Fall 2016 & IAP 2017
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 Stellar site

Concourse 18.01A Stellar site

Concourse 18.02A 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 9):
Lecture: Daily, Monday through Friday, from 12:00pm to 1:00pm in 16-160 (subject to change).
Recitation takes place informally as needed immediately following class.

The Final Exam will take place on Friday, February 3 in 16-160 from 1:00pm to 4:00pm.

Practice exams from mainstream 18.02:
Practice Final Exam A     Solutions to Practice Final Exam A
Practice Final Exam B Solutions to Practice Final Exam B

(Some topics on these practice exams might not appear on Friday's exam, but they are a reasonable approximation.)

Mega-List of Math 18.02 techniques     Math 18.02 Useful Facts

You will be permitted to bring to the Final Exam ONE standard sheet of paper (single side) with information of your choosing. A good choice might be the "Math 18.02 Useful Facts" sheet listed above.

Problem Set #14 (due Thurs, Feb 2)
References: Notes V10 (The Divergence Theorem); Notes V13 (Stokes' Theorem); Notes V14 (Some Topological Questions); Notes V15 (Relations to Physics); RW-Orthogonal Curvilinear Coordinates - Div, grad, curl, and Laplacian

Problem Set #13 (due Mon, 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)

Late HW will not be accepted.

Exam #5 took place on Monday, January 23 during class. The exam covered 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; and applications of integration (area, volume, mass, average value, weighted averages, centroids, center of mass, moment of inertia, etc.).

Approximate letter grades for Exam #1
Total points on exam was 48. Median score was 34.5.
Mean score was 34.9. Standard deviation was 4.4.
score grade   score grade
42+ A 28+ C+
39+ A– 26+ C
36+ B+ 24+ C–
33+ B 22+ D
30+ B– 0-21 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 52.5.
Mean score was 47.6. Standard deviation was 13.8.
score grade   score grade
68+ A 48+ C+
64+ A– 44+ C
60+ B+ 40+ C–
56+ B 36+ D
52+ B– 0-35 F
Practice Questions     Solutions
Exam #2 solutions
Approximate letter grades for Exam #3
Total points on exam was 60. Median score was 39.5.
Mean score was 39.5. Standard deviation was 6.9.
score grade   score grade
54+ A 37+ C+
52+ A– 34+ C
50+ B+ 31+ C–
45+ B 28+ D
40+ B– 0-27 F
Exam 3 Practice Questions    Solutions
Exam #3 solutions
Approximate letter grades for Exam #4
Total points on exam was 50. Median score was 36.
Mean score was 37.6. Standard deviation was 5.8.
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
Practice Exam #4 Questions     Solutions
Exam #4 solutions
Approximate letter grades for Exam #5
Total points on exam was 45. Median score was 36.
Mean score was 33.1. Standard deviation was 10.8.
score grade   score grade
41+ A 27+ C+
38+ A– 25+ C
35+ B+ 23+ C–
32+ B 21+ D
30+ B– 0-20 F
Exam #5 Practice Questions      Solutions
Exam #5 solutions

Problem Set #12 (due Tues, Jan 24)
References: Simmons text, sections 20.5, 20.6, and 20.7; Notes CV (Changing Variables in Multiple Integrals); Notes G (Gravitational Attraction); Lecture #10 Notes; Lecture #11 Notes

Late HW will not be accepted.

Problem Set #11 (due Wed, Jan 18)
References: Simmons text, section 21.3; Notes V3 (Two-dimensional Flux); Notes V4 (Green's Theorem in Normal Form)

Late HW will not be accepted.

Problem Set #10 (due Fri, Jan 13)
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); and Simmons text, sections 21.1 and 21.2.

Problem Set #9 (due Tues, Dec 13)
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)

Problem Set #8 (due Thurs, Dec 1)
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)

Lecture Notes on Extrema of Functions of Several Variables

Problem Set #7 (due Fri, Nov 18)
References: Read 19.1; 19.2 (Partial Derivatives); SN-TA (Tangent Approximation); 19.6 (Chain Rule);
Notes on Differentials, the Chain Rule, Gradients, Directional Derivative, and Normal Vectors;
Notes on Gradients, Chain Rule, Implicit Differentiation, and Higher Order Derivatives

Problem Set #6 (due Wed, Nov 9)
References: Read text: 17.1, 17.2, 17.4, 17.5; SN-M (Matrices and Linear Algebra); Supplement: Vector and matrix forms of a system of linear equations, matrix algebra, inverse matrices, and related facts; RW-ParamCurves (Supplement on parametrized curves and vector-valued functions); SN-K (Kepler's 2nd Law)

Problem Set #5 (due Monday, Oct 31)
References: Text sections 17.3, 18.1, 18.2, 18.3, 18.4; RW-Vectors (Supplement on Vectors, dot product, projections, cross product); 18.02 SN-D (Determinants); SN-M (Matrices and Linear Algebra); Supplement on Solving Systems of Linear Equations via Row Reduction

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, 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 scheduled for 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: Wednesday, February 1, 2017 7:34 PM