Take-Home Exam 1: pick up on Thursday, June 8, return Monday,

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1 SYLLABUS FOR Overview This course is a review of calculus. We will start with a week-long review of single variable calculus, and move on for the remaining five weeks to multivariable calculus. For a further (tentative) lecture-by-lecture description, see the more detailed at the end of the syllabus. 2. Instructors There are two instructors for this course. Either of us is happy to answer any questions you may have, though each of us taking the lead in different halves of the term, so you might get a quicker response if you that person. Weeks 1-3: Tudor Padurariu (tpad(at)mit.edu) Weeks 4-6: Kevin Sackel (ksackel(at)mit.edu) 3. Meetings All meetings are in Room The schedule is as follows: Week 1: Monday - Thursday, June 4-7, 9AM - 12:30 PM. There is no class Friday! Weeks 2-6: Monday - Friday, June 11 - July 13, 12:30-2:30 PM. 4. Website The main resource for the class is the following website. All materials, including lecture notes, PSets, exams, and solutions, will be posted on the website. Additionally, all of this information can be found on the website. 5. Problem sets and Exams All exams are take-home, and should be picked up and returned in class, with the exception of the last exam, which will be returned in a yet-to-bedetermined manner. Please do not collaborate with other classmates on the exams. Problem sets are also due in class - collaboration is encouraged, but please list your collaborators if you choose to do so. There will be an optional problem set for the first week - you do not need to submit it. For all other assignments, here is a schedule: 1

2 SYLLABUS FOR Take-Home Exam 1: pick up on Thursday, June 8, return Monday, June 11 Problem set 1: due Friday, June 15 Problem set 2: due Friday, June 22 Problem set 3: due Friday, June 29 Take-home Exam 2: pick up on Friday, June 29, return Monday, July 2 Problem set 4: due Friday, July 6 Problem set 5: due Friday, July 13 Take-home Exam 3: pick up on Friday, July 13, return Monday, July Other Resources The Stellar site is where your grades will be posted. The url is There are supplementary notes available. For single variable calculus, suppnotes/suppnotes01-01a/index-01.html. For multi-variable calculus, suppnotes/suppnotes02/ A quick google search will also reveal past versions of the class from 2009, 2013, and The urls are on the website. There you can find past problem sets and exams. 7. Detailed schedule Subject to change, the lecture schedule is below. Week 1. Lecture 1 Monday, June 4th. Continuous functions; differentiation, computing the slope of the tangent line at a given point on y = f(x); properties of differentiation- linearity, product rule, chain rule, quotient rule, derivative of x n, higher derivatives; using the first and second derivative to graph a function in the xy plane, critical points, the second derivative test; max-min problems; implicit differentiation. Lecture 2 Tuesday, June 5th. Exponential, logarithm, and their derivatives, the constant e; trigonometric functions and their derivatives; linear approximation, fundamental limits, l Hopital rule, examples. Lecture 3 Wednesday, June 6th. Fundamental problems from antiquity, Archimedes solution to computing areas, Fundamental Theorem of Calculus, antiderivatives, examples of antiderivatives, techniques for computing antiderivatives: substitution, partial fractions, and integration by

3 SYLLABUS FOR parts. Lecture 4 Thursday, June 7th. Computing the length of a curve; computing volumes by the disk and shell methods; surface areas, examples; indefinite integrals, comparison test; infinite series, convergence and divergence, exercises. Pick up your Take home midterm 1 on Thursday, June 8th, after class, and bring it back Monday, June 11th, before class. Week 2. Lecture 5 Monday, June 11th. More on infinite series, the integral test, alternating series test, the root and ratio tests; powers series: examples, radius of convergence, basic operations, Taylor series, examples. Lecture 6 Tuesday, June 12th. Vectors, addition, multiplication by a scalar, dot product, cross product, geometric significance, geometric applications. Lecture 7 Wednesday, June 13th. Matrices as linear transformations, composition/ multiplication, determinants, and inverses of matrices; lines in 2D space, lines and planes in 3D space: parametric and implicit equations. Lecture 8 Thursday, June 14th. Curves in 2D and 3D space: parametric and implicit equations, velocity, speed, tangent vector, acceleration, curvature. Lecture 9 Friday, June 15th. Normal and tangent vectors to a curve; tangential and normal components of acceleration; 2D polar coordinates; going from cartesian to polar coordinates and back; arclength and area for curves in polar coordinates. Problem set 1 due in class, Friday, June 15th. Week 3. Lecture 10 Monday, June 18th. Area inside a curve with polar coordinates; functions of two variables: examples, domain, graph; partial derivatives; implicit differentiation; equation of tangent plane at a point. Lecture 11 Tuesday, June 19th. Differentiability for functions of two variables; tangent plane as approximation to the graph of the function;

4 total derivative; gradient, directional derivatives. SYLLABUS FOR Lecture 12 Wednesday, June 20th. Exercises with gradient, directional derivatives and tangent planes; gradient as normal to level curves; generalizations to functions of more than two variables. Lecture 13 Thursday, June 21st. Total derivative and generalization of the chain rule for functions of several variables, applications; critical points; generalization of second derivative test to functions of two variables. Lecture 14 June 22nd. Exercises on the material from week 2 and week 3. Problem set 2 due in class, Friday, June 22nd. Week 4. Lecture 15 Monday, June 25th. Max-min problems; method of least squares; Lagrange multipliers. Lecture 16 Tuesday, June 26th. More Lagrange multipliers; double integrals; finding volume; writing down bounds. Lecture 17 Wednesday, June 27th. Switching order of integration; mass; center of mass; integration in polar coordinates; changing coordinates (2D). Lecture 18 Thursday, June 28th. Triple integrals; bounds for triple integrals; mass; moment of inertia; possibly other applications. Lecture 19 Friday, June 29th. Cylindrical coordinates; spherical coordinates; triple integrals in cylindrical and spherical coordinates. Problem set 3 due in class, Friday, June 29th. Pick up your Take home midterm 2 on Friday, June 29th, after class, and bring it back Monday, July 2nd, before class. Week 5. Lecture 20 Monday, July 2nd. Changing coordinates for 3D integrals; vector fields (2D and 3D); line integrals (2D and 3D). Lecture 21 Tuesday, July 3rd. Fundamental theorem of calculus for line integrals (2D and 3D); gradient fields; path-independence; potential

5 SYLLABUS FOR functions for gradient fields; simply connected regions (2D). Lecture 22 Wednesday, July 4th. form); flux in 2D. Green s Theorem (tangential Lecture 23 Thursday, July 5th. Curl (2D); Green s Theorem (normal form); extensions and applications of Green s Theorem. Lecture 24 Friday, July 6th. 2D significance of divergence and curl; relations of div, grad, and curl. Problem set 4 due in class, Friday, July 6th. Week 6. Lecture 25 Monday, July 9th. Surface area, including along spherical and cylindrical regions. Lecture 26 Tuesday, July 10th. Flux in 3D; the divergence theorem. Lecture 27 Wednesday, July 11th. gravitational attraction. Curl (3D); Stokes Theorem; Lecture 28 Thursday, July 12th. 3D significance of divergence and curl; more on div, grad, and curl; review of FTC, Green, Divergence, Stokes. Lecture 29 Friday, July 13th. Maxwell s Equations; other applications to physics. Problem set 5 due in class, Friday, July 13th. Pick up your Take home midterm 3 on Friday, July 13th, after class. It is to be submitted on Monday, July 16th - we will let you know how to submit as the date gets closer.