What follows below reflects uncategorized recent activity in the courses I am teaching, with the most recent activity at the top. This information can be categorized by course and type by selecting the "courses" link above.

Final Exam Review

Review for the final exam is attached.

Solutions have been posted.

Slow Pitch on the Mathematics of Photography

On Wednesday, December 8, I will be giving a talk in the Slow Pitch Colloquium on "The Mathematics of Photography." The abstract is listed below. I'll post the slides/photos from the talk once it is done.

Location: MATH350
Time: Wednesday, December 8 at 4:00pm

Abstract: As we're all scrambling to pull the end of the semester together, it's time for something easy and fun for this week's Slow Pitch. I gave a similar talk several years ago on this topic, and was asked by some to bring it back. This week, we'll look at the history, use and exploration of color, light and perspective from the mathematician's viewpoint. The two main topics we'll cover are: (1) How famous names in math and science have made photography of the visible and invisible world possible, and (2) how to create stereographic projections with a point-and-shoot camera and open-source math software.

Web Survey

Due Date: 
Tuesday, December 7, 2010 - 04:00

As at the beginning of the semester, this online survey about the course structure is worth a quiz grade. Plus, we're always looking for input on the course structure.

I don't see what anyone specifically writes, so the survey is anonymous. However, I do receive a list of students who took the survey, for the purpose of giving them the quiz credit.

http://www.zoomerang.com/Survey/WEB22B5EXTNC25

Book Assignment 14

Due Date: 
Monday, December 6, 2010 - 16:00

Section 16.1: problems 25, 29
Section 16,2: problems 29, 42
Section 16.3: problems 21, 22

Book Assignment 13

Due Date: 
Wednesday, December 1, 2010 - 16:00

Section 14.2: Problems 39 and 40

Fall Break Worksheet

Due Date: 
Friday, December 3, 2010 - 16:00

This worksheet provides some extra practice for sections 16.1 and 16.2. It is entirely optional and is being provided for those that want to get some extra practice during/after break. It hasn't been entirely decided what will happen with the grading of this worksheet, but it will most likely count as a quiz replacement or two.

It is due the Friday after break.

Midterm 3 Solutions

Solutions to Midterm 3 are attached.

Review Quiz 2

Math 2300 Section 005 – Calculus II – Fall 2010

Quiz – Monday, November 15, 2010

We will be having a review in class every day until the next midterm exam on Wednesday of this week. For this quiz, you are given 10 minutes to decide which problems you think you can solve and which ones you want to see solved during our review session today.

  1. Use the Lagrange Error Bound Formula for P_{n}(x) to find a reasonable bound for the error in approximating the quantity e^{{0.40}} with a third-degree Taylor polynomial for the function f(x)=e^{x} about x=0. Choose the best error estimate.

    Solution: We use the formula

    |E_{3}(x)|\le\frac{M}{4!}|x|^{4}

    where M\ge|f^{{(4)}}(x)| on the interval [0,0.4] (i.e., the interval between where our Taylor polynomial is centered and where we're approximating the value). Since f^{{(4)}}(x)=e^{x} has a maximum value of e^{{0.4}} on the interval [0,0.4], we obtain

    |E_{3}(0.4)|\le\frac{e^{{0.4}}}{4!}0.4^{4}.
  2. Use the Lagrange Error Bound Formula for P_{n}(x) to find a reasonable bound for the error in approximating the quantity 17/\sqrt{3} with a third-degree Taylor polynomial for the function

    g(x)=\frac{17}{\sqrt{4-x}}

    about x=0. Choose the best error estimate.

Review Quiz 1

Math 2300 Section 005 – Calculus II – Fall 2010

Quiz – Friday, November 12, 2010

We will be having a review in class every day until the next midterm exam on Wednesday of next week. For this quiz, you are given 10 minutes to decide which problems you think you can solve and which ones you want to see solved during our review session today.

  1. If f(2)=g(2)=h(2)=0, f^{{\prime}}(2)=h^{{\prime}}(2)=0, g^{{\prime}}(2)=22, f^{{\prime\prime}}(2)=3, g^{{\prime\prime}}(2)=5 and h^{{\prime\prime}}(2)=7, calculate

    \lim _{{x\rightarrow 2}}\frac{f(x)}{h(x)}.

    Solution: Using L'Hopital's rule we have

    \lim _{{x\rightarrow 2}}\frac{f(x)}{h(x)}=\lim _{{x\rightarrow 2}}\frac{f^{{\prime}}(x)}{h^{{\prime}}(x)}=\lim _{{x\rightarrow 2}}\frac{f^{{\prime\prime}}(x)}{h^{{\prime\prime}}(x)}=\frac{3}{7}.
  2. Find the first four nonzero terms of the Taylor series about 0 for the function

    \frac{t}{1+t}.

    Solution: Notice that

    \frac{d}{dt}\frac{t}{1+t}=\frac{1}{(1+t)^{2}}=(1+t)^{{-2}}.

    Since this is a binomial series with p=-2 we will expand this series and then integrate it term by term to obtain the series that we need. We have

    \displaystyle(1+t)^{{-2}} \displaystyle=1-2t+\frac{(-2)(-3)}{2!}t^{2}+\frac{(-2)(-3)(-4)}{3!}t^{3}+\cdots
    \displaystyle=1-2t+\frac{6}{2!}t^{2}-\frac{24}{3!}t^{3}+\cdots
    \displaystyle=1-2t+3t^{2}-4t^{3}+\cdots

    Now integrate term by term and find

    \displaystyle\int(1+t)^{{-2}}\, dt \displaystyle=\int 1-2t+3t^{2}-4t^{3}+\cdots\, dt
    \displaystyle\frac{t}{1+t} \displaystyle=t-t^{2}+t^{3}-t^{4}+\cdots

    Since we're only looking for the first four terms, we're done.

Midterm 3 Review

The attached file is a review for the third midterm. Many thanks to instructor Anca Radulescu for typing up solutions.

© 2011 Jason B. Hill. All Rights Reserved.