# Pub114

# Math 135, Applied Calculus, Spring 2012

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May | 3 |

# Semester Schedule

## Mon Jan 23

##### Topic:

What is calculus? Some modeling exercises. RStudio.

**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 New York 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 that shows your face clearly.

**On your own** and for your own edification ... a newspaper article about study skills

[ back ]

## Wed Jan 25

##### Topic:

Basic Modeling Functions in one variable. Introduction to graphing in R

**Readings**:

- AC §1.2, §1.9, §1.10
- A short video introduction to AcroScore, the system for handing in most homeworks in the course

**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
- Cut and paste this command into your R session to install the course software:
source("http://dl.dropbox.com/u/5098197/math135.r")

- Cut and paste this command into your R session to install the course software:

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

**On your own**:

## Fri Jan 27

##### Topic:

Basic modeling functions: exponentials and logs. Graphing mathematical functions in R

**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

**In Class**:

- Since we've just started, we will
**not**have a Friday quiz. That will start next week. - Doubling-time and half-life activity

**On your own**:

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

## Mon Jan 30

##### Topic:

Using exponentials and logs. Brief introduction to functions of two variables

**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
- Finish the in-class plotting exercise:

- Plot out the line <math>2 x + 7</math> for <math>x</math> over the range -5, 5
- Plot out the power-law <math>4 x^3</math> for <math>x</math> over the range -2 to 3
- Plot out the power-law <math>t^n</math> for each of <math>n=0,1,2,3,4</math> for <math>t</math> over the range -1 to 1
- Repeat the above, but put them all on the same graph (using
`add=TRUE`

and the colors "red", "blue", "green" "aquamarine" "tomato"- Make the curves thick with the line-width argument,
`lwd=5`

and setting the y-axis limits to`ylim=range(-1,1)`

- You can get a list of named colors with
`colors( )`

- Make the curves thick with the line-width argument,
- The above functions are power laws. Contrast them to the exponentials by plotting out, say <math>2^t</math> on the same graph. Note how fundamentally it differs. Try <math>3^t</math> and so on.
- Plot out <math>2 \sin( \frac{2\pi}{3} t )</math> for <math>t</math> from 0 to 15.
- How many cycles are there?
- What is the period?
- What is the amplitude from max to min?

- Add in a line <math>y=2</math> in green.
- Add in a line <math>y=-2</math> in blue.
- Add in a sine of half the amplitude and twice the period. How can you see that you have it right?
- Add in a sine of the original amplitude and one-half the period. How can you see that you have it right?

**In-class**: Doubling-Times and Half-Lives.

**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

Instructor Notes [ back ]

## Wed Feb 1

##### Topic:

Review of Exponentials, Power-Law, and Logs. Functions of Two Variables

**Readings**:

- Review AC §9.1, §9.2

**Assignment**:

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

**In Class**:

- Using Log Axes
- A first contour-drawing problem. Draw the contours at 0, 30, 60, 90, 120.

**On Your Own**:

## Fri Feb 3

##### Topic:

Functions of Two Variables. Solving equations with functions.

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

**Assignment**:

- Catch up on assignments from the start of the semester. Make sure to hand them in on AcroScore by Sunday.
- StartR: Solving equations with R

**In-class activities**:

- Drawing Contours from Tabular Data
- Solving Equations with Functions
- The first of our regular Friday 15-minute quizzes.

[ back ]

## Mon Feb 6

##### Topic:

Units and Dimensions

**Readings**:

**Assignment**:

- Finish the worksheet on drawing contours from tabular data (Hand in the paper in class.)
- Finish the worksheet on Solving Equations with Functions (Hand in the paper in class.)
- Units Problems 100, 104, 105, 107,109 --- note after semester: these have been consolidated into one file.

**In Class**

- Organizing Units by Dimension

**On your own**:

- Prof. Topaz mini-lecture Introducting Units and Dimensions

## Wed Feb 8

##### Topic:

Units and Dimensions.

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

**Assignment**:

**In Class**:

- Using Google to convert units.
- Energy and dimensional analysis

**On Your Own**:

## Fri Feb 10

##### Topic:

Dimensional Analysis. Dimensions and Power Laws. Introducing Fermi Problems.

**Readings**:

**In Class**:

- Dimensions of Light and Temperature
- Power law patterns in scaling: Internal Combustion Engines.
- Friday Quiz

**Assignment**:

**On Your Own**:

- An XKCD cartoon about scale and the metric system.

### Group Project 1: Dimensions and the Atomic Bomb (due Feb. 20)

- Instructions and grading rubric for the project
- Sign up for groups of three here

#### 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.

There was substantial experience during World War II with "large" bombs, for instance the 10-ton Grand Slam bomb designed to destroy heavily fortified bunkers and tunnels. The MOAB bomb, deployed to the the US Air Force since 2003 is 20 tons. Such sizes are at the limit of what's practical for launch from aircraft. The the "calibration test" done at at the Trinity site was much larger: 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.

Enrico Fermi's estimation, on site, of the yield: [1]

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

## Mon Feb 13

**Topic**:

**Readings**:

- Review the two readings about Fermi problems.

**Assignment**:

- Dimensions of Light and Temperature (started in class)
- In late September, 2011, the re-entry of a satellite was in the news. 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.
- 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!)

**In Class**:

**On Your Own**:
A Fermi problem

FIN : Instructor Notes : Instructor Background on Fermi Problems

## Wed Feb 15

**Topic**: Start on Linear Algebra.

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

**Assignment**:

- StartR on constructing functions from data Splines and Smoothers

**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."

## Fri Feb 17

**Topic**: Linear Algebra

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

**Assignment**:

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

## Mon Feb 20

**Topic**: Linear Algebra
**Readings**:
Linear Algebra Reader §5

**Assignment**:

- StartR on Linear Algebra and Projection
- 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!

**In Class**:

- Worksheet on linear combinations of functions. In addition to drawing the linear combinations by hand, for the "Other functions" section, create the functions in R to confirm your hand drawing. Hand-in the sheet on Wednesday.
- Using
`project()`

## Wed Feb 22

**Topic**: Curve Fitting and Linear Algebra

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

**Assignment**:

- StartR: Linear Algebra and Curve Fitting
- Hand in the worksheet from Monday on linear combinations of functions.

**On your own**:

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

## Fri Feb 24

**Topic**: Nonlinear Curve Fitting

**Readings**:

- See the assignment

**Assignment**:

**In Class**:

## Mon Feb 27

**Topic**: Rise over Run: Three ambiguities

**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

**In-Class**: Data files `fetchData("water940.csv")`

and `fetchData("water1050.csv")`

**On your own**:

- Strogatz on derivatives.
- Prof. Topaz mini-lectures: Average Rate of Change, Instantaneous Rate of Change, Derivative Function, Interpretations of the Derivative
- Try the derivative graphing software in R:

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

## Wed Feb 29

**Topic**: From the derivative to the gradient

**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**:

## Fri Mar 2

**Topic**: Change and paths: directions, gradients and the gradient vector

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

**In-Class**:

**Assignment**:

- Eyeballing partial derivatives (again)
- Walking uphill and downhill
- Direction of the Gradient
- 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

**On Your Own**:

- Practice sketching derivatives. In this applet, you drag the mouse from left to right in the bottom panel to sketch out the derivative of the function shown in the upper panel. You can edit your drawing by pressing "auto" (to turn it dark), then "smooth" and dragging the cursor over your graph. Check your answer to see how you did, and make sure you understand how your answer differs from the displayed answer. Practice this a few times on different functions until you feel good about your ability to visualize the derivative of a function. (Admittedly, the sketching software is a bit awkward.)

## Mon Mar 5

**Topic**: Introduction to Symbolic Derivatives

**Readings**:

- Chapter 3 of
*Calculus Made Easy*"On Relative Growings" especially the "Note to Chapter III" on p. 17 - AC §3.1, §3.2

**Assignment**:

- StartR Computing Derivatives
- AC §3.1 problems 2, 6, 8, 20, 30, 38, 46
- AC §3.2 problems 2, 12, 16, 26

**On your own**:

- Prof. Topaz mini-lecture on derivatives of polynomials

## Wed Mar 7

**Mid-term Exam**: 1 hour

- 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.

### Study Guide through Mid-Term

#### Review Problems

A study-guide set of 50 review problems of practice problem for the whole semester is available for your use. Make note of the disclaimers at the top of the document.

The following outline of topics in Math 135 lists each study guide problem as it relates to the course topics up through the mid-term. The problems are denoted **SG**. Other topics are covered by the **StartR** lessons.

##### Simple modeling functions

- Familiarity with the basic functions and their parameters: linear, exponential, power-law, logarithmic, periodic (sine)
- Pick an appropriate functional form and estimate parameters from "data" about the function.
**SG 1, 2, 3, 4, 5, 6, 7, 14** - Interpret correctly the various forms of the exponential function, e.g., <math>2^t, 10^t, e^t, (1+r)^t</math>
- Produce and interpret graphs of functions
- Describe the role of parameters and how they differ from variables
- Interpret functions of more than one variable via contour diagrams and cross sections.
**SG 23, 24** - Combine functions of a single variable to produce functions of multiple variables by addition, multiplication, combining domains, composition (e.g., track pos -> speed -> fun), etc.
**StartR** - Modern kinds of functions driven by data
**StartR**- Smoothers, splines
- Bumps, S-curves

##### Linear Algebra

- Adding, subtracting, and scaling of vectors,
**Linear Algebra Reader. SG 47** - Calculations relating to the dot product, e.g. dot product itself, angle, length, projection
**SG 45** - Properties of the projection operation,
**SG 42, 43, 47** - Linear dependence and independence, span.
- Finding and interpreting residuals,
**SG 44, 46** - Setting up curve fitting as a vector projection,
**SG 50** - Perform projection (using software) and interpret results
- signs of linear dependence and independence,
**SG 48, 49** - create the function that implements the fitted curve
**StartR**

- signs of linear dependence and independence,
- Principles of nonlinear least-squares fitting using optimization techniques,
**StartR**

##### Units and Dimensions

- Dimensional consistency and the allowed operations
- Algebra of exponentiation
- Use of dimensions to figure out allowed relationships (e.g., Buckingham Pi theorem)

##### Scale and Estimation

- Logarithmic spacing
- Fermi Problems

##### The Fundamental Focus of Calculus

- Derivatives
- Sketching/estimating derivatives from graphs,
**SG 8, 11, 24, 25, 26, 28, 30**

- Sketching/estimating derivatives from graphs,

## Fri Mar 9

**Topic**: Deriving simple forms of derivatives. Handling limits. General rules for symbolic derivatives. Linear combinations, products, compositions ("chain rule")

**Readings**

- Get started for after Spring Break: AC §9.4

**Assignment**:

- Due after Spring Break: AC §9.4 Problems 12, 14, 18, 28, 34, 35, 36

**In class**:

- R and Wolfram Alpha

**On your own**:

- Prof. Topaz mini-lecture on derivatives of polynomials

### Spring Break: March 10-18

## Mon Mar 19

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

**Readings**:

- Review AC §3.1, §3.2
- AC §3.3
- Chapter 2 of
*Calculus Made Easy*"On Different Degrees of Smallness"

**Assignment**: From before break

- 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**:

- Derivative Drill Hand in on Friday

**On your own**:

- Prof. Topaz mini-lectures on derivatives of exponentials and logs and the chain rule
- Another flavor of <math>\pi</math>. Is it right?

## Wed Mar 21

**Topic**: Symbolic Partial Differentiation

**Readings**:

- AC §3.4, §3.5
- AC §9.4
- Chapter 4 of
*Calculus Made Easy*"Simplest Cases"

**Assignment**:

- Hand in Ideal Gas Activity sheet from March 2.
- 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"

## Fri Mar 23

**Topic**: Continuity and differentiability, 2nd-order and mixed partials.

**Readings**:

- Short biography of Jean Baptiste Joseph Fourier
- An introduction to Fourier Analysis. Work through "Experiment E2." You don't need to go further.
- A revised model of celebrity divorce and the original.

**Assignment**:

- Hand in drill on symbolic ordinary derivatives from Monday's class.
- Hand in drill on symbolic partial derivatives from Wednesday's class.

**In class**:

P = makeFun( 50*( (NYT/ENQ)*((Ah+Aw)/(Sc+5))*Md* (Md/(Md+2))^(T^2))^(1/15) ~ T&NYT&ENQ&Ah&Aw&Sc&Md )

**On your own**:

- Prof. Topaz mini-lecture on derivatives of polynomials

## Mon Mar 26

**Topic**: Integration

**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
- Mini-modeling project on engine power and speed

**In Class**:

- Modeling Project 2: Designing the Super-Slide Due 16 April

**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 Mar 28

**Topic**: Integration continued.

**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
- Hand in Engine speed and power mini-modeling project

**On your own**:

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

**In class**:

## Fri Mar 30

**Topic**: Using Integrals

**Readings**:

- AC §5.5 and "Focus on Theory" p. 271-272
- AC §6.1, §6.3, §6.4
- Those interested in economics should read §6.2

**Assignment**:

- AC §5.5 problems 1, 2, 4, 10, 11, 13
- AC §6.1 problems 2, 5, 6, 13, 14, 17, 21, 22
- AC §6.3 problems 4, 8, 16
- Limits, Circles, and Ellipses

**In-class activities**:

Note to instructor: See 2011-11-11.

## Mon Apr 2

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

**Readings**:

- AC §4.7, §4.8

**Assignment**:

- §4.7: Problems 4, 10, 16, 22
- The AcroScore problems for last Friday 30 March are now posted (see above). Please hand them in by the end of this week.

**In-class**:

**On your own**:

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

## Wed Apr 4

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

**Readings**:

- Review AC §9.3, §9.4

**Assignment**:

- Hand in The Backyard Heat Engine
- AC §9.4 Problems 20, 21, 22, 24, 25
- Roller-blading Polynomials

**In Class**:

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

**On your own**:

- The first few minutes of this YouTube video on 2nd-order partial derivatives
- Derivatives in the economy

## Fri Apr 6

**Topic**: Polynomials in two variables. Selecting model terms.

**Readings**:

- Review AC §9.3, §9.4

**Assignment**:

- Catch Up!

**In Class**:

## Mon Apr 9

**Topic**: Finish: Second-Order Partials. Start: Newton's method for solving and for optimization, gradient ascent method

**Readings**:

- AC §9.5
- The Gradient Reader

**Assignment**:

- §9.5 problems 2, 6, 8, 20

**In-class activities**:

- Eyeballing Partials in Interaction Plots
- Optimization in 2 variables with
`mGradSearch()`

## Wed Apr 11

**Topic**: Multiple Objectives and constraints, trade-offs, Lagrange multipliers ("Shadow Costs")

**Readings**:

**Assignment**:

- §9.6 problems 6, 8, 12, 18

**In Class**:

- Constraints Activity
- Grading rubric for the group project.

**On your own**:

- An optimization problem: What should the price be to park? How do you find it?
- Half-lives in the news: Measuring the age of underground water

## Fri Apr 13

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

**Readings**:

- AC §10.1, §10.2

**Assignment**:

- Graphing Constrained Optimization
- Extra Credit: Try a modeling problem in development -- kayak tipping

**In-Class**:

- Graphing Constrained Optimization
- Optimizing Health Care Expenditures
- Example superslide script: superslide.R

**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?)

## Mon Apr 16

**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

## Wed Apr 18

CAPSTONE DAY.

Go to two of the capstone presentations. Link to Schedule.

**Assignment**: Write a paragraph or two describing each of the talks. Be sure to include the name of the presenter. You can enter your paragraphs here on Moodle.

## Fri Apr 20

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

**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 Apr 23

**Topic**: Interacting populations

**Readings**:

- Review AC §10.6

**Assignment**:

- AC §10.6 #12, 16, 17, 18

**On your own**:

## Wed Apr 25

**Topic**: Oscillation and exponentials. SIR models

**Readings**:

- AC §10.7

**Assignment**:

**On your own**:

## Fri Apr 27

**Topic**: Equilibrium and stability

**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 Apr 30

**Topic**: Finishing up Differential Equations. Semester review

**Readings**:

**Assignment**:

- Modeling Project 2 due: Designing the Super-Slide Hand it in here.
- On-line course evaluation This is required, but anonymous. When you have completed the survey, enter a short note saying so here on Moodle, so the instructor will know that you've done it. Thanks!

**On your own**:

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

## Thursday 3 May

Final Exam 4:00 to 6:00 pm in Olin/Rice 250

### Study Guide for Final Exam

#### Textbook Review Problems

A set of review problems for Chapters 5, 7, 9, and 10 of the textbook. You may wish to focus your attention on the following subset of problems:

- §5.1 problem 2
- §5.2 problems 1,2
- §5.3 problems 1,2,3,4,5
- §5.4 problems 1,2
- §5.5 problems 1,2
- Second Fundamental Theorem of Calculus, problems 1,2,3,4,5,6,7,8 (these are all pretty much the same)
- §7.1 problems 1,2,3
- §7.4 problems 1,2,3,4,5,6,7,8
- §9.1 problems 1,2,3,4,5,6
- §9.2 problems 1,2,3,4
- §9.3 problems 1,2,3,4,5,6
- §9.4 problems 2,3,4,5
- §9.5 problems 1,2,3,4,5,6,7,8
- §9.6 problems 1,2,3,4
- §10.1 problems 1,2,3,4,7,8
- §10.2 problems 1,2,3
- §10.5 problems 1,2
- §10.7 problem 1

## Review Problems

A study-guide set of 50 review problems is available for your use. Make note of the disclaimers at the top of the document.

The following outline of topics in Math 135 lists each study guide problem as it relates to the course topics. These are denoted **SG**. Other topics are covered by the **StartR** lessons.

## Topics in Math 135

##### Simple modeling functions

- Familiarity with the basic functions and their parameters: linear, exponential, power-law, logarithmic, periodic (sine)
- Pick an appropriate functional form and estimate parameters from "data" about the function.
**SG 1, 2, 3, 4, 5, 6, 7, 14** - Interpret correctly the various forms of the exponential function, e.g., <math>2^t, 10^t, e^t, (1+r)^t</math>
- Produce and interpret graphs of functions
- Describe the role of parameters and how they differ from variables
- Interpret functions of more than one variable via contour diagrams and cross sections.
**SG 23, 24** - Combine functions of a single variable to produce functions of multiple variables by addition, multiplication, combining domains, composition (e.g., track pos -> speed -> fun), etc.
**StartR** - Modern kinds of functions driven by data
**StartR**- Smoothers, splines
- Bumps, S-curves

##### Linear Algebra

- Adding, subtracting, and scaling of vectors,
**Linear Algebra Reader. SG 47** - Calculations relating to the dot product, e.g. dot product itself, angle, length, projection
**SG 45** - Properties of the projection operation,
**SG 42, 43, 47** - Linear dependence and independence, span.
- Finding and interpreting residuals,
**SG 44, 46** - Setting up curve fitting as a vector projection,
**SG 50** - Perform projection (using software) and interpret results
- signs of linear dependence and independence,
**SG 48, 49** - create the function that implements the fitted curve
**StartR**

- signs of linear dependence and independence,
- Principles of nonlinear least-squares fitting using optimization techniques,
**StartR**

##### Units and Dimensions

- Dimensional consistency and the allowed operations
- Algebra of exponentiation
- Use of dimensions to figure out allowed relationships (e.g., Buckingham Pi theorem)

##### Scale and Estimation

- Logarithmic spacing
- Fermi Problems

##### The Fundamental Focus of Calculus

- Derivatives
- Sketching/estimating derivatives from graphs,
**SG 8, 11, 24, 25, 26, 28, 30** - Serivatives in symbolic form of the basic modeling functions: power-law, polynomial, sine, exponential, log, sum rule, simple chain rule (<math>\frac{d}{dx} \sin(a x)</math> and <math>\frac{d}{dt} e^{kt}</math>:
- Review problem with answers here

- Average rate of change and approximating derivatives using tabular data.
**SG 9, 10, 20** - Using derivatives to make "predictions" (e.g. to extrapolate or find the function value at a nearby value of the input),
**SG 15, 16** - Approximating derivatives by fitting functions (e.g. interpolating splines, smoothers, etc.) and differentiating them using software.
- Second derivatives
- Partial Derivatives
- Interpretation of derivatives in terms of slopes and "curvature"
**SG 10, 18, 20, 22** - Derivatives in physicsand familiar contexts: e.g. "velocity" and "acceleration"
**SG 14, 17, 27** - Choosing an appropriately small <math>h</math> when needing to approximate <math>\lim h \rightarrow 0</math>

- Sketching/estimating derivatives from graphs,
- Anti-derivatives
**StartR**Also, a few multiple choice questions- "from" and "to" as arguments
- Accumulation
- The S-functions and their derivatives: the bump functions
- Ideas for new exercises:
- What's the surplus in the social security fund? See the data here

##### Polynomial Approximations

- Local linear approximation by tangent line,
**SG 13, 15, 21** - Matching derivatives and the Taylor Series
- Least squares approximation and the "root mean square"
- The interaction term and mixed partial derivatives
- Choice of terms in <math>a_0 + a_1 x + a_2 y + a_3 x y + a_4 x^2 + a_5 y^2</math>

##### Optimization

- Identifying extrema from a graph (the "method of exhaustion")
- Local versus global optima, maxima, minima
- "Guess, check and refine"
**SG 12, 19, 29, 31**- Using the derivative and its multivariate generalization, the gradient vector, for checking and refining

- Contraints
- Constraints as a function
- Constraints reflecting multiple objectives
- Parallel contours at optima => parallel gradients
- Lagrange multiplies and shadow prices,
**SG 32, 33**

Additional exercises: [2]

##### Differential Equations

- The idea of a "state" and <math>\frac{d}{dt}</math>state.
- Translate DEs into verbal descriptions about rates of change.
**SG 41** - Translate verbal descriptions into sketches of DEs,
**SG 34, 35, 36** - Flow field on a line
- Equilibrium location and stability for <math>\frac{d}{dt} x = f(x)</math>,
**SG 37, 38, 39** - The symbolic solution to <math>\frac{d}{dt} x = k x</math>: that is, <math>x(t) = A e^{kt}</math>
- The relationship of the sign of <math>k</math> to the stability.
- Simple extension to <math>\frac{d}{dt} x = k x + b</math>, that is <math>x(t) = (A+k/b) e^{kt} - k/b</math>,
**SG 40** - Interpretation using the solution of <math>t=0</math> and <math>t \rightarrow \infty</math>

- How the Euler method works, and how Euler integration is different from "anti-differentiation",
**SG 39** - The phase plane and the flow field on a plane.
- Equilibria and stability of equilibria in flows of two variables: center, saddle, focus (rotation).
- Basic sorts of models in the plane, including the roles and physical interpretation of interaction terms.

## Other Themes in Math 135

### Modeling

- Approximation
- Breaking into parts
- "Here and now" and modeling the dynamics

### Computation

- Operators and arguments produce values
- Syntax and notation is important
- The mechanics of syntax,
`= , ( ) ~ &`

etc. - Broadening notation beyond the classical algebraic notation (in statistics, we'll add "randomization" and "repete/collect")

- The mechanics of syntax,

## Course Information

**Mark the Date**: Final exam on **Thursday 3 May**, 4pm-6pm

**Preceptors**: Jie Shan, Denghui Sun

**Prof. Kaplan's Office Hours**: (Olin-Rice 231 --- the corner of the building by the wind turbine)

- Wed. 2-3pm, Fri. 1:30-2:30pm
- By appointment or just come in when you see me in my office.

**Preceptor Office Hours** Denghui Sun

- Tues. 4:00-5:30pm MAX center
- Thurs. 9:00-11:00pm Olin-Rice 254 (in the central corridor parallel to the classroom corridor)

**Help with R** Andrew Rich, a Mac student, will hold office hours specifically for R questions

- Mon and Tues: 9-10pm, Olin-Rice 256. You should also feel free to use the assignment discussion forum for posting questions.

**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**

- Post a question to the Assignment Discussion Forum
- MAX Center (Kagin Commons)
- Hours: Mon-Fri 9am-4:30pm, Sun-Thurs 7-10pm

- Preceptor help sessions (MAX Center)
- to be announced

**Computer Resources**

- R through the web-based RStudio system: beta.rstudio.org
- Crib sheet for R commands
- StartR tutorials: language, RStudio, graphing, creating functions, scatterplots from data, graphing functions of two vars, solving equations, splines and smoothers, linear algebra and projection, curve fitting, nonlinear curve fitting, taking derivatives, integrals and anti-differentiation,
- 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 [3]
**AcroScore**: Your account info ::changing your password ::LOGIN for homework::A short video introduction

### Should You Take This Course?

Different people naturally have different goals for a course. Depending on your own goals, this course might not be appropriate for you.

First, you should know that for students with no previous background in calculus, *Applied Calculus* is the appropriate starting point. But, in fact, the majority of students who take *Applied Calculus* have already had some calculus in high school. Those students don't have much of an advantage, and perhaps this is because Macalester's *Applied Calculus* course is very different from a conventional introductory calculus course.

*Applied Calculus* is different for a reason. For students preparing for a career that connects to the natural or social sciences, the conventional introductory Calc I & II provide very few usable skills. It's only if you progress on to Calc III, Linear Algebra, or Differential Equations that Calc I & II make sense for the student whose primary interest relates to the natural or social sciences.

*Applied Calculus* is designed to cover the important topics from Calc III, linear algebra, and differential equations that are behind many of the methods and topics found throughout the natural and social sciences. It has a strong emphasis on modeling and will provide a solid start in the effective use of computers for scientific and statistical work. The course has become something of a model nationwide for mathematics education reform. Its development and dissemination is funded by the National Science Foundation. (Start up funding came from the Howard Hughes Medical Institute.) *Applied Calculus* is the basis for new professional development workshops offered by the Mathematical Association of America.

Course surveys show that most students get a lot out of *Applied Calculus.* Most take it as a lead-in to *Introduction to Statistical Modeling*, and many course graduates decide to continue on for a math or statistics major or minor.

There is, however, one group of students who consistently report **dissatisfaction** with *Applied Calculus*. These are students who are looking to repeat or extend their experience with high-school calculus, students who have enjoyed the symbolic manipulation techniques of differentiation and integration that are at the core of a conventional calculus course. If that's your goal, you should seriously consider taking a different course, perhaps *Single Variable Calculus* or a Calc I course at another ACTC school. (But make sure this is consistent with your plans for a major, since majors such as biology specifically require *Applied Calculus*.) Historically, the symbolic manipulation techniques were very important because, in the days before computers, they were the only practical way to get things done. This was the situation for so long that the symbolic techniques and the algebraic manipulations that support them came themselves to be regarded as the core of calculus. In *Applied Calculus*, taking advantage of modern computing, we shift the emphasis to the concepts and techniques needed to deal with real-world problems in the current era.

## Instructor Wiki Resources

If revising the template, replace the rest of the text of this page with the following

{{subst:Template:Spring 2012 Macalester|Course=Math135}}

Each daily class schedule can be constructed with this template:

{{subst:Math 135 Spring 2012 Daily Outline}}

Template:Math 135 Spring 2012 Daily Outline**Study Guide for Final Exam**

Review Sessions for Final Exam

- Tuesday May 1, 12:20 - 1:20, Olin/Rice 245
- Wednesday May 2
- 2:00 - 3:00 pm
- 7:00 - 8:30 pm

- Thursday May 3, 11:00am-noon

**Professor**: Danny Kaplan, **Preceptors**: Jie Shan, Denghui Sun, **Important links**: Clickers, Assignment Discussion Forum, R Crib Sheet

Command to install the course software:
`source("http://dl.dropbox.com/u/5098197/math135.r")`

Once it's installed, you may occasionally be asked to give this command: `update.the.math135.software()`

**Prof. Kaplan's Office Hours**: (Olin-Rice 231 --- the corner of the building by the wind turbine)

- Wed. 2-3pm, Fri. 1:30-2:30pm
- By appointment or just come in when you see me in my office.

**Preceptor Office Hours** Denghui Sun

- Tues. 4:00-5:30pm MAX center
- Thurs. 9:00-11:00pm Olin-Rice 254 (in the central corridor parallel to the classroom corridor)

**Help with R** Andrew Rich, a Mac student, will hold office hours specifically for R questions

- Mon and Tues: 9-10pm, Olin-Rice 256. You should also feel free to use the assignment discussion forum for posting questions.

## Homework Answers

- StartR: RStudio, Graphing, scatterplots, math functions, income and housing, graphing with two variables, Solving Eqs,Splines and Smoothers, Linear Algebra and Projection, Nonlinear curve fitting
- Units and Dimensions:Problem 5 6 7 100 102 104 105 107 108 109
- Linear Algebra Reader

# Upcoming Days

## Mon Apr 16

**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

## Wed Apr 18

CAPSTONE DAY.

Go to two of the capstone presentations. Link to Schedule.

**Assignment**: Write a paragraph or two describing each of the talks. Be sure to include the name of the presenter. You can enter your paragraphs here on Moodle.

## Fri Apr 20

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

**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

manipulate( {plot( TwoVehicles ~ Income, data=housing); plotFun(C - A*exp(-Income/Y0) ~ Income, Income=c(0,70000),C=offset,A=A,Y0=Y0,add=TRUE)}, offset = slider(80,120), A=slider(80,120), Y0=slider(10000,50000))