Fall** 2012**

**Meetings:** TR 9:30AM-10:45PM, ~~Ingraham 222~~
Van Vleck B119

**Instructor:** Benedek Valkó**
Email:** valko at math dot wisc dot edu

Office:

TA: Jo Nelson (nelson at
math dot wisc dot edu)

**Webpage:** http://www.math.wisc.edu/~nelson/teaching/

**Discussion Sections: **MW 8:50AM-9:40AM (301), MW
11:00AM-11:50AM (302)

I will use the class email list to send out
corrections, announcements, please check your wisc.edu email
regularly.

**Course description:**

This course is the first semester of the Calculus Honors sequence
developed by the Mathematics Department at the UW. The goal of the
sequence is to provide highly motivated and well-prepared students
with an opportunity to go beyond the traditional approach to the
subject to develop a deeper understanding of this fundamental area
of mathematics and to appreciate its power and beauty. The
material covers essentially the same topics as the standard first
semester calculus course, but the material is discussed in greater
depth, and with much more emphasis on mathematical ideas. The
course will be challenging, and the student might find it
surprisingly different at the beginning. But it is also meant to
be a lot of fun and to provide the students with the kind of clear
and precise thinking that is characteristic of mathematics and
that will be useful for them in almost any subject they pursue.

**Prerequisites:** Personal invitation or invitation from
the instructor.

- Real and complex numbers
- Integral calculus
- Limits of functions, continuous functions
- Differentiation
- The logarithm and exponential function, inverse trig functions

**Evaluation:** Course grade will be based on work in the
discussion section (20%), two evening
midterm exams (20%) and a final exam (40%). The final grade will
be computed according to the following scale

A: [100,89), AB: [89,87), B: [87,76), BC: [76,74), C: [74,62), D:
[62,50), F: [50,0]

There will be no curving in the class, but the instructor reserves
the right to modify the final grade lines.

Discussion section grades consist of homework and quizzes.
Attendance in one of the weekly discussion sections is required.

You will not be allowed to use calculators during the quizzes and
exams.

**Exams:**

1st midterm: Wednesday, October 10, 7:15PM-8:45PM, Social
Sciences 6102 review
sheet

2nd midterm:
Wednesday, November 14, 7:15PM-8:45PM, Social Sciences 6102
review
sheet

Final exam: December 18, Tuesday, 12:25PM - 2:25PM, Ingraham 120 review
sheet

There will be no make-up exams in the course. If you know that
you cannot make it for one of the evening exams then you should
contact me *ahead of time*.

**Homework and quizzes:** Homework will be assigned every week
and it should be handed in at the Wednesday
discussion section ~~at the start of the lecture
on Thursday~~ (starting with the 2nd week). Your TA will
decide how much of the assignment will be graded. Your TA may also
decide to give quizzes during the discussion section.

The assignments will contain *bonus problems*. These will be
more challenging than the regular homework and will have flexible
deadlines. You will need to present the solution of the bonus
problem in person (send me an email to set up an appointment). You
may schedule a presentation up to the last week of the semester.

**Instructions for the homework assignments:**

- Please put the problems in order and staple your homework in the upper left corner.
- Justify your steps. In most cases the numerical answer is the least significant part of the solution, it is much more important to show how you got there.
- You must present an original solution to each problem you are
asked to hand in. Cheating/
plagiarism will not be tolerated. Instances of academic
misconduct will lead to disciplinary action.

- Please use complete sentences. Your solution should not be
just a sequence of equations and formulas.

- Make sure that your presentation is clean. Do not submit solutions with crossed out parts. Recopy problems if necessary.

How to succeed in this course: It helps immensely if you keep up with the material during the semester by reading the textbook and following the lectures. I will post section numbers of the textbook for upcoming lectures, it is beneficial to read ahead.

Although reading the textbook is important, the best way to master the material is through problem solving. This of course includes the homework assignments, but you shouldn't stop there, there are more than enough practice problems in the text book. (If you run out of problems to solve, I'm happy to supply more!)

For some students studying in groups helps a lot, let me know if you need help with setting up a study group.

If you feel lost, do not hesitate to ask for help. I'm available in my office hours (see above) or by appointment. Do not wait until the end of the semester if you think you are in trouble!

**Schedule:**

Dates | Covered topics | Homework | Suggested reading for next week |

Week 1 (9/4, 9/6) |
Field and order axioms of real numbers, integers and
rational numbers (I 3.2-3.7) complex numbers: basic operations, conjugate, modulus, argument, geometric representation, polar coordinates (9.1-9.6) |
HW1 |
Parts 3 and 4 from the Introduction |

Week 2 (9/11, 9/13) |
Least upper bound, induction (I. 3.8-3.11) |
HW2 | Part 4 from the introduction, 1.1-1.2 |

Week 3 (9/18, 9/20) |
Sums and products, inequalities (triangle
inequality, Cauchy-Schwarz inequality) (I. 14.6-14.8) functions, area (1.2-1.6) |
HW3 | 1.8-1.14 |

Week 4 (9/25, 9/28) |
Intervals, partition, step functions,
integral for step functions, properties of the integral
(1.8-1.14) |
HW4 | 1.16-1.20 |

Week 5 (10/2, 10/4) |
Integrals of more general functions, upper
and lower integrals, monotonic functions are integrable, the
integral of x^p (1.16-1.25) |
Extra bonus problems | 2.1-2.16 |

Week 6 (10/9, 10/11) |
Basic properties of integrals, applications
of integration (area between graphs, volume, average value)
(2.1-2.19) First midterm exam solutions |
HW5 | 3.1-3.4 |

Week 7 (10/16, 10/18) |
Limits, continuity, basic limit theorems
(3.1-3.5) |
HW6 | 3.5-3.9 |

Week 8 (10/23, 10/25) |
Composite functions and continuity, Bolzano's
theorem, intermediate value theorem, inverse functions (3.7-3.13) write-up on infinite limits |
HW7 | 3.15-3.19 |

Week 9 (10/30, 11/1) |
Extreme values of continuous functions,
uniform continuity, integrability of continuous functions,
mean value theorem (3.14-3.19) |
HW8 | 4.1-4.7 |

Week 10 (11/6, 11/8) |
Derivative of a function, algebra of
derivatives, the derivative as a slope (4.1-4.7) |
no hw this week |
4.8-4.13 |

Week 11 (11/13, 11/15) |
Chain rule, implicit differentiation,
derivative of the inverse, relative min/max (4.8-4.13) Second midterm exam solutions |
HW9 | 4.14-4.16 |

Week 12 (11/20) |
Rolle's theorem, Mean Value theorem for
derivatives, finding relative extrema 4.14-4.16 |
no HW this week |
4.17-4.20, 5.1 |

Week 13 (11/27, 11/29) |
Convexity, Curve sketching, extremum problems Fundamental theorem of calculus 4.17-4.20, 5.1-5.6 |
HW10 | 5.7-5.9, 6.1-6.7 |

Week 14. (12/4, 12/6) |
Integration by substitution, integration by
parts, the logarithm function 5.7-5.9, 6.1-6.8 |
HW11 | 6.12-6.16 |

Week 15. (12/11, 12/13) |
The exponential function, hyperbolic
functions, inverse trigonometric functions, integration by
partial fractions |
no HW |
review for the final exam |