# Pub111

Mon | Wed | Fri | Mon | Wed | Fri | Mon | Wed | Fri | Mon | Wed | Fri | Mon | Wed | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Sept | 7 | 9 | 12 | 14 | 16 | 19 | 21 | 23 | 26 | 28 | 30 | ||||||||

Oct | 3 | 5 | 7 | 10 | 12 | 14 | 17 | 19 | 21 | 24 | 26 | 28 | 31 | ||||||

Nov | 2 | 4 | 7 | 9 | 11 | 14 | 16 | 18 | 21 | 23 | 25 | 28 | 30 | ||||||

Dec | 2 | 5 | 7 | 9 | 12 | 16 |

# Applied Calculus: Day-by-Day

**NEW** Course topic outline and study guide.

General Information :: StartR Series :: Problem Answers :: ... outline for instructors :: Day-by-day Instructor notes for semester

### Announcements

- Software Updates: Give this command:
`update.the.math135.software( )`

, or, if that doesn't work, start from scratch:

source("http://dl.dropbox.com/u/5098197/math135.r")

## General Information about the Course

Moodle Assignment Discussion Forum

**Preceptors**

- Vidarith Chan -- vchan@macalester.edu
- Nicole Miller -- nmiller5@macalester.edu
- Andrew Rich -- arich@macalester.edu
- Shuang Zheng -- szheng@macalester.edu

**Course materials**

- Required: Applied Calculus, 4th ed. Hughes-Hallett et al. Information about getting a copy of the text
- Recommended: Laptop computer for use in class
- Course Policies and Work Flow

**Getting help**

- MAX Center (Kagin Commons)
- Hours: Mon-Fri 9am-4:30pm, Sun-Thurs 7-10pm

- Preceptor help sessions (MAX Center)
- Hours: 7-8:30pm on Sun (Zheng), Mon (Chan), Wed (Rich), Thurs (Miller)

**Office Hours**

- Flath:
- Mon 2:20-3:20 pm
- Tue 1:15-2:10 pm
- Wed W 2:20-3:20 pm
- by appointment.

- Kaplan: (Olin-Rice 231 --- the corner of the building by the wind turbine)
- Wed. 2-3 pm
- Fri. 1:30-2:30 pm
- by appointment or just come in when you see me.

**Computer Resources**

- R through the web-based RStudio system: beta.rstudio.org
- Give this command in RStudio to install the software

source("http://dl.dropbox.com/u/5098197/math135.r")

- Google Docs: Instructions for handing in collaborative work [1]
- AcroScore: Your account info ::changing your password ::LOGIN for homework

# Class Schedule

## Wed 7 Sept

**Topic**: What is calculus? Some modeling exercises. RStudio.
(one-minute essay)

**Readings**:

- Applied Calculus (course textbook): AC §1.1
- Chapter 1 of
*Calculus Made Easy*a 100-year old calculus book. *Change we can believe in*by Steven Strogatz in the New York Times.- A recent Times Op-Ed about mathematics education:
*How to fix our math education* - Starting R: StartR -- Why a Language?

**Assignment**: Upload a picture of yourself to Moodle. To do this, go to the *Profile* item on your Moodle page, then select "edit profile" and upload a photo file in the usual way. Since the point of the photo is to help other people recognize you by sight, please pick a relatively recent photo.

**On your own**: For fun and your own edification ... a newspaper article about study skills

## Fri 9 Sept

**Topic**: Basic Types of Functions. Introduction to graphing in R (OME)

**Readings**: AC §1.2, §1.9, §1.10

**Assignment**:

- AC §1.2: #6, #8, #16, #28, #30 AcroScore Link
- AC §1.9: #2-12 (evens), #16, #18
- AC §1.10: #12, #14, #18, #20, #22, #28
- StartR -- Starting with RStudio

**In-class activities**: Using RStudio and collaborating in Google Docs. Notes for instructors

**On your own**:

## Mon 12 Sept

**Topic**: Basic modeling functions: exponentials and logs. Graphing mathematical functions in R (OME)

**Readings**:

- AC §1.5, §1.6, §1.7
- StartR -- Graphing Mathematical Functions

**Assignment**:

- Exercises in StartR -- Graphing Mathematical Functions
- Book Exercises. Do these as best you can, then revise them over the week.
- §1.5: probs 6, 14, 16, 18, 24
- §1.6: probs 2, 24, 28 (Heads up: next week you'll see why
`a^t`

is not as good a form as`exp(b*t)`

for writing exponential functions.) - §1.7: probs 2, 10

**On your own**:

- Why "2 pi" is a choice: Why not "tau" instead?
*Forbes*magazine article and blog entry on R- Prof. Topaz mini-lectures on

## Wed 14 Sept

**Topic**: Functions of two variables
(OME)

**Readings**: AC §9.1, §9.2

**Assignment**:

- StartR -- Creating Mathematical Functions
- StartR -- Making Scatterplots from Data
- Exercises §9.1: #1, #2, #4, #12, #16 (not on AcroScore)
- Exercises §9.2: #16, #18, #22, #24, #2

In-class: Drawing contours from tables.

**On your own**:

- Get a gut feeling for exponential growth. If you have a NetFlix account, watch
*Hoarders*: Episode 20, Glen and Lisa, Season 3 (2010) Glen's story, especially. - Prof. Topaz mini-lectures on

## Fri 16 Sept

**Topic**: Solving equations with functions. (OME)

**Readings**: Review AC §1.5, §1.6, §1.7, §9.1, §9.2

**Assignment**:

- Final revision of your AcroScore exercises from this week (due by class time).
- Fitting exponentials to data. ("Income" activity to be posted.)

**In-class activities**:

- The first of our regular Friday 15 minute quizzes.

## Mon 19 Sept

**Topic**: Finish exponentials and power-law functions. Continue with functions of Two Variables
(OME)

**Readings**:

- Review AC §9.1, §9.2

**Assignment**:

- Exponentials and Data, an example about income and housing.
- Exercises for §9.1
- Posted after class: StartR: Graphing Functions of Two Variables

**In Class**:

**On Your Own**: Sept. 19 is International "Talk Like a Pirate Day". We'll be using ARRRR, the computer language of pirates!

## Wed 21 Sept

**Topic**: Units and Dimensions

**Readings**:

**Assignment**:

**In Class**

- Using Google to convert units.
- Energy and dimensional analysis Document to be posted.

**On your own**:

- Prof. Topaz mini-lecture Introducting Units and Dimensions

## Fri 23 Sept

**Topic**: Solving equations with functions. (OME)

**Readings**:
Review Quick Notes on Units
and Excerpt from Giordano and Weir

**Assignment**:

**In-class activities**:

**On Your Own**:

### Group Project 1: Dimensions and the Atomic Bomb (due Oct. 10)

#### Some Background

A news article with some movie footage of the explosions.

- Some videos: one and two
- There are many videos on YouTube, many of which are civil defense training movies from the 1950s.

Two mini-lectures by Prof. Chad Topaz to orient you to the project:

- Lecture 1: Dimensional Analysis and the "Buckingham Pi Theorem"
- Lecture 2: An example problem --- The Volume of a Crater.

A "calibration test" done at at the Trinity site of 100 tons of TNT. Near the end of this video, there is an aerial view comparing the Trinity crater to that of the 100 ton calibration test. And this video of a cold-war era test of 500 tons in the 1960s, after the test-ban treaty came into effect.

The original paper by Sir G.I. Taylor on ``The formation of a blast wave by a very intense explosion

## Mon 26 Sept

**Topic**: Dimensional Analysis. Introducing Fermi Problems. (OME)

**Readings**:

**Assignment**:

## Wed 28 Sept

**Topic**: (OME)

**Readings**:

- Review the two readings about Fermi problems.

**Assignment**:

- Start to form your groups for the Atomic Bomb project. Register your group here.
- Make your own estimation to compare to NASA's of 1/3200 as the probability of a person being hit by one of the items of space debris in the recent uncontrolled satellite re-entry. You can find estimates of the number of pieces of debris in the newspaper.
- At the very beginning of this video, the climber, Catherine Destivelle, is dangling from a rope. Make an estimate of how long the rope is, based on principles of dimensional analysis. (Hint: You might have to do an measurement with a short string and a small mass in your room.
**No rock climbing**required or recommended!)

Instructor Notes Instructor Background on Fermi Problems

## Fri 30 Sept

**Topic**: Fermi Problems (OME)

**Readings**:

**Assignment**:

**In Class**:

## Mon 3 Oct

**Topic**: Finishing Fermi Problems. Start on Linear Algebra. (OME)

**Notice**: SOFTWARE UPDATE NEEDED. Give this command:

source("http://dl.dropbox.com/u/5098197/math135.r")

**Readings**:
Linear Algebra Reader §1 and §2.

**Assignment**:

- StartR on constructing functions from data Splines and Smoothers
- Do two of the Fermi problems from the handout in class. Your choice. But be prepared to report the results of your two problems in class.

**On your own**:

- Prof. Topaz mini-lecture on vectors
- A telephone conversation involving a dispute about decimals and dimensions and more about the story. "They're both the same, if you look at them on paper-wise."

## Wed 5 Oct

**Topic**: Linear Algebra (OME)

**Readings**:
Linear Algebra Reader §1, §2, §3, and §4.

**Assignment**:

- All the exercises contained in the reading. (Not in AcroScore.)

**On your own**:

- Prof. Topaz mini-lecture on linear equations

## Fri 7 Oct

**Topic**: (OME)

**Readings**:
Linear Algebra Reader §5

**Assignment**:

- StartR on Linear Algebra and Projection You might have to repeat the software update from 3 Oct.

**In-class activities**:

## Mon 10 Oct

**Topic**: Curve Fitting and Linear Algebra (OME)

**Readings**: Review the Linear Algebra Reader.

**Assignment**:

- Hand in your atomic-bomb project. Print out ONE COPY per group and hand it in during class. ALSO, EVERYONE in the group should cut-and-paste a link to their Google Doc report here on Moodle. Remember to do both!
- StartR: Linear Algebra and Curve Fitting

**On your own**:

- Prof. Topaz mini-lecture Vector Projection and Curve Fitting

## Wed 12 Oct

**Topic**: Nonlinear Curve Fitting (OME)

**Readings**:

**In Class**:

manipulate(plotFun( A*sin(2*pi*t/3) +C ~ t, t=range(-5,10),A=two,C=one,ylim=c(-10,10)),one=slider(-10,10),two=slider(-10,10))

**Assignment**:

- StartR: Nonlinear curve fitting (to be posted)

**On your own**:

## Fri 14 Oct

**Topic**: Nonlinear Curve Fitting (OME)

**Readings**:

- See the assignment

**Assignment**:

**In-class activities**:

## Mon 17 Oct

**Topic**: Rise over Run: Three ambiguities (OME)

**Readings**: AC §1.3, §2.1, §2.2, §2.3

**Assignment**: only some are on AcroScore

- AC §2.1: Problems #1, 2, 4, 6, 8, 24
- AC §2.2: Problems #8, 10, 21 to 24, 27, 30

**On your own**:

- Strogatz on derivatives.
- Prof. Topaz mini-lectures: Average Rate of Change, Instantaneous Rate of Change, Derivative Function, Interpretations of the Derivative
- Software update again! Then try the derivative graphing software:

> mDerivs( sin(x^2)~x)

## Wed 19 Oct

**Topic**: From the derivative to the gradient (OME)

**Readings**: AC §2.4, §2.5, **§9.3** (Note: it's Nine.three)

**Assignment**:

- AC §2.3: Problems #2,4,6,14,38,40
- AC §2.4: Problems #2,4,6,8,26,28
- AC §2.5: Problems #2,8,10,12
- AC §9.3: Problems #2,10,20,24

**On your own**:
Prof. Topaz mini-lectures: Product and Quotient Rules, Derivatives of Periodic Functions, Partial Differentiation

**In class**:
Sketching derivatives of simple functions

## Fri 21 Oct

**Mid-Term Exam**: 1 hour in class.

- You can bring one sheet of paper (any size), with notes. The notes
**must**be written or typed by you, yourself. No copying from others! No cutting-and-pasting on the computer to patch together notes! - You will be able to use R or a calculator, as you wish. But you may not access any other materials, notes, etc. beyond the one sheet mentioned above.
- Remember to write
*something*for each question. If you write nothing, it's not possible to give partial credit.

## Mon 24 Oct

**Topic**: (OME)

**Readings**: Review AC §2.3, §2.4, §2.5, and §9.3

**Assignment**: Practicing with interactive software

- Practice sketching the derivative. Use the applet from the US Naval Academy
**Instructions:**- To start drawing your curve, click and drag your mouse from the small region just to the left of the y-axis --- it's only about 2mm wide, and you
**must**start in that region. Then drag your mouse (while holding the button down) toward the right to draw your curve. - Press "Original start" to erase your curve and start over.
- Press "New start" to generate a new curve
- Once you have your curve, press "Answer" and "Error" to see how your guess compares to the mathematical derivative

- Do this for several "new start" curves until you feel pretty comfortable in your ability to sketch the derivative. Don't worry about small errors --- getting the exact curve is what the mathematical formalism and software is for. If you're close, you've got it.
- Try to identify the features of the blue function that make it easy to guess the sign (positive, negative, exactly zero) of the derivative.

- To start drawing your curve, click and drag your mouse from the small region just to the left of the y-axis --- it's only about 2mm wide, and you

- Differences between tangent lines and chords (secant lines). Use another Navy applet.
- Compare the tangent-line slope at t=0.4 to the secant-line slope between t=0.4 and t=0.6 (that is, with dt=0.2).
- Same thing, but make dt=-0.2.

- Partial derivatives from still another Navy applet:
- When you click on the "floor" of the graph, you are selecting a value for x and y. A yellow point will be drawn to show you the values of x and y you are selecting, and a yellow line will run vertically up to the function f(x,y). By checking the "slice" and "tangent" check boxes, you'll see a faux-3d line that indicates the slope.

- Find values of x and y that do each of the following:
- Produce similar and large slopes in the direction of x and in y.
- Produce similar but
**small**slopes in the x and y directions - Produce a large slope with respect to x but a small slope with respect to y
- Produce a small slope with respect to x but a large slope with respect to y

And ... if you haven't done them yet ...

- AC §2.3: Problems #2,4,6,14,38,40
- AC §2.4: Problems #2,4,6,8,26,28
- AC §2.5: Problems #2,8,10,12
- AC §9.3: Problems #2,10,20,24

## Wed 26 Oct

**Topic**: (OME)

**Readings**:

- Chapter 3 of
*Calculus Made Easy*"On Relative Growings" especially the "Note to Chapter III" on p. 17

**Assignment**:

**On your own**:

## Fri 28 Oct

Fall Break

## Mon 31 Oct

**Topic**: Continuity and differentiability, change along paths, intro to symbolic methods (OME)

**Readings**:

- Applied Calculus text: §3.1, §3.2

**Assignment**:

- Welcome back from break! If you have time, get ahead on the assignment for Wednesday's class, which relates to the reading for today.

**In class**:

- Designing a roller coaster --- continuity and differentiability
- Derivatives in the economy
- Heat engines and the ideal gas law.

**On your own**:

- Prof. Topaz mini-lecture on derivatives of polynomials

## Wed 2 Nov

**Topic**: Symbolic derivatives of power law, polynomial, and exponential functions (OME)

**Readings**:

- Review AC §3.1, §3.2
- AC §3.3
- Chapter 4 of
*Calculus Made Easy*"Simplest Cases"

**Assignment**:

- AC §3.1 problems 2, 6, 8, 20, 30, 38, 46
- AC §3.2 problems 2, 12, 16, 26
- AC §3.3 problems 2, 4, 6, 12, 14, 42

**In class**:

- R and Wolfram Alpha

**On your own**:

- Prof. Topaz mini-lectures on derivatives of exponentials and logs and the chain rule

## Fri 4 Nov

**Topic**: Symbolic Differentiation II (OME)

**Readings**:

- AC §3.4, §3.5
- AC §9.4
- Chapter 2 of
*Calculus Made Easy*"On Different Degrees of Smallness"

**Assignment**:

- AC §3.4 Problems 4, 6, 8, 38
- AC §3.5 Problems 8, 10, 16, 22, 28, 30
- AC §9.4 Problems 12, 14, 18, 28, 34, 35, 36

**In-class activities**:

**On your Own**:

- Prof. Topaz mini-lectures product and quotient rules, derivatives of periodic functions, partial differentiation
- Chapter 9 of
*Calculus Made Easy*"Introducing a Useful Dodge"

## Mon 7 Nov

**Topic**: Integration (OME)

**Readings**:

- Applied Calculus §5.1, §5.2
- Strogatz on Integration

**Assignment**:

- StartR on Integration and Anti-Derivatives
- AC §5.1 Problems 4, 8, 16
- AC §5.2 Problems 5,10,16,18

**In Class**:

- Modeling Project 2: Designing the Super-Slide Due 23 Nov.

**On your own**:

- Wikipedia on the "Fundamental Theorem of Calculus." Two important characters are the first two holders of the Lucasian chair at Cambridge University: Isaac Barrow and Isaac Newton.
- Chapter 17 of
*Calculus Made Easy*"Integration"

## Wed 9 Nov

**Topic**: Integration continued. (OME)

**Readings**:

- Applied Calc §5.3, §5.4.
**Warning!**Don't be misled into thinking that an integral is an area. That just happens to be a simple setting in which integrals have an intuitive meaning. Thinking that an integral is an area is like thinking that a wind sprint is football.

**Assignment**:

- Form a group for your Super-Slide Design Project and sign-up here.
- AC §5.3 Problems 10, 18, 24, 26
- AC §5.4 Problems 4, 10, 12
- Some integration drill

**On your own**:

- Chapter 18 of
*Calculus Made Easy*"Integrating as the Reverse of Differentiating" -- through p. 196 will do.

**In class**:

## Fri 11 Nov

**Topic**: (OME)

**Readings**:

**Assignment**:

**In-class activities**:

## Mon 14 Nov

**Topic**: Bumps and S-functions. Polynomial approximations in one variable. (OME)

**Readings**:

- AC §4.7, §4.8

**Assignment**:

- §4.7: Problems 4, 10, 16, 22 (Not on AcroScore)

**In-class**:

- Comparing Taylor and Least Squares function with
`mTaylor( )`

- Estimating Road Capacity

**On your own**:

- Taylor Theorem revisited --- through example 4. The material on "Error in Taylor Series" is entirely optional.
- For after class:

## Wed 16 Nov

**Topic**: Polynomials in two variables. Higher-order partial derivatives. (OME)

**Readings**:

- Review AC §9.3, §9.4

**Assignment**:

- AC §9.4 Problems 20, 21, 22, 23, 24, 25 (Not yet on AcroScore)
- Roller-blading Polynomials

**In Class**:

- Grading rubric for the group project.
- Modeling with polynomials in two variables

**On your own**:

- The first few minutes of this YouTube video on 2nd-order partial derivatives

## Fri 18 Nov

**Topic**: Newton's method for solving and for optimization, gradient ascent method(OME)

**Readings**:

- AC §9.5
- The Gradient Reader

**Assignment**:

- §9.5 problems 2, 6, 8, 20
- In draft. See instructor's notes.

**In-class activities**:

## Mon 21 Nov

**Topic**: Constrained Optimization and Lagrange Multipliers ("Shadow Costs") (OME)

**Readings**:

**Assignment**:

- §9.6 problems 6, 8, 12, 18

**In Class**:

**On your own**:

## Wed 23 Nov

**Topic**: Multiple Objectives and constraints, trade-offs (OME)

**Readings**:

**Assignment**:

- Hand in Designing the Super-Slide via Google Docs. Each member of the group should hand in a link to your Google Doc here on Moodle and the group should hand in
**one**printed copy of the report.

**On your own**:

- Half-lives in the news: Measuring the age of underground water

## Fri 25 Nov

Thanksgiving Break!

## Mon 28 Nov

**Topic**: Wrapping up Constrained Optimization. Start on Modeling with Differential Equations: Exponential growth and decay. (OME)

**Readings**:

- AC §10.1, §10.2

**Assignment**:

**In-Class**:

**On your own**:

- Prof. Topaz mini-lecture Introduction to Differential Equations
- Constrained optimization video: Safety Third from Dirty Jobs if you have Netflix. (Explanation: When people say, "Safety first," do they mean it? Is it the first of priorities, or just a constraint.

Instructor notes 135 DE software

## Wed 30 Nov

**Topic**: Logistic growth (OME)

**Readings**:

- AC §10.3, §10.4

**Assignment**:

- AC §10.4 #2, 4, 6

**In Class**:

- Activity on Constrained Optimization
- Hysteresis in Ecology

**On your own**:

- Prof. Topaz mini-lecture Confirming Solutions to Differential Equations

## Fri 2 Dec

**Topic**: Dynamics in two variables. Compartment models. The harmonic oscillator. (OME)

**Readings**:

- AC §10.5, §10.6

**Assignment**:

- AC §10.5 #2, 4, 6, 14, 16
- Constrained Optimization (revisited): Allocating Health-Care Funds

**In-class activities**:

**On Your Own**:

- Read a classic paper from
*Nature*on dynamics in ecology

## Mon 5 Dec

**Topic**: Interacting populations (OME)

**Readings**:

- Review AC §10.6

**Assignment**:

- AC §10.6 #12, 16, 17, 18

**On your own**:

## Wed 7 Dec

**Topic**: Oscillation and exponentials. SIR models (OME)

**Readings**:

- AC §10.7

**Assignment**:

**On your own**:

## Fri 9 Dec

**Topic**: Equilibrium and stability (OME)

**Readings**:

- Review AC §10.7
- Strogatz on Love and Differential Equations
- More about love here and elaborations here

**Assignment**:

- AC §10.6 problems 12, 16, 17, 18

**In-Class**:

spring = function(x,v){ k=1; m=1; return(c(dx=v,dv=-(k/m)*x) ) } mPP(spring, xlim=c(-5,5),ylim=c(-5,5))

**On your Own**:

## Mon 12 Dec

**Topic**: Finishing up Differential Equations. Semester review (OME)

**Readings**:

**Assignment**:

- On-line course evaluation This is required, but anonymous. When you have completed the survey, enter a short note saying so on Moodle here, so the instructors will know that you've done it. Thanks!

**On your own**:

- The "Better Explained" series. Let us know if it is indeed better explained.

## Fri 16 Dec - Final Exam

4:00-6:00 pm.

# Final Exam

- Time: Fri. 16 Dec. 4-6pm
- Location:
- Prof. Kaplan's sections: O/R 250
- Prof. Flath's sections: O/R 350

- You can bring and use any static written material (notes, the book, etc.), but you cannot use the Internet or other forms of communication. You can also bring and use a calculator. If you want to use R, you can use the professor's computer in the exam room.

Review Sessions:

- Wednesday from 4:00-5:00 pm in O/R 245 (Kaplan)
- Thursday from 3:30-4:30 pm in O/R 245 (Kaplan)
- Friday from 11:00-12:00 am in O/R 245 (Kaplan)

# Study Guide - Answers to Assigned Problems

Answers to problems (available by login to instructors and distributed to students at appropriate times in the semester)

- Course outline and some study guide review questions
- A good attitude toward calculus expressed in the "Epilogue and Apologue" of
*Calculus Made Easy*pp. 249-250