**Instructor:** Hemanshu
Kaul

**Office:** 125C, Engineering 1

**Phone:** (312) 567-3128

**E-mail:** kaul [at]
math.iit.edu

**Lecture Time and Place:** 10am, Monday-Wednesday-Friday at 204, Siegel Hall.

**Lab Time and Place:** 3:15pm, Wednesday at 112E Stuart Bldg.

**Recitation Time and Place:** 3:15pm, Wednesday at 222, Alumni Memorial Hall. ** <-- Note the Change **

**Office Hours:** 1:45-3pm Monday, and 4:30-5:30pm Wednesday, and by appointment. Emailed questions are also encouraged.

**Teaching Assistant:** Aleksandr Borkovskiy, aleks_bork [at] yahoo.com.

**TA Office Hours: ** 12:30-1:30pm, Tuesday at 120, Engineering 1 Bldg. (primarily for Mathematica labs and recitation).

**Tutoring Service:** Mathematics tutoring at the Academic Resource Center.

**Online Problem Practice:** Calculus II at WebAssign and/ or at COW (Calculus on Web).

|Course Information| |Advice| |Announcements| |Examinations| |Labs and Recitations| |Homework| |Class Log| |Links|

The

The official course syllabus.

How to read Mathematics? The Basics, More Details.

How to study and learn Math.

Practical advice on studying Math and taking tests.

Understanding Mathematics - a study guide.

How to ace Calculus!

You need to be familiar with the material from PreCalculus and Calculus I, and earlier.

Use the following to review and remind yourselves of all that you already should know:

Paul Dawkin's Algebra Review Sheet

Paul Dawkin's Trignonometry Review Sheet

Paul Dawkin's Calculus I and II Review Sheet

*Wednesday, 10/19*: The due date for Lab#4 has been postponed to 10/26. See below.*Tuesday, 8/30*: Change in Recitation room has been announced above.*Monday, 8/29*: All the Exam dates have been announced below.*Friday, 8/26*: TA office hour has been announced above, and info for downloading Mathematica has been updated below.*Monday, 8/22*: Check this webpage regularly for homework assignments, announcements, etc.

*Exam #1*: Wednesday, October 5th. Topics: All the topics corresponding to the HW#1, HW#2, HW#3, HW#4, and HW#5.*Exam #2*: Wednesday, November 16th. Topics: All the topics corresponding to the HW#6, HW#7, HW#8, HW#9, and HW#10.*Final Exam*: Thursday, December 8th, 8am to 10am. Topics: Everything covered in class during the semester.

You have to prepare the solution for at least two problems among all the listed problems for a particular recitation. In the recitation class, the TA will ask for volunteers to present the solution of each of the assigned problems and/or distribute other problems to work on. Each student has to volunteer to present at least one long problem during the semester. The solution will be graded on the clarity of presentation, details of the intermediate steps, intermediate reasons, and ability to answer any questions or doubts the classmates/ TA might have. A poor performance is replaceable by presenting an additional problem (if possible) after everyone has had an opportunity.

You start the Mathematica assignment in the lab on the day of its assignment and then finish it over the one week before the next Wednesday. The TA can help you with the assignment during the lab and during his office hours on Tuesday. You can obtain a free version of Mathematica for your personal use from OTS at IIT through the "Training and Support" tab on your 'my.iit.edu' account (there is a "OTS downloads" window in that tab and at bottom of that window are the links for downloading Mathematica and activating the license).

You must submit the Lab report in the following format (with appropriate changes) as an .nb file through email to the TA (lab#2 onwards).

Mathematica Tutorials

A Beginner's guide to Mathematica

A short Mathematica Command Reference.

*Wednesday, 8/24*: Mathematica Lab #1: Introduction to Mathematica. Due Wednesday, 8/31.

*Wednesday, 8/31*: Recitation section at 222, AM (note the change classroom) - In-class problem solving, including the following problems: Problems Plus (on page 487, end of Chapter 7) problem #2, #12; Chapter 7 Review (on pages 483-485) Exercises #122abc.

*Wednesday, 9/7*: Mathematica Lab #2: Inverse, Log, and Exponential Functions. Due Wednesday, 9/14.

*Wednesday, 9/14*: Recitation section at 222, AM - In-class problem solving, including the following problems: Problems Plus (on page 487, end of Chapter 7) problem #6, #9; Chapter 7 Review (on pages 483-485) Exercises #120, #116.

*Wednesday, 9/21*: Mathematica Lab #3: Integration. Due Wednesday, 9/28.

*Wednesday, 9/28*: Recitation section at 222, AM - In-class problem solving, including the following problems: Problems Plus (on page 558-559, end of Chapter 8) problem #4, #11, #16; Chapter 8 Review (on pages 554-556) Exercises #60.

*Wednesday, 10/5*: Recitation section at 222, AM - Discussion of exam problems. In-class problem solving using problems to be distributed in class.

*Wednesday, 10/12*: Mathematica Lab #4: Differential Equations. Due Wednesday,~~10/19~~10/26.

*Wednesday, 10/19*: Recitation section at 222, AM - In-class problem solving, including the following problems: Chapter 8 Review (on pages 554-556) Exercises #62, 79; Chapter 10 Review (on pages 650-653) Exercise #17; Problems Plus (on page 654-655, end of Chapter 10) Problem #2.

*Wednesday, 10/26*: Finish up Lab#4 which will be due at the end of the lab. Start with Mathematica Lab #5: Parametric Equations and Polar Coordinates. You have to submit 5 out the following 6 Assignment/Problems #1, #2, #3, #6, #7, #8. Due Wednesday, 11/9.*Wednesday, 11/2*: Recitation section at 222, AM - In-class problem solving, including the following problems: Chapter 11 Review (on pages 706-707) Exercise #29, #35; Problems Plus (on page 708-709, end of Chapter 11) Problem #1, #5abc, #5efg.

*Wednesday, 11/9*: Recitation section at 222, AM - The TA will lead a review session for Exam#2 (scheduled on 11/16), you can send him any requests for topics or HW problems to review.

*Wednesday, 11/16*: Mathematica Lab #6: Sequences and Infinite Series. You have to submit 3 out of the 4 problems. Assignment/Problem #4 is compulsory. Due Wednesday, 11/30.

*Wednesday, 11/30*: Last Recitation section, at 222, AM. Priority will be given to the students who have not presented yet. - In-class problem solving, including the following problems: Chapter 12 Review (on pages 794-796) Exercise #23, #25, #28, #32, #34, #39.

Homework will be assigned on WebAssign every week, mostly on Friday.

You will be required to set up your own account using the "class key" that I will give you in class. Go to WebAssign. Click on the Log In link on the right hand side of the page. Click on the I Have A Class Key link. The Class Key is composed of three fields. In the first field you will enter the word iit (lower case) followed by two four digit numbers unique to your section. Click Submit. If the information about your section is correct, click Yes, This Is My Class. At this point, follow the instructions to create a username and password. Once your account is set up, you will not have to use the Class Key again.

You will be given a grace period of approximately 14 days from the first day of classes to access WebAssign before you need to purchase your account online.

You should first attempt each assigned problem on paper like regular paper and pen assignments (log off the WebAssign website).

Carefully read the WebAssign Student Guide to understand how to format your solutions.

When you have finished entering your answers, click on "Submit" and WebAssign will immediately grade your solutions. You can then re-work your incorrect problems and re-submit them to improve your score. You can submit your assignment up to 5 times without penalty.

After two incorrect attempts, look at the book again, review your notes and re-do the problem from scratch. Do not guess an answer. Remember you have only 5 attempts for each problem, so do not waste your attempts on frivolous trial and error solutions.

Do not enter an equivalent expression when your answer is marked wrong. x*(2*x-5) is the same as (2*x-5)*x, changing from one form to another will not change the validity of your answer. The software behind Webassign matches equivalent expressions, so do not send an email to your instructor that "WebAssign is marking your correct answer wrong". Its extremely rare for such errors to occur.

Avoid simple mistakes like: wrong syntax; incorrect usage of parentheses, x*(2*x-5) is not the same as x*2*x-5; using the incorrect letter for a parameter or a variable in a symbolic solution.

If you have technical difficulties with the WebAssign website, please contact their technical support directly online.

Homework Solutions will be visible soon after the due time. Use these solutions to prepare for the in-class quiz on Friday (next class). Especially note the way of writing and describing the solutions. I use the same descriptive details when I solve examples in class, so pay attention in class also.

For the quizzes and exams, you need to be able to describe the intermediate steps/ calculations/ substitutions for the solution. Just writing the final answer without justifying it will lead to deduction of points.

*Monday, 8/22*: Discussion of course aim and grading, etc. Inverse functions, one-to-one functions, how to find inverse of a one-to-one function, how the domain of a function affects it one-to-one/inverse properties, relation between the graphs of f and inverse of f. (From Section 7.1)*Wednesday, 8/24*: Distribution of the course information sheet and related discussion. Formula for the derivative of the inverse of a differentiable one-to-one function, exponential function and its definition for integer, rational and irrational exponents, the different limits of the exponential function depending on its base, natural exponent function, logarithmic function. (From Sections 7.1, 7.2 and 7.3)*Friday, 8/26*: Properties of the exponential function and the log function and how they are related to each other, derivative of a^x and its relation to its derivative at zero, definition of Euler's number e and the natural exponential function, derivative and integral of e^x, ln: natural log, using properties of log and exp to solve expressions in x, change of base formula, derivative of ln and integral of 1/x, logarithmic differentiation. (From Sections 7.2, 7.3 and 7.4)*Monday, 8/29*: Derivatives of general log and general exponential functions and the corresponding derivatives, Using definition of derivative log to express e as an limit, examples of log differentiation, differential equation for natural law of growth or decay and its only solution, How to make sine into a one=to-one function, inverse sine and its domain and range. (From Sections 7.4, 7.5 and 7.6)*Wednesday, 8/31*: Inverse sine and its domain and range, Finding the derivative of arcsin, Inverse cosine - domain, range and derivative, Inverse tan - domain, range, derivative, asymptotes, examples for domain and differentiation, Domain, range and derivative of remaining three trigonometric functions, proving a trigonometric identity using calculus. (From Section 7.6)*Friday, 9/2*: Quiz#1 and the discussion of solutions. Integration rules based on arcsin and arctan derivatives, indeterminate forms of limits (0/0 and infinity/infinity) and L'Hospital's rule with an example. (From Sections 7.6 and 7.8)*Monday, 9/5*: Labor Day Holiday*Wednesday, 9/7*: Examples for L'Hospital's rule, Indeterminate forms of limits - product form: infinity.0, difference form: infinity - infinity, power form: 0^0, infinity^0, 1^infinity - with examples and strategies. Distribution of graded quiz#1. (From Section 7.8)*Friday, 9/9*: Quiz#2 and the discussion of solutions. Hyperbolic functions - sinh, cosh, tanh - their properties, derivatives, etc. Integration by parts formula and its derivation from the product rule, examples. (From Sections 7.7 and 8.1)*Monday, 9/12*: More examples for integration by parts, thumb rule for choosing "u" in integration by parts, reducing an integral to itself, Trigonometric integrals, substitution strategy for integrals of product of powers of sine and cosine with at least one of the powers odd. (From Sections 8.1 and 8.2)*Wednesday, 9/14*: Strategy and Examples for integrals of product of powers of sine and cosine with even powers, strategy and examples for integrals of product of powers of tan and sec, strategy and example for integrals of product of powers of cot and cosec, Trigonometric identities for integral of product of sin(mx) and cos(nx), etc. (From Section 8.2)*Friday, 9/16*: Quiz#3 and the discussion of solutions. Inverse substitution - meaning and validity, example. (From Section 8.3)*Monday, 9/19*: More examples for trigonometric/inverse substitutions, polynomial division and the idea for integrating rational functions. (From Sections 8.3 and 8.4)*Wednesday, 9/21*: The method for integrating rational functions: polynomial division, factorization of denominator into linear and irreducible quadratic factors, partial fractions; partial fractions - for distinct linear factors, for repeated linear factors, for distinct quadratic factors, two methods for finding the constants of partial fractions. (From Section 8.4)*Friday, 9/23*: Quiz#4 and the discussion of solutions. How to integrate an fraction with an irreducible quadratic polynomial in the denominator, partial fractions for repeated quadratic factors, examples.**Read the overview of integration techniques in Section 8.5**(From Sections 8.4 and 8.5)*Monday, 9/26*: Examples of when to make a substitution to avoid partial fractions and when to make a substitution to convert the integrand into a rational function to apply partial fractions, Overview and examples for methods learned so far, Examples of elementary functions that do not have elementary antiderivatives, Approximate integration and its underlying idea, Left-endpoint approximation and its geometric interpretation, Right-endpoint approximation, Midpoint Rule. (From Sections 8.5 and 8.7)*Wednesday, 9/28*: Trapezoidal rule, Examples for midpoint and Trapezoidal rules, Definition of Error, Error bounds on midpoint rule and trapezoidal rule, how to find n to guarantee a low error, how to bound the values of a function over an interval, Simpsons rule and its error bound. (From Section 8.7)*Friday, 9/30*: Quiz#5 and the discussion of its solutions. Improper integrals - the idea and two basic examples. (From Section 8.8)*Monday, 10/3*: Review of L'Hospital's rule with examples (per student request). Improper integral of type 1 and its variants, convergent and divergent improper integrals, When is Integral(1 to infinity) of 1/x^p convergent?. (from Section 8.8)*Wednesday, 10/5*: Mid-term Exam #1.*Friday, 10/7*: Examples of improper integral of type 1, Improper integral of type 2 and its variants, examples and non-example, Comparison theorem and how to apply it, examples. (From Section 8.8)*Monday, 10/10*: Fall Break*Wednesday, 10/12*: Distribution of Mid-term Exam#1 and score distribution (solutions discussed last week in recitation). Models of population growth, family of solutions, specifying one solutio by a condition, Logistic differential equation and behavior of its solutions, Equilibrium solutions, General differential equation and its order, integration as solving a differential equation, initial value problem, verifying the solutions of a D.E., solution curve and direction field, the idea behind Euler's method for finding approximate solution for a D.E. (from Sections 10.1 and 10.2)*Friday, 10/14*: Quiz#6 and the discussion of its solutions. The geometric idea behind Euler's method, the algebraic formulation of Euler's method, Separable equations and how to solve them. (From Sections 10.2 and 10.3)*Monday, 10/17*: Examples for Separable equations including the population growth models and logistics equation, Implicit vs. Explicit solutions, 1st order linear differential equations, Concept of integrating factor, how to find the integrating factor, examples. (From Sections 10.3, 10.4, and 10.5)*Wednesday, 10/19*: Parametric Equations - examples, how to convert a parametric equation into a usual equation in x and y, traversal of a parametric curve - examples where underlying curve is same but the parametric curve goes around the curve different number of times, differentiation formula for dy/dx, formula for d/dx(dy/dx), when a curve has horizontal tangent, vertical tangent, what to do when derivative is of the form 0/0, equation of a tangent line. (From Sections 11.1 and 11.2)*Friday, 10/21*: Quiz#7 and the discussion of its solutions. Equation of a tangent line, how to convert answers in terms of parameter into x-y coordinates, checking for concavity, how to convert an x-y coordinate point into a parameter, checking for increasing and decreasing behavior, Area under a parametric curve, a parametric form of ellipse and its area, length of a x-y curve and length of a parametric curve. (From Section 11.2)*Monday, 10/24*: Examples for length of a parametric curve, examples that show length is measured with respect to how many times its traced, Polar coordinates - definition, meaning of negative theta and negative r, two different polar coordinates for the same point in the plane, relationship between x-y and r-theta coordinates, meaning of constant theta and constant r, reducing a polar curve to cartesian curve, tracing polar curves using table of points and using a (theta, r)-cartesian-sketch of the function. (From Sections 11.2 and 11.3)*Wednesday, 10/26*: Tangents to polar curves, horizontal and vertical tangents, examples illustrating different possibilities when the derivative in 0/0 form, finding area under a polar curve. (From Sections 11.3 and 11.4)*Friday, 10/28*: Quiz#8 and the discussion of its solutions. More examples of areas under a polar curve and inbetween two polar curves, finding all points of intersection of two polar curves. (From Section 11.4)*Monday, 10/31*: Another example for finding all points of intersection of two polar curves, length of a polar curve: new formula from old formula for parametric curves. Definition of a sequence, limit of a sequence, examples of limiting behavior, relation between limit of a sequence and limit of the corresponding function, squeeze theorem, limit laws. (From Sections 11.4 and 12.1)*Wednesday, 11/2*: More examples of limits of sequences, Principle of Mathematical induction, an example for applying Induction as proof, Increasing and decreasing sequences, Monotone sequences, Bounded sequences, Monotonic Sequence Theorem, Investigating a recursive sequence - using induction and monotonic seq. theorem. (From Section 12.1)*Friday, 11/4*: Quiz#9 and the discussion of its solutions. Sum of infinite series, sequence of partial sums, finding a formula for the partial sum and its limit, Sum of geometric series and how to find it. (From Section 12.2)*Monday, 11/7*: Necessary condition for convergence of a series - limit of the sequence must be zero, Integral test - checking the conditions for the function in the improper integral, When the p-series "sum(1/n^p)" is convergent, approximating the sum of a series, Error of an series approximation using the integral test. (From Sections 12.2 and 12.3)*Wednesday, 11/9*: Remainder estimate for the integral text and estimating the value of the series using this test, calculating the value of n that guarantees a small error, Comparison test and finding the correct inequalities to apply the comparison test, Ratio test or the limit comparison test (all three possibilities) and applying it when comparison test fails. (From Section 12.3 and 12.4)*Friday, 11/11*: Quiz#10 and the discussion of its solutions. More examples for application of comparison test and ratio test including how to find the series b_n for comparison, Alternating series, Alternating series test and its application. (From Sections 12.4 and 12.5)*Monday, 11/14*: Review of some topics for Exam#2 as requested by students. More examples for AST, Estimating the error/ remainder using the AST, Absolute convergence and Conditional convergence, Using absolute convergence to show convergence of a series. (From Sections 12.5 and 12.6)*Wednesday, 11/16*: Mid-term Exam #2.*Friday, 11/18*: Ratio test and Root test for absolute convergence with examples, Overview of Section 12.7**Read the discussion and examples in Section 12.7**, Power series, Geometric series and its convergence, Power series centered at a, Using ratio test to find where a power series is convergent and divergent, Checking convergence and divergence when ratio test fails, The three possibilities for convergence of a power series and definition of radius of convergence. (From Sections 12.6, 12.7, and 12.8)*Monday, 11/21*: Distribution of Mid-term Exam#1 and score distribution, and discussion of its solutions. Definitions and examples for radius of convergence and interval of convergence. (From Section 12.8)*Wednesday, 11/23*: Thanksgiving Break.*Friday, 11/25*: Thanksgiving Break.*Monday, 11/28*: Quiz#11 and the discussion of its solutions. Discussion of solutions of Mid-term Exam#2 (continued).*Wednesday, 11/30*: Discussion of solutions of Mid-term Exam#2 (concluded). Finding power series for a given function using geometric series, Differentiation and integration of power series, Using differentiation and integration to powers series representation of functions, Analytic functions and their Taylor series representation. (From Sections 12.9 and 12.10)*Friday, 12/2*: Every Analytic function can be differentiated infinitely many times but not every function that can be differentiated infinitely many times is analytic, HOw to check whether a function is analytic, nth degree Taylor polynomial and the remainder, the theorem for estimating the remainder, Examples of e^x, sin x and cos x, showing the three steps for expressing a function in terms of its McLaurin series. (From Section 12.10)

- Student companion site for the textbook.
- Calculus II notes by Paul Dawkins.
- Common Math Errors by Paul Dawkins.
- Ask Dr. Math
- Review Sheets:

Algebra Review Sheet by Paul Dawkins

Trignonometry Review Sheet by Paul Dawkins

Calculus I and II Review Sheet by Paul Dawkins

- Mathematica Help:

Mathematica Tutorials

A Beginner's guide to Mathematica

A short Mathematica Command Reference.