics); and Lecture Notes.

The Final Exam will take place on Friday, February 2 in 16-160 from 1:00pm to 4:00pm.
Two (mainstream) Practice Exams are posted.
Practice Final Exam A    Practice Final Exam B
Solutions to Practice Final Exam A      Solutions to Practice Final Exam B

Exam 5 will take place on Monday, January 22. The exam may cover vector fields, curl and divergence of a vector field; integration along curves, line integrals, Green's Theorem; triple integrals in Cartesian, cylindrical, and spherical coordinates; applications of integration (area, volume, mass, average value, weighted averages, centroids, center of mass, moment of inertia, etc.); and change of variables in multiple integrals.
Exam #5 Practice Questions (18.02A - IAP)      Solutions

Problem Set #14 (due Wed, Jan 24)
References: Simmons text sections 20.5, 20.6, and 20.7; Notes CV (Changing Variables in Multiple Integrals); Notes G (Gravitational Attraction); and Lecture Notes.

Problem Set #13 (due Thurs, Jan 18)
References: Simmons text, section 21.3; Notes V3 (Two-dimensional Flux); Notes V4 (Green's Theorem in Normal Form); and Lecture #12 Notes

Problem Set #12 (due no later than noon on Fri, Jan 12)
References: Notes V1 (Plane Vector Fields); Notes V8 (Vector Fields in Space); Notes V11 (Line Integrals in Space); Notes V2 (Gradient Fields and Exact Differentials); Notes V12 (Gradient Fields in Space); Simmons text, sections 21.1 and 21.2; and posted Lecture Notes.

Problem Set #11 [Do these problems but don’t turn them in. We’ll go over some of them in class.]
References: Read 20.1 (Volumes); 20.2 (Double Integrals and Iterated Integrals); SN-I (Limits in Iterated Integrals); 20.3 (Applications of Double Integrals); 16.1 (Polar Coordinates); 16.2 (Graphs of Polar Equations); 20.4 (Double Integrals in Polar Coordinates); SN-CV (Change of Variables in Double Integrals); and Lecture Notes.

Exam #4 took place on Thurs, Dec 7. Topics included partial derivatives, linear approximation, differentials, gradient vector, normal vectors to curves and surfaces, directional derivative, the Chain Rule, implicit differentiation, stationary points, second derivative test, optimization, Method of Lagrange Multipliers, constrained optimization with one or more constraints, partial derivatives in the case of non-independent (constrained) variables.

Approximate letter grades for Exam #1 (18.01)
Total points on exam was 50. Median score was 40.
Mean score was 39.6. Standard deviation was 3.1.
score grade   score grade
45+ A 30+ C+
42+ A– 27+ C
39+ B+ 25+ C–
36+ B 23+ D
33+ B– 0-22 F
Exam 1 Practice Questions    Solutions
Exam #1 solutions
Approximate letter grades for Exam #2 (18.01A Final)
Total points on exam was 76. Median score was 53.
Mean score was 54.3. Standard deviation was 7.8.
score grade   score grade
67+ A 47+ C+
63+ A– 43+ C
59+ B+ 39+ C–
55+ B 37+ D
51+ B– 0-36 F
18.01A Final Practice Exam     Solutions
Exam #2 solutions
Approximate letter grades for Exam #3 (18.02)
Total points on exam was 60. Median score was 39.5.
Mean score was 41. Standard deviation was 6.6.
score grade   score grade
53+ A 38+ C+
50+ A– 35+ C
47+ B+ 32+ C–
44+ B 30+ D
41+ B– 0-29 F
Exam 3 Practice Questions     Solutions
Exam #3 solutions
Approximate letter grades for Exam #4 (18.02)
Total points on exam was 50. Median score was 40.5.
Mean score was 40.9. Standard deviation was 4.6.
score grade   score grade
45+ A 32+ C+
43+ A– 30+ C
40+ B+ 28+ C–
37+ B 26+ D
34+ B– 0-25 F
Practice Exam #4 Questions     Solutions
Exam #4 solutions

Problem Set #10 (due Wed, Dec 6)
References: 19.7 (Maximum and Minimum Problems); 19.8 (Constrained Maxima and Minima, Lagrange Multipliers); 19.10 (Implicit Functions); RW-Chain Rule and Implicit Differentiation; SN-LS (Least Squares Interpolation); SN-N (Non-independent Variables, sections 1-3); and Lecture Notes.

Problem Set #9 (due Wed, Nov 29)

Exam #3 took place of Thursday, Nov 16.

Problem Set #8 (due Thurs, Nov 16)

Problem Set #7 (due Wed, Nov 8)
References: 17.1, 17.2, 17.4, 17.5; 18.02 Supplementary Notes: SN-D (Determinants), SN-M (Matrices and Linear Algebra); Supplement on Solving Systems of Linear Equations via Row Reduction; Supplement: Vector and matrix forms of a system of linear equations, matrix algebra, inverse matrices, and related facts

Problem Set #6 (due Tuesday, Oct 31)
References: Read text: 17.3, 18.1, 18.2, 18.3, 18.4.
RW-Vectors (Supplement on Vectors, dot product, projections, cross product); Lecture Notes #1, Lecture Notes #2, Lecture Notes on Coordinates & Vectors, Lecture Notes on Dot Product, Cross Product, Planes, Area, and Volumes

Exam #2 (a.k.a. the 18.01A Final Exam) took place on Tuesday, Oct 24. Math 18.01A Essentials (PDF)

Problem Set #5 (due Monday, Oct 23)
References: Read relevant portions of text: 14.1-14.5;
Supplement on Integral Test and Comparison Tests;
Supplement on Alternating Series; Absolute vs. Conditional Convergence; Ratio Test; Strategies

Exam #1 took place on Tues, Oct 3.

Problem Set #4 (due Friday, Oct 13)
References: 12.4 (Improper integrals); SN-INT (Improper Integrals); and class notes on numerical integration.

Problem Set #3 (due Wed, Oct 4)
References: Text 10.2-10.4, 10.6-10.7, 10.9; SN: F.

Problem Set #2 (due Tues, Sept 26)
References: SN-PI (Properties of Integrals); SN-FT (Second Fundamental Theorem); text: 6.3-6.6, 7.1-7.7; SN-AV (Average Value)

Problem Set #1 (due Thurs, Sept 14)

This sequence, intended for students who have had a full year of high school calculus, begins with 18.01A, a six-week review of one-variable calculus, emphasizing integration techniques and applications, polar coordinates, improper integrals, sequences, and infinite series. Prerequisite is a score of 4 or 5 on the Advanced Placement Calculus AB exam or a passing grade on the first half of the 18.01 Advanced Standing Exam, covering differentiation and elementary integration. Most students completing 18.01A continue directly into 18.02A, in which the remaining weeks of the fall term is devoted to the material in the first half of Calculus II. 18.02A is taught at the same pace as 18.02. Concourse students complete the second half of Calculus II during Independent Activities Period (IAP) in January.

Syllabus for Concourse Math 18.01A/02A     Printable syllabus (PDF)

Supplementary Notes:
18.01 Supplementary Notes authored by Prof. Arthur Mattuck of the MIT Mathematics Department, exercises by David Jerison.
18.02 Supplementary Notes authored by Prof. Arthur Mattuck of the MIT Mathematics Department.

Text: Calculus with Analytic Geometry, 2nd Edition by George F. Simmons (ISBN 9780070576421), published by McGraw-Hill

18.01/02 Mathlets (to appear)

Homework: Homework will be posted on the course website and will be due approximately weekly. Typical assignments will include some exercises that are to be turned in as well as additional practice problems. Homework may be submitted in class or at my office, but it should be completed by the posted due date. Additional time will only be given if requested before the due date and if appropriate for the circumstances. You should not consult any solutions manual in preparing your assignments. You are encouraged to work with your fellow students on the homework, but your written solutions must be your own. Solutions will be made available (as PDF files) on the course website shortly after they are due.

Class during IAP is expected to be at the same time as the mainstream class: Daily, Mon-Fri, 12:00-1:00pm. We could meet for longer on some days in order to create a free day, but only if everyone agrees.

We'll also schedule a few recitation times during the week based on the preferences of the class. Alternatively, we can hold informal meetings on several afternoons around the conference table in the Concourse Lounge.

Condensed Syllabus: (See the Calendar for day-by-day details and assignments, updated as the course proceeds.)

18.01A Topics:

  • Review of basic ideas of Differential Calculus. (Chaps. 2-7)
  • Applications of Integration: area, volume, volume of solids of revolution, arclength, area of a surface of revolution, work and energy, hydrostatic force. (Chap. 7)
  • Techniques of integration: substitution, trigonometric integrals, trigonometric substitutions, partial fractions, integration by parts, miscellaneous methods, numerical integration and Simpson's Rule. (Chap. 10)
  • Further Applications of Integration: Center of mass, centroids, moment of inertia. (Chap. 11)
  • Indeterminate forms, L'Hôpital's Rule, improper integrals.
  • Sequences, infinite series, convergent vs. divergent series, comparison tests, integral test, ratio and root tests, alternating series, absolute vs. conditional convergence. (Chap. 13)
  • Power series, interval of convergence, differentiation and integration of power series, Taylor Series and Taylor's Formula, applications to differential equations. (Chap. 14)
  • Probability. (Supplementary Notes)

18.02A Topics:

  • Coordinates, vectors and vector algebra in R2 and R3; dot product, cross product, projection, equations of lines and planes, matrix methods. (Chaps. 17-18 and Notes)
  • Parametric equations of curves in R2 and R3; coordinates, derivatives of vector-valued functions, velocity and acceleration, tangent vectors, arclength; curvature and unit normal vector, tangential and normal components of acceleration, Kepler's Laws and Newton's Law of Gravitation. (Chaps. 17-18)
  • Cylinders and surfaces of revolution, cylindrical and spherical coordinates; parameterized surfaces in R2 and R3. (Chap. 18)
  • Functions of several variables - limits, continuity, and differentiabilty; partial derivatives, gradients, linear approximation, directional derivatives, Chain Rule. (Chap. 19)
  • Optimization - unconstrained and constrained; implicit functions and implicit differentiation. (Chap. 19)
  • Multiple integrals, integration over regions in R2 and R3 and their applications using Cartesian, polar, cylindrical, and spherical coordinates, gravitational attraction. (Chap. 20)
  • Vector fields and their applications. (Notes)
  • Integration over curves in R2 and R3 by parameterization; work integrals, and applications; independence of path and conservative vector fields; Green's Theorem. (Chap. 21)
  • Integration over surfaces in R3 by parameterization - flux integrals, surface area, and applications. (Chap. 21)
  • Calculus of vector fields; curl and divergence of vector fields; Stokes’ Theorem, Divergence Theorem; Maxwell's equations. (Chap. 21)

Topics and Assignments are posted in the Course Calendar.

Mega-List of Math 18.02 techniques     Math 18.02 Useful Facts

Singular Sensations - Steve Strogatz in the New York Times

If ever the MIT mail servers are not accessible from the outside world and you need to get in touch, you can also contact me at either robert@math.rwinters.com or Robert@rwinters.com.

Here's something:  http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html

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Please send comments to Robert Winters.
URL: http://math.rwinters.com/18012A
Last modified: Thursday, February 1, 2018 1:13 PM