Math 18.01A-18.02A Concourse Calendar of topics and HW assignments -- Fall 2018 and IAP 2019
last updated Tuesday, February 5, 2019 9:35 AM
The topics and assignments will change as the course proceeds!! Check back frequently.

N Date Topics Assignments & References [Solutions]
1 Wed, Sept 5 Review of differentiation [Simmons Chapters 1-3]
Linear and quadratic approximations.

Read: SN-A (Approximation);
Read text: 2.6 (through pg. 77);
Read SN-MVT (Mean Value Theorem).
Read text: 12.2, 12.3 (examples 1-3, remark 1).

Problem Set #1 (due Thurs, Sept 13)

2 Thurs, Sept 6 Higher order (Taylor and Maclaurin) polynomial approximations.
3 Mon, Sept 10 Extended Mean Value Theorem and error estimation.
4 Tues, Sept 11 Indeterminate forms, L'Hospital's rule, growth rate of functions.
5 Wed, Sept 12 Definite integral; summation notation. First Fundamental Theorem of Calculus, Second Fundamental Theorem, ln x as an integral, Leibnitz' Rule.

Read: SN-PI (Properties of Integrals)
Read: SN-FT (Second Fundamental Theorem).
Read text: 6.3-6.7; 7.1-7.6
Read SN-AV (Average Value).

Problem Set #2 (due Fri, Sept 21)

6 Thurs, Sept 13 Problem set questions; functions defined by integrals. Geometric applications: area, signed area, average value of a function; area between curves.
7 Mon, Sept 17 Integral applications: volumes, volumes of solids of revolution, arclength, surface area.
8 Tues, Sept 18 Examples of volumes, surface area, arclength using integration.
9 Wed, Sept 19 Integral applications: work, mass, pumping. Integration techniques: basic integrals, substitution, trig. integrals,

References: Text sections 7.7 (work and energy); 10.1-10.6 (Integration: substitution, trig. integrals, completing the square, partial fractions and rational functions.)

Problem Set #3 (due Fri, Sept 28)

10 Thurs, Sept 20 Problem set questions; Integration techniques: basic integrals, substitution, trig. integrals, powers and products of trig. functions.
11 Mon, Sept 24 Integration techniques: powers and products of trig. functions; dealing with quadratic expressions in integrals (trigonometric substitutions), completing the square; integration of rational functions via partial fractions.
12 Tues, Sept 25 Integration techniques: partial fractions.
13 Wed, Sept 26 Integration techniques: partial fractions, integration by parts, reduction formulas.

Read text: 10.7-10.9 (integration by parts, miscellaneous other methods, numerical integration); SN: F; 12.4 (improper integrals); Read SN-INT (Improper Integrals)

Problem Set #4 (due Fri, Oct 5)

14 Thurs, Sept 27 Numerical integration; improper integrals (infinite domain, discontinuous integrands), comparison test, limit comparison test, p-test for integrals. Problem set questions.
15 Mon, Oct 1 Numerical integration; improper integrals (infinite domain, discontinuous integrands), comparison test, limit comparison test, p-test for integrals.

Exam 1 Practice Questions    Solutions
(use same username/password as solutions)

16 Tues, Oct 2

Exam 1 - Topics covered may include:     Solutions

1) Linear, quadratic, Taylor (nth order) approximations and series: a) by formula; b) by manipulating known series;

2) L'Hôpital's Rule and indeterminate forms (limits);

3) Definite integrals in the calculation of area, volume, arclength, average value, work;

4) First and Second Fundamental Theorems of Calculus, and Leibnitz' Rule

5) Integration techniques: a) substitution; b) trig. substitutions; c) powers and products of trig. functions; partial fractions; integration by parts; reduction formulas.

17 Wed, Oct 3 Sequences of real numbers, convergence of a sequence. Infinite series, geometric series

Read relevant portions of text: 13.1 – 13.8 (sequences and series).

Supplement on Integral Test and Comparison Tests

Supplement on Alternating Series; Absolute vs. Conditional Convergence; Ratio Test; Strategies

Problem Set #5 (due Mon, Oct 15)

18 Thurs, Oct 4 Problem Set questions
19 Wed, Oct 10 Harmonic series, convergence tests (divergence test, p-test, comparison test, integral test, limit comparison test, ratio test); Alternating series, absolute vs. conditional convergence.
20 Thurs, Oct 11  
21 Mon, Oct 15 Ratio Test for absolute convergence. Power series, radius and interval of convergence.

Read relevant portions of text: 14.1-14.5

Problem Set #6 (due Monday, Oct 22)

22 Tues, Oct 16 Taylor series revisited.  
23 Wed, Oct 17

Error estimation using Taylor's Theorem with Remainder.

Practice Questions for 18.01A Final Exam (Exam #2)

Solutions to 18.01A Exam #2 Practice Questions

Math 18.01A Essentials (PDF)

24 Thurs, Oct 18

Taylor's Theorem with Remainder, continued.

25 Mon, Oct 22

Coordinates, vectors (start of 18.02A); coordinate-free vector proofs, parameterized lines.

Lecture Notes #1

Read text: 17.3, 18.1, 18.2, 18.3, 18.4.
RW-Vectors (Supplement on Vectors, dot product, projections, cross product)

Problem Set #7 (due Wednesday, Oct 31)

26 Tues, Oct 23 18.01A Final Exam, a 2-hour exam
27 Wed, Oct 24

Coordinates, vectors; coordinate-free vector proofs, parameterized lines; dot product.
[Lecture Notes on Coordinates & Vectors]

28 Thurs, Oct 25 Dot product, scalar and vector projection, equation of a plane; introduction to cross-product.
[Lecture Notes on Dot Product, Cross Product, Planes, Area, and Volumes]

Lecture Notes #2

Read text: 17.1, 17.2, 17.4, 17.5; 18.02 Supplementary Notes: SN-D (Determinants), SN-M (Matrices and Linear Algebra)

RW-Vectors (Supplement on Vectors, dot product, projections, cross product)

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 #8 (due Thurs, Nov 8)

29 Mon, Oct 29

Cross product; lines and planes; determinants, distance, area, volume.

30 Tues, Oct 30  
31 Wed, Oct 31

Matrix methods, row reduction, inverse matrices.
[Lecture Notes on vector and matrix forms of a system of linear equations, matrix algebra, inverse matrices, and related facts]

32 Thurs, Nov 1
33 Mon, Nov 5

Parametric equations and vector derivatives; velocity vector, acceleration vector, speed, unit tangent vector, curvature.
[Lecture Notes on Parameterized Curves and Vector-Valued Functions]

Lecture Notes #3

Read text: 17.1-17.2, 17.4-17.6; RW-ParamCurves (Supplement on parametrized curves and vector-valued functions); SN-K (Kepler's 2nd Law)

Problem Set #9 (due Thurs, Nov 15)


Exam 3 Practice Questions     Solutions

34 Tues, Nov 6

Parametric equations, continued.
Kepler's second law (read the supplement, not covered in class)

35 Wed, Nov 7

Graphs of surfaces, cross-sections. Functions of several variables, graph vs. level sets (contours). [Lecture Notes on Paramaterized Surfaces, Tangent Vectors, Partial Derivatives]

Lecture Notes #4

36 Thurs, Nov 8

Functions of several variables, level curves and surfaces. Limits and continuity, differentiability, and partial derivatives. Tangent plane, linear approximation.

37 Mon, Nov 12
(to be rescheduled)

Tangent plane, linear approximation, increments and differentials, rate of change of a function along a parameterized curve: basic Chain Rule. Gradient and directional derivative. [Notes on limits, continuity, differentiability and linear approximation]

Lecture Notes #5

Read 19.1, 19.2 (Partial Derivatives)
SN-TA (Tangent Approximation)
19.6 (Chain Rule)

Notes on Gradients, Chain Rule, Implicit Differentiation, and Higher Order Derivatives

Problem Set #10 (due Thurs, Nov 29)

38 Tues, Nov 13 Chain Rule, gradient vector, directional derivatives. Find normal vectors using the gradient vector.
39 Wed, Nov 14

Implicit differentiation.
[Lecture Notes on Differentials, the Chain Rule, Gradients, Directional Derivative, and Normal Vectors]

Lecture Notes #6

40 Thurs, Nov 15 Exam 3
Topics include: Coordinates, vectors, vector operations and tools (adding, scaling, dot product, cross product), lines, planes, intersections, distance, area, volume; matrix methods for solving systems of linear equations, row reduction, parameterization, matrix inverse; parameterized curves, velocity, acceleration, unit tangent vector, unit normal vector, curvature, differentiation of vector-valued functions.
41 Mon, Nov 19

General Chain Rule, implicit differentiation, higher order derivatives, equality of mixed partials (Clairaut's Theorem).

42 Tues, Nov 20

Quadratic approximation; critical points, second derivative test; unconstrained optimization.

Lecture Notes #7

43 Mon, Nov 26

Lecture Notes on Extrema of Functions of Several Variables

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)

Problem Set #11 (due Thurs, Dec 6)

44 Tues, Nov 27


45 Wed, Nov 28

Max-min problems, Method of Least Squares. Constrained optimization, Method of Lagrange Multipliers.

Lecture Notes #8

46 Thurs, Nov 29

Method of Lagrange Multipliers, continued - economics applications.

47 Mon, Dec 3

Constrained optimization w/multiple constraints; Non-independent variables.

Practice Exam #4 Questions     Solutions
48 Tues, Dec 4

Double integrals and their applications - volume, mass, area, averaging, centroid. Evaluation of double integrals via iterated single integrals, the Fubini Theorem.

Lecture Notes #9

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 #12 (due 1st week IAP)

49 Wed, Dec 5

Polar coordinates, double integrals in polar coordinates; additional applications. Weighted averages, center of mass; change of variables in double integrals.

50 Thurs, Dec 6 Exam 4: Topics may include 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.
51 Mon, Dec 10 Polar coordinates, double integrals in polar coordinates; additional applications. Weighted averages, center of mass; change of variables in double integrals.

Lecture #10 Notes

52 Tues, Dec 11 Examples and techniques for change of variables; problem set questions  
53 Wed, Dec 12

Weighted averages, center of mass; change of variables in double integrals.

Lecture #11 Notes (first part re: change of variables in double integrals)

Holiday Break

IAP Schedule (starting Monday, January 7):
Lecture/Recitation: Mon, Wed 12:00pm to 1:00pm; Tues, Thurs, 1:00pm to 2:30pm in 16-160.
Additional recitation time informally afternoons in Concourse Lounge or the classroom.

IAP Schedule (Mon, Jan 14 onward):
Lecture: Mon, Wed 12:00pm to 1:30pm; Tues 1:00pm to 3:00pm; Recitation: Thurs 1:00pm to 2:30pm.
Additional recitation time informally afternoons in Concourse Lounge or the classroom.

Pset 13: Vector fields, integration along curves, line integrals, work, conservative vector fields, and work.

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 #13 (due Fri, Jan 11)

54 Mon, Jan 7

Vector fields; integration along curves in R2 and R3. Line integrals and work.

Lecture #11 Notes

    Here's a website that has a good java-based tool for showing vector fields and flows: The current version requires you to download a Java executable file to your own computer and to run it locally on your own machine. You can customize various options. You can also print the graphs. [How Java is called varies on computer and operating system, so we may have to provide some additional documentation.]
55 Tues, Jan 8 Gradient and conservative fields.
56 Wed, Jan 9

Potential functions; test for exactness; algebraic definitions of curl and divergence.

Lecture #11 Notes and Lecture #12 Notes

Pset 14: Green's Theorem (ordinary and normal form), its corollaries and applications; algebraic definitions of divergence and curl of a vector field.

Simmons text, section 21.3
Notes V3 (Two-dimensional Flux)
Notes V4 (Green's Theorem in Normal Form)

Problem Set #14 (due Wed, Jan 16)

57 Thurs, Jan 10 Green's Theorem; corollaries and applications of Green's Theorem.

2D-Flux: normal form of Green's Theorem.

Lecture #12 Notes

58 Mon, Jan 14 Review of double integrals; introduction to triple integrals and applications.
Triple integrals in rectangular, cylindrical, and spherical coordinates.

Lecture #10 Notes

Pset 15: Integration of functions over regions in space using Cartesian, cylindrical, and spherical coordinates; applications in physics.

Simmons text, sections 20.5, 20.6, and 20.7
Notes CV (Changing Variables in Multiple Integrals)
Notes G (Gravitational Attraction)

Problem Set #15 (due Tues, Jan 22)

59 Tues, Jan 15 Triple integrals (again) in rectangular, cylindrical, and spherical coordinates.

Exam #5 Topics & Practice Questions (18.02A - IAP)      Solutions

60 Wed, Jan 16

Triple integrals applications - moment of intertia; gravitational attraction.
Applications, continued. Change of variable in triple integrals and Jacobian determinants.

Lecture #11 Notes

61 Thurs, Jan 17 Exam 5 covering 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.).

Pset 16: Parameterized surfaces, surface area, surface integrals and their applications, flux of a vector field through a surface, Divergence Theorem.

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)

Problem Set #16 (due Tues, Jan 29)

62 Tues, Jan 22 Parameterization of surfaces; surface integrals; flux of a vector field through a surface.
Supplement on surface integrals

Lecture #13 Notes

63 Wed, Jan 23 Surface integrals, continued; Divergence Theorem.
64 Thurs, Jan 24 Applications and interpretations of the Divergence Theorem. Stokes' Theorem.

Lecture #14 Notes

Pset 17: Divergence Theorem, Stokes' Theorem, applications to physics.

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 #17 (due Thurs, Jan 31)

65 Mon, Jan 28 More Stokes' Theorem.
66 Tues, Jan 29 Applications and interpretations of Stokes' Theorem.
67 Wed, Jan 30 Div, grad, curl, and Laplacian in curvilinear coordinates
68 Thurs, Jan 31 Generalized Stokes' Theorem in terms of forms; Final Exam review.
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

  Fri, Feb 1 Final Exam - Friday, February 1 in 16-160 from 9:00am to 12:00pm

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