# Math 221 — Lectures 001 and 002, Spring 2013

Professor: Autumn Kent, 615 Van Vleck Hall

Lecture times & location: MWF 8:50–9:40am and 11:00–11:50am in Van Vleck B130.

Office hours: Wednesdays 9:50–10:50am and 2:00–3:00pm, and by appointment.

Communicating with me: The best way to communicate with me is to come to my office hours. If you cannot make it to these, try your TA's office hours, or make an appointment with me. Email is a very inefficient way to ask questions about mathematics, and due to the large number of students, I am unable to answer mathematical questions via email. I am open to questions during lecture, so bring them to class!

Topics and approximate lecture schedule: Below is an approximate list of topics covered in each lecture. Topics might change during the semester. It is your responsibility to keep track of any updates.

### Supplementary material

UW–Madison Calculus II Textbook (Integration by parts is covered in Section I.5.)

### The textbook

We will follow the UW Madison Calculus text, which is an essential part of the course, and is meant to be read in it entirety. Reading assigned sections of the text before lecture will help you get most out of the lectures. Here is an electronic version of the textbook, including selected answers as an appendix:

The text is very new, and is essentially in a trial stage. Please let me or your TA know of any comments you have on the text.

### Midterms and Final

There will be two in-class midterm exams and one final exam.

First midterm — Wednesday, March 6.

Second midterm — Wednesday, April 17.

Final Exam — May 17 from 10:05am–12:05pm in a location to be determined. By registering in this class you agree to be present at the final exam on this date and at this time.

### Homework and quizzes

There will be two other kinds of graded assignments.
• Online homework through Moodle, assigned weekly. Login here.
• Written quizzes roughly every other week in discussion, graded by your TA.

### Supplemental homework assignments.

Here are some problems from the text that you should complete. These are not to be submitted and will not be graded. Feel free to ask about them in discussion or lecture. Chapters are indicated by Roman numerals, and sections in chapters by Arabic numerals. So IV.14 means Section fourteen of Chapter 4.

Assignment 1.
I.7: 1, 2, 4, 5, 10, 14, 15, 20
II.6: 1, 2, 3, 4, 11
Assignment 2.
III.4: 4–12
III.14: 10–15, 16, 18, 21, 23, 24, 25
Assignment 3.
III.18: 1–4, 6
IV.5: 1–12
Assignment 4.
1. (Think about this before we talk about the chain rule.) When $$x$$ is measured in radians, we have $$\frac{\mathrm{d}}{\mathrm{d} x}(\sin(x)) = \cos(x)$$. What should the derivative be if we measure $$x$$ using degrees?
IV.8: 1–15, 18
IV.12: 1–11, 13
IV.14: 6, 8, 9, 11, 13
Assignment 5.
IV.16: 1–17
IV.17: 1, 2, 10, 14, 17
V.3: 1–10
Assignment 6.
V.12: 8–14, 15–18, 23–29, 33, 34
V.14: 1, 2, 4, 7, 16
Assignment 7.
V.18: 1–6
VI.9: 1–22, 25, 27–32, 46–48, 51, 52
Assignment 8.
VII.5: 5–36
V.16: 1 (but don't find the inflection points)
Assignment 9.
VII.10: 2–6, 11–52, 62–76, 84–97
Assignment 10.
VIII.6: 3–12
Assignment 11. (last assignment)
The problems are HERE (pdf).

Your final grade will be based on the midterms, final, homework, and quizzes weighted as follows:
 Final 35% Midterm 1 25% Midterm 2 25% Homework and quizzes 15%
The curve, which translates your score into a letter grade, will be determined at the end of the semester.

### Technology

A note about cell phones. Please keep your phone stowed and muted during class. This shows me a modicum of respect, and gives you a fifty minute break from Facebook. If you're bored, resist the urge to text your pal to say how bored you are. I find that I have my most creative moments while daydreaming during boring lectures. I hope not to be boring, but give daydreaming a try if you find me so.

Calculators and computers: You can use calculators and/or computer while working on your homework (to double check answers, investigate further, etc). Twenty years ago a graphing calculator represented the very best a hand held device could offer in computation. Today however, any computer, or even your smart phone combined with the internet, is a much more powerful device. Many answers to calculus problems can be found on Wolfram-alpha, or simply by asking Google. Try entering '' y=x^2-x^3  '' in Google (just cut-and-paste without the quotes.) Note that asking the internet to solve your problems can also lead to completely wrong answers, or ''correct'' answers that are so poorly written up that they might as well be wrong. A similar functionality is given by the on-line Desmos Graphing Calculator.

Calculators and computers on the exams: You will NOT be allowed to use a calculator, computer or cell phone during the exams. All exams will be written so that you will not need a calculator to solve the problems. You should always show your work and your understanding of the material, both on exams and in homework. Even a correct answer may not receive credit if there is no justification for it.

### Further help

There are several ways students can get further help.

The math department has a drop-in tutorial service, the Math lab. It runs afternoon/evening, Monday through Thursday. It will help with small questions in your homework, etc, but it will not tutor extensively (link to the mathlab page.)

The Math Tutorial service does more extensive tutoring, but you need to commit several hours a week. If you do not attend, they will stop your tutoring. You need a referral from your TA to enter the Tutorial service.

Finally, if in the beginning of the semester you already feel you are in real trouble in the class, you can ask your TA to refer you to the Early Alert System during the third week of classes. The system will have an adviser review your background and issue a recommendations on what else you can do to improve your situation.

All these services are free of charge.

### Approximate lecture schedule

Below is a rough timeline of the course that will change frequently as we go along. I will adjust the list as we go to indicate where we are. The listed topics are actually section titles in the text.

#### Week 1 (January 20 – January 26)

• Chapter 1. Numbers and Functions
• Functions (I.3)
• Implicit functions (I.4)
• Inverse functions (I.5)

#### Week 2 (January 28 – February 1)

• Chapter 2. Derivatives
• The tangent to a curve (II.1)
• An example -- tangent to a parabola (II.2)
• Instantaneous velocity (II.3)
• Rates of change (II.4)
• Examples of rates of change (II.5)
• Chapter 3. Limits and continuous functions
• Informal definition of limits (III.1)
• The formal, authoritative, definition of limit (III.3)

#### Week 3 (February 3 – February 9)

• Properties of the Limit (III.6)
• Examples of limit computations (III.7)
• When limits fail to exist (III.8)
• Limits that equal $$\infty$$ (III.9)
• Limits and Inequalities (III.11)
• Variations on the limit theme (III.5)

#### Week 4 (February 10 – February 16)

• Variations on the limit theme (III.5 continued)
• Asymptotes (III.17)
• Continuity (III.12)
• Substitution in Limits (III.13)
• Chapter 4. Derivatives.
• Derivatives Defined (IV.1)
• Direct computation of derivatives (IV.2)
• Differentiating Trigonometric functions (IV.11 (and III.15))
• Differentiable implies Continuous (IV.3)
• Some non-differentiable functions (IV.4)
• The Differentiation Rules (IV.6)

#### Week 5 (February 17 – February 23)

• The Differentiation Rules (continued)
• Differentiating powers of functions (IV.7)
• The Chain Rule (IV.13)
• Implicit differentiation (IV.15)
• Higher Derivatives (IV.9)

#### Week 6 (February 25 – March 1)

• Chapter 5. Graph sketching an Max-min problems
• Tangent and Normal lines to a graph (V.1)
• The Intermediate Value Theorem (V.2)
• Finding sign changes of a function (V.4)
• Increasing and decreasing functions (V.5)
• Examples (V.6)
• Maxima and Minima (V.7)
• Examples -- functions with and without maxima or minima (V.9)
• Must a function always have a maximum? (V.8)

#### Week 7 (March 3 – March 9), Midterm 1 on Wednesday

• General method for sketching the graph of a function (V.10)
• Convexity, Concavity and the Second Derivative (V.11)

#### Week 8 (March 10 – March 16)

• Applied Optimization (V.13)
• Parametrized Curves (V.15)
• l'Hopital's rule (V.17)
• Chapter 7. Exponentials and Logarithms
• Exponents (VI.1)
• Logarithms (VI.2)
• Properties of logarithms (VI.3)

#### Week 9 (March 17 – March 23)

• Graphs of exponential functions and logarithms (VI.4)
• The derivative of $$a^x$$ and the definition of $$e$$ (VI.5)
• Derivatives of Logarithms (VI.6)
• Limits involving exponentials and logarithms (VI.7)
• Exponential growth and decay (VI.8)

#### Week 10 (March 31 – April 6)

• Chapter 8. The Integral
• The summation notation (VII.4)
• Area under a Graph (VII.1)
• When f changes its sign (VII.2)
• The Fundamental Theorem of Calculus (VII.3)

#### Week 11 (April 7 – April 13)

• The definite integral as a function of its integration bounds (VII.8)
• The Fundamental Theorem of Calculus, revisited (VII.3)
• The indefinite integral (VII.6)
• Properties of the Integral (VII.7)
• Substitution in Integrals (VII.9)

#### Week 12 (April 14 – April 20), Midterm 2 on Wednesday

• Chapter 9. Applications of the Integral.
• Areas between graphs (VIII.1)

#### Week 13 (April 21 – April 27)

• Cavalieri's principle and volumes of solids (VIII.3)
• Three examples of volume computations of solids of revolution (VIII.4)
• Distance from velocity (VIII.7)
• The arclength of a curve (VIII.8)

#### Week 14 (April 28 – May 4)

• Area of a surface of revolution (not in text)
• Improper integrals
• An amazing surface of infinite area and finite volume
• Additional topic. Integration by parts.

• Review