Course website for 18.089, Summer 2009

Instructor information and contact

Name: Joel Lewis
This page is the archived version of a class taught in 2009. For information about current or future versions of this class, please contact the appropriate office in the math department.

Course meeting place/time

June 2 to June 5: 9 AM - 12:30 PM in room 12-142
June 8 to July 10: MTWR: 12:30 - 2:30 PM, F: 12 - 2 PM, in room 2-132

Note that class meets every weekday except Friday, July 3.

I will hold office hours every Tuesday and Thursday from 4 - 5 PM in my office, 2-333.

Basic course data

Overview: This course is intended as a review of single- and multi-variable calculus for students who already have some familiarity with calculus. The first week will be a fast-paced review of single-variable calculus (18.01). The remaining five weeks will cover multi-variable calculus (18.02).
Text: George F. Simmons, Calculus with Analytic Geometry, 2nd edition.

Assignments

There will be four exams and regular homework assignments. The first exam was Friday, June 5, 9 AM. The second was Wednesday, June 17. The third will be Tuesday, June 30 and the last will be Friday, July 10 (the last day of class).

Brief solutions to the penultimate problem set are available here.

The current (7/08) version of the problems for homework, beginning with question 88, is here (.pdf). The older problems are available here (.pdf) (numbers 1 to 49) and here (.pdf) (numbers 50 to 87).

Problem set 17 (to be turned in Friday, July 10, 12 PM): Questions 99-101. (There will be no additional problems assigned on Thursday, so you're welcome to hand this in Thursday to have it graded and returned on Friday.)
Problem set 16 (to be turned in Wednesday, July 8, 12:30 PM): From Monday: Questions 93-96.
From Tuesday: Questions 97-8.
Problem set 15 (to be turned in Monday, July 6, 12:30 PM): There are 10 possible positions for the extra half hour of class: from 12 to 12:30 or from 2:30 to 3 on any day next week. Send me an e-mail stating those times with which you have a conflict, or that you have no conflict with any of those times. Then do questions 88-92.
Problem set 14 (to be turned in Thursday, July 2, 12:30 PM): Questions 84-87.
Exam 3: Questions and solutions in .pdf format.
Problem set 13 (to be turned in Monday, June 29, 12:30 PM): Questions 78-82. There is a typo in question 81: the words "below" and "above" should be switched.
Problem set 12 (to be turned in Friday, June 26, 12 PM): From Wednesday: Questions 71-75. Also re-do question 67 if you would like to submit it for credit.
From Thursday: Questions 76, 77.
Problem set 11 (to be turned in Wednesday, June 24, 12:30 PM): From Monday: Questions 59, 60, 61, 62.
From Tuesday: 65, 66, 67, 68. There is a typo in question 67. It should read, "what is the equation of the plane passing through (1, 2, 3) that cuts a tetrahedron from the first octant of minimal volume?" The first octant is the set of points in space such that x, y, z are all greater than or equal to 0. The 2-dimensional analogue of this problem would be, "What is the equation of the line passing through (1, 2) that cuts off a triangle from the first quadrant of minimal area?"
Problem set 10 (to be turned in Monday, June 22, 12:30 PM): Questions 56, 57, 58.
Problem set 9 (to be turned in Friday, June 19, 12 PM): Questions 50, 51, 52, 53, 55. Hint for 53: the direction vector of the line is parallel to both planes and so is perpendicular to the normal vectors of both planes. Hint for 55: ellipses are a lot like circles.
Problem set 8 (to be turned in Wednesday, June 17, 12:30 PM): Questions 48 and 49. For number 49, substitute a well-chosen value into one of the power series formulas you know in order to compute the series in question.
Come prepared on Tuesday with any review questions for the upcoming exam.
Exam 2: Questions and solutions in .pdf format.
Problem set 7 (to be turned in Monday, June 15, 12:30 PM): Questions 42, 44, 45, 46, 47.
Problem set 6 (to be turned in Friday, June 12, 12:30 PM): From Wednesday: questions 36, 37.
Added later Wednesday: 40, 41. For question 41, leave your answer in terms of an inverse trig function.
Added Thursday: 38, 39, 43.
Problem set 5 (to be turned in Wednesday, June 10, 12:30 PM): From Monday: questions 19, 30, 32. (For number 19, the final answer should consists of the 18 functions in order using the symbols ">>," ">," "~," and "=" as defined in the problem.
From Tuesday: questions 34, 35.
Exam 1: Questions and solutions in .pdf format.
Problem set 4 (to be turned in Moday, June 8, 12:30 PM): Questions 27-29. Also, send me an e-mail from your preferred e-mail account with your score from problem set 2. Also, please fill out the semi-anonymous comment form I handed out in class.
Problem set 3 (to be turned in Friday, June 5, 9 AM): Questions 15-18, 22-24, 26. In question 15, leave your answer in terms of the functional inverse of the cosh function. In question 18, you may at some point encounter an integral from 1 to cosh u of (x^2 - 1)^(1/2). (Here carrot means exponentiation.) If you do, try the substitution x = cosh w. Note that when x = 1, this makes w = 0, and when x = cosh u, this makes w = u.
Problem set 2 (to be turned in Thursday, June 4, 9 AM): Questions 2, 5 (use two different methods), 11, 13, 14, 20, 21.
Problem set 1 (to be turned in Wednesday, June 3, 9 AM): Questions 1, 3, 4, 6-9, 12. I neglected to define "continuous function" in class: a function f is continuous at x = a if the limit as x goes to a of f(x) is equal to f(a). For example, all polynomials are continuous at every point, but any function with a "jump" or a "hole" in its graph will not be continuous at the point.

Lecture summaries and notes

If possible, I will keep this updated with a brief summary of topics covered in each lecture.

1. Functional notation. Limits. Definition of a derivative. Computing derivatives from the definition. Differentiation rules (nth power, constant multiple, sum, product, chain, quotient). Higher derivatives. Graphing (increasing, decreasing, local extrema, inflection points). 2nd derivative test. Tangent lines/linear approximations. Implicit differentiation. Max-min problems. Related rates problems.
Simmons 1.1-6, 2.1-4, 3.1-3, 3.5-6, 4.1-5, 5.1-2.

Lecture notes 1 (.pdf)
2. Basic properties exponential functions and logarithms. The number e, the exponential function and the natural logarithm. Derivatives of exponential and logarithmic functions. Radian measure. Basic properties of trigonometric functions. A few special values and identities. Trigonometric limits and derivatives. Inverse trigonometric functions and their derivatives. Antidifferentiation basics. Antidifferentiation by substitution (relationship to chain rule). Trigonometric substitutions. Completing the square. Polynomial long-division.
Simmons 1.7, 2.5, 3.4, 5.3, 8.1-4, 9.1-5, 10.1-5 (but you can ignore the bits about definite integration for now).

Lecture notes 2 (.pdf)
3. Derivative and integral of the secant function. The method of partial fractions. Definite integration and the Fundamental Theorems of Calculus. Basic properties of definite integrals. Definite integrals and integration by substitution. Computing area between a single curve and the x-axis. Computing areas bounded between two curves. Arclength.
Simmons 6.1-7, 7.1-2, 7.5, 10.1-5 (just for the bits about definite integration, especially 10.2), 10.6.

Lecture notes 3 (.pdf) (I'm still trying to figure out how to make the scan quality better -- if I succeed, I may update these files & let you know in class.)
4. Areas by integrating with respect to y. Solids of rotation. The disk, shell and washer methods. Pappus' Theorem. Hyperbolic functions.
Simmons 7.3-4, 9.7, 11.1-3.

Lecture notes 4 (.pdf)
5. Surface area of solids of revolution. Integration by parts and reduction formulas. Indeterminant forms and L'Hopital's rule.
Simmons 7.6, 10.7, 12.1-3.

Lecture notes 5 (.pdf) (Are these scans more readable? Let me know in class if it's an improvement.)
6. Improper integrals: definition as a limit; p-test; comparison test and limit comparison test; infinite height versus infinite width. Series: the geometric series (finite and infinite); the n-th term test; the harmonic series; the comparison test and limit comparison test; telescoping series.
Simmons 12.4, 13.1-5.

Lecture notes 6 (.pdf)
7. Vectors: geometric and algebraic definition (2 and 3 dimensions). Addition, multiplication by a scalar. Unit vectors. Dot product: definition and basic properties. Determinants. Checkerboard rule and computing determinants of small sizes. Determinants and area/volume. Cross product: definition and basic properties.
Simmons 17.3, 18.1-3.

Lecture notes 7 (.pdf)
8. Matrices. Matrix operations (multiplication, addition, determinant). The identity matrix. Computing matrix inverses.
Simmons doesn't have any content on this topic. You might try looking at OCW to see if they have relevant materials in 18.02 if you're interested. (The linear algebra classes 18.06 should cover what we discussed in much greater depth.)

Lecture notes 8 (.pdf)
9. Integral test. Alternating series test. Root and ratio tests. Introduction to power series.
Simmons 13.6-8, 14.1-2.

Lecture notes 9 (.pdf)
10. Power series. Radius of convergence. Computing power series of a given function. Computing with power series (addition, multiplication, substitution, differentiation and integration of power series). Applications of power series. Taylor series (power series not centered at 0).
Simmons 14.2-4, 14.6-7.

Lecture notes 10 and 11 and 12 (.pdf)
11. Exam review. Equations of lines in 3D space: parametric and symmetric. Equations of planes in 3D space. Normal vector to a plane. Finding the intersection of a line with a plane.
Simmons 18.4.

Lecture notes 10 and 11 and 12 (.pdf)
12. Parametric equations of curves on the plane and in space. Vector functions. Velocity. Speed.
Simmons 17.1, 17.4.

Lecture notes 10 and 11 and 12 (.pdf)
13. Acceleration. Getting rectangular equations from parametric equations. The cycloid. Brief discussion of functions of more than one variable.
Simmons 17.2, 19.1. Completely optional but possibly of interest: 17.6-7. (17.7 explains how Newton's theory of universal gravitation can be used to prove Kepler's laws of planetary motion.)

Lecture notes 13 and 14(.pdf)
14. Functions of several (usually 2) variables. Graphing. Level curves and contour plots. Partial derivatives. Second partials. The equality of mixed partials. The chain rule for partial derivatives. The gradient. Tangent planes.
Simmons 19.1-3, 19.5-6.

Lecture notes 13 and 14 (.pdf)
15. Linear approximations. Directional derivatives. Maximizing and minimizing functions of several variables. The second derivative test.
Simmons 19.5, 19.7.

Lecture notes 15 and 16 (.pdf)
16. Best-fit lines. Maximizing and minimizing functions subject to constraints. Lagrange multipliers.
Simmons 19.8.

Lecture notes 15 and 16 (.pdf)
17. Iterated and double integrals. Applications.
Simmons 20.1-3.

Lecture notes 17 (.pdf)
18. Polar coordinates.
Simmons 16.1-5.

Lecture notes 18 (.pdf)
19. Double integrals in polar coordinates. Triple integrals. Cylindrical coordinates.
Simmons 20.4-6.

Lecture notes 19 (.pdf)
20 and 21. Spherical coordinates and gravitational attraction.
Simmons 20.7.

Lecture notes 20, 21 and 22 (.pdf)
22. Line integrals. Path independence. Conservative fields. Fundamental Theorem of Line Integrals.
Simmons 21.1-2.

Lecture notes 20, 21 and 22 (.pdf)
23. Green's Theorem.
Simmons 21.3.

Lecture notes 23 (.pdf)
24. Line integrals and flux. Conservative fields in three dimensions. Computing the scalar field associated to a conservative field. A "checkable" condition for conservative fields.
Lecture notes 24 (.pdf) (don't exist yet)
25. Div and curl. The differential operator "del." Flux and the Divergence Theorem.
Simmons 21.4.

Lecture notes 25 (.pdf) (don't exist yet)