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.
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.||
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
|16||Tues, Oct 2||
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).
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.
Read text: 17.3, 18.1, 18.2, 18.3, 18.4.
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.
|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]
RW-Vectors (Supplement on Vectors, dot product, projections, cross product)
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.
|32||Thurs, Nov 1|
|33||Mon, Nov 5||
Parametric equations and vector derivatives; velocity vector, acceleration vector, speed, unit tangent vector, curvature.
Problem Set #9 (due Thurs, Nov 15)
Exam 3 Practice Questions Solutions
|34||Tues, Nov 6||
Parametric equations, continued.
|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]
|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]
Read 19.1, 19.2 (Partial 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||
|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.
|43||Mon, Nov 26||
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.
|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.
Read 20.1 (Volumes)
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.|
|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)
IAP Schedule (starting Monday, January 7):
IAP Schedule (Mon, Jan 14 onward):
Pset 13: Vector fields, integration along curves, line integrals, work, conservative vector fields, and work.
Notes V1 (Plane Vector Fields)
Problem Set #13 (due Fri, Jan 11)
|54||Mon, Jan 7||
Vector fields; integration along curves in R2 and R3. Line integrals and work.
|Here's a website that has a good java-based tool for showing vector fields and flows: http://math.rice.edu/~dfield/dfpp.html. 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.
Pset 14: Green's Theorem (ordinary and normal form), its corollaries and applications; algebraic definitions of divergence and curl of a vector field.
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.
|58||Mon, Jan 14||Review of double integrals; introduction to triple integrals and applications.
Triple integrals in rectangular, cylindrical, and spherical coordinates.
Pset 15: Integration of functions over regions in space using Cartesian, cylindrical, and spherical coordinates; applications in physics.
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.
|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.
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
|63||Wed, Jan 23||Surface integrals, continued; Divergence Theorem.|
|64||Thurs, Jan 24||Applications and interpretations of the Divergence Theorem. Stokes' Theorem.||
Pset 17: Divergence Theorem, Stokes' Theorem, applications to physics.
Notes V10 (The Divergence Theorem)
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.|
(Some topics on these practice exams might not appear on Friday's exam, but they are a reasonable approximation.)
|Fri, Feb 1||Final Exam - Friday, February 1 in 16-160 from 9:00am to 12:00pm|
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